Daily Papers

On-policy distillation (OPD) supervises a student on its own trajectories with token-level signals from a frozen teacher, yet how a sampled loss allocates updates across tokens remains poorly understood. We analyze the gradient of the per-token K2 estimator of reverse KL with respect to the student logits. The ell_1 norm of this gradient factorizes into the absolute teacher--student log-probability gap and a student-side softmax factor that grows as the sampled token becomes less likely under the student. In our math-distillation runs, these per-token norms are highly non-uniform: low-student-probability tokens account for a disproportionate share of their sum and are also enriched in large teacher--student gaps. As a lightweight intervention suggested by this analysis, we study Surprise-aware Reweighting (SuRe), a detached, bounded weighting rule that further amplifies this existing allocation. Across two Qwen3 student scales, SuRe improves several math metrics over vanilla OPD and shows no clear degradation on the selected out-of-domain benchmarks. Our primary contribution is therefore a gradient-level characterization of reverse-KL OPD trained with the K2 estimator, with SuRe as one empirical instantiation.

Chain-of-thought (CoT) is a standard approach for eliciting reasoning capabilities from large language models (LLMs). However, the common CoT paradigm treats thinking as a prerequisite for answering, which can delay access to plausible answers and incur unnecessary token costs even when the model is able to identify an answer before extended thinking, a behavior known as performative reasoning. In this paper, we introduce CopT, a reformulated reasoning pipeline that reverses the usual order of thinking and answering. Instead of thinking before answering, CopT first elicits a draft answer and then invokes subsequent on-policy thinking conditioned on its own draft answer for reflection and correction. To assess whether the draft answer should be trusted, CopT recasts continuous embeddings as inference-time contrastive verifiers. Specifically, it contrasts the model's support for the same generated tokens under discrete-token inputs and continuous-embedding inputs, yielding a sequence-level reverse KL estimator for answer reliability. Our analysis shows that under certain assumptions, the expected estimate equals the mutual information between the unresolved latent state and the emitted answer token, explaining why it captures answer-relevant uncertainty rather than arbitrary uncertainty in the latent state. When the answer is deemed insufficiently reliable, CopT performs further on-policy thinking, where a second KL estimator dynamically controls draft-answer visibility, preserving useful partial information while reducing the risk of being misled by unreliable content. Across mathematics, coding, and agentic reasoning tasks, CopT improves peak accuracy by up to 23% and reduces token usage by up to 57% at comparable or higher accuracy, without any additional training. The code is available at https://github.com/sdc17/CopT.

On-Policy Distillation (OPD) is a fundamental technique for efficient post-training of large language models (LLMs), with broad applications in agent learning, multi-task enhancement, and model compression. However, OPD training becomes unstable when the teacher and student distributions differ substantially, as teacher supervision on student-generated tokens may yield unreliable policy gradients and even cause optimization failure. This work addresses reliable on-policy token-level supervision through credit assignment strategies, and proposes Trust Region On-Policy Distillation, TrOPD. It features the following characteristics: 1) Trust-Region On-Policy Learning: TrOPD performs OPD only in regions where the teacher provides reliable supervision, mitigating the optimization difficulty of the K1 reverse-KL estimator under distribution mismatch. 2) Outlier Estimation: For outlier regions, we explore gradient clipping, masking, and forward-KL estimation to reduce the adverse effects of unreliable supervision. 3) Off-Policy Guidance: The student continues generation from teacher prefixes and uses forward KL to imitate off-policy guidance, encouraging on-policy exploration toward reliable regions. Experiments show that TrOPD consistently outperforms SoTA OPD baselines, including OPD, EOPD, and REOPOLD, across mathematical reasoning, code generation, and general-domain benchmarks.

On-policy knowledge distillation transfers reasoning from large teachers to compact students, but existing approaches suffer three compounding failure modes: (i) cold-start collapse, where a fresh student assigns near-zero mass to teacher-preferred tokens; (ii) state-agnostic divergence scheduling, where time-only forward/reverse-KL interpolation ignores the student's coverage state; and (iii) binary reward sparsity, where pass/fail signals discard information from partially correct traces. We present CADENCE, a unified framework with a targeted fix for each. Its DRIFT mechanism schedules a per-token convex mixture of forward-KL and reverse-KL surrogate objectives on student-sampled trajectories (per-token surrogates, not sequence-level KL gradient estimators). Six components extend it: (A) COVA, a coverage-adaptive β schedule accelerating the forward-to-reverse transition; (B) FTB, a forking-token boost concentrating gradient at high-entropy positions via a globally-normalized entropy reference; (C) CCD, a dense reward adding numerical-proximity partial credit for incorrect-but-close traces; (D) LAP, brevity-preferential correct-rollout reinforcement; (E) EMR, an entropy-matching calibration regularizer; (F) BSD, a bootstrapped self-distillation phase. On GSM8K and MATH-500 (corrected 512-token protocol, 5 seeds, reported std), CADENCE distills a 0.5B student from a 1.5B teacher to 69.8 pm 0.5% GSM8K pass@1 (from 48.7% pretrained; 63.2% of the teacher gap closed) and to 72.1 pm 0.4% with a 3B teacher (76.2% closed), beating the strongest matched-compute label-using baseline (DRIFT+binary reward) by +4.4 pm 0.7 points. All experiments run on a single Apple Mac Studio (M-series, 64GB unified memory), showing principled distillation reaches strong reasoning quality without datacenter-scale hardware.

On-policy distillation (OPD) trains a student on its own rollouts guided by teacher feedback and is becoming increasingly important for large language model (LLM) post-training. Like reinforcement learning (RL), however, OPD faces an on-policy systems bottleneck, as rollouts can dominate training time for reasoning workloads. Asynchronous training pipelines can alleviate this bottleneck by decoupling rollout generation from learner updates, but doing so introduces stale-policy data. While prior work has studied stale data in asynchronous RL, its effects in OPD remain underexplored. We present the first systematic study of staleness in asynchronous OPD, focusing on a practical setting where teacher feedback is implemented through local KL losses and full-vocabulary teacher logits are too expensive to store or transfer, necessitating finite teacher-score caches. We first show that KL direction changes the stale-data problem: teacher-weighted forward KL is more robust to stale rollouts, whereas student-weighted reverse KL is vulnerable. Second, for this vulnerable reverse-KL case, we study whether methods designed to stabilize asynchronous RL can mitigate OPD staleness. In our experiments, they do not improve over a simpler OPD-specific surrogate: recomputing the reverse-KL signal under the current student at learner time. Third, we analyze how finite teacher-score caches create a bias-variance tradeoff for sparse and sampled reverse-KL OPD estimators. This motivates multi-sample Monte Carlo (MC), which preserves MC correctability while reducing one-sample variance. Finally, we present and open-source AsyncOPD, a fully asynchronous OPD training pipeline built from these estimator choices. Experiments show that AsyncOPD improves training throughput by 1.6times to 3.8times over strict synchronous training while reaching comparable accuracy.

The reasoning performance of large language models (LLMs) can be substantially improved by training them with reinforcement learning (RL). The RL objective for LLM training involves a regularization term, which is the reverse Kullback-Leibler (KL) divergence between the trained policy and the reference policy. Since computing the KL divergence exactly is intractable, various estimators are used in practice to estimate it from on-policy samples. Despite its wide adoption, including in several open-source libraries, there is no systematic study analyzing the numerous ways of incorporating KL estimators in the objective and their effect on the downstream performance of RL-trained models. Recent works show that prevailing practices for incorporating KL regularization do not provide correct gradients for stated objectives, creating a discrepancy between the objective and its implementation. In this paper, we further analyze these practices and study the gradients of several estimators configurations, revealing how design choices shape gradient bias. We substantiate these findings with empirical observations by RL fine-tuning Qwen2.5-7B, Llama-3.1-8B-Instruct and Qwen3-4B-Instruct-2507 with different configurations and evaluating their performance on both in- and out-of-distribution tasks. Through our analysis, we observe that, in on-policy settings: (1) estimator configurations with biased gradients can result in training instabilities; and (2) using estimator configurations resulting in unbiased gradients leads to better performance on in-domain as well as out-of-domain tasks. We also investigate the performance resulting from different KL configurations in off-policy settings and observe that KL regularization can help stabilize off-policy RL training resulting from asynchronous setups.

Knowledge Distillation (KD) is a promising technique for reducing the high computational demand of large language models (LLMs). However, previous KD methods are primarily applied to white-box classification models or training small models to imitate black-box model APIs like ChatGPT. How to effectively distill the knowledge from white-box generative LLMs is still under-explored, which becomes more and more important with the prosperity of LLMs. In this work, we propose MiniLLM that distills smaller language models from generative larger language models. We first replace the forward Kullback-Leibler divergence (KLD) objective in the standard KD approaches with reverse KLD, which is more suitable for KD on generative language models, to prevent the student model from overestimating the low-probability regions of the teacher distribution. Then, we derive an effective optimization approach to learn this objective. Extensive experiments in the instruction-following setting show that the MiniLLM models generate more precise responses with the higher overall quality, lower exposure bias, better calibration, and higher long-text generation performance. Our method is also scalable for different model families with 120M to 13B parameters. We will release our code and model checkpoints at https://aka.ms/MiniLLM.

Group Relative Policy Optimization (GRPO) is widely used for critic-free Large Language Model (LLM) post-training, but its KL regularization is usually implemented as a local loss-side token penalty. We show that this misses the policy-gradient signal induced by autoregressive KL regularization. Unlike standard KL-regularized Reinforcement Learning (RL) objectives, GRPO's group normalization induces a non-linear prompt-level utility; for binary verifier rewards, this utility is 2arcsinsqrt p. As a result, reward and KL cannot be fused before normalization without changing the implicit objective. We derive the on-policy gradient of GRPO-style objectives with token-wise f-divergence regularization. The reward term recovers the standardized GRPO advantage, while the regularizer term includes a causal future-regularization return-to-go omitted by local KL losses. For reverse KL, this yields a simple future KL correction: add a reverse cumulative sum of per-token log ratios after advantage construction. The resulting method, Future-KL Regularized Policy Optimization (FRPO), requires no critic or extra model passes. On mathematical reasoning tasks, FRPO improves pass@16 in our main large-model setting while maintaining higher entropy and lower policy drift than conventional loss-side KL baselines.

Kullback-Leiber divergence has been widely used in Knowledge Distillation (KD) to compress Large Language Models (LLMs). Contrary to prior assertions that reverse Kullback-Leibler (RKL) divergence is mode-seeking and thus preferable over the mean-seeking forward Kullback-Leibler (FKL) divergence, this study empirically and theoretically demonstrates that neither mode-seeking nor mean-seeking properties manifest in KD for LLMs. Instead, RKL and FKL are found to share the same optimization objective and both converge after a sufficient number of epochs. However, due to practical constraints, LLMs are seldom trained for such an extensive number of epochs. Meanwhile, we further find that RKL focuses on the tail part of the distributions, while FKL focuses on the head part at the beginning epochs. Consequently, we propose a simple yet effective Adaptive Kullback-Leiber (AKL) divergence method, which adaptively allocates weights to combine FKL and RKL. Metric-based and GPT-4-based evaluations demonstrate that the proposed AKL outperforms the baselines across various tasks and improves the diversity and quality of generated responses.

On-policy distillation (OPD) learns from teacher feedback on student-generated responses and has shown promise in reducing forgetting relative to supervised fine-tuning (SFT). However, its benefits and fragility remain incompletely understood. We study sequential distillation from multiple teachers, where the student minimizes its average divergence from the teachers. Forward Kullback--Leibler (KL) divergence yields a weighted arithmetic mixture, while reverse KL yields a normalized weighted geometric aggregate. We develop algorithms that learn these targets under off-policy and on-policy feedback, respectively, establishing logarithmic regret bounds in the tabular setting and extending the analysis to function approximation. By analyzing these aggregation targets, we identify mechanisms that help explain both the benefits and fragility of OPD. Relative to forward KL, reverse KL can better retain a confident expert's preferences under uninformative feedback, but is more sensitive to teachers that assign very low probabilities to correct responses. Its token-level conditionals also reveal a dependence on continuation distributions that can favor incorrect prefixes over long horizons.

Knowledge distillation (KD) has been widely adopted to compress large language models (LLMs). Existing KD methods investigate various divergence measures including the Kullback-Leibler (KL), reverse Kullback-Leibler (RKL), and Jensen-Shannon (JS) divergences. However, due to limitations inherent in their assumptions and definitions, these measures fail to deliver effective supervision when few distribution overlap exists between the teacher and the student. In this paper, we show that the aforementioned KL, RKL, and JS divergences respectively suffer from issues of mode-averaging, mode-collapsing, and mode-underestimation, which deteriorates logits-based KD for diverse NLP tasks. We propose the Sinkhorn Knowledge Distillation (SinKD) that exploits the Sinkhorn distance to ensure a nuanced and precise assessment of the disparity between teacher and student distributions. Besides, profit by properties of the Sinkhorn metric, we can get rid of sample-wise KD that restricts the perception of divergence in each teacher-student sample pair. Instead, we propose a batch-wise reformulation to capture geometric intricacies of distributions across samples in the high-dimensional space. Comprehensive evaluation on GLUE and SuperGLUE, in terms of comparability, validity, and generalizability, highlights our superiority over state-of-the-art methods on all kinds of LLMs with encoder-only, encoder-decoder, and decoder-only architectures.

On-policy distillation (OPD) has emerged as a promising post-training technique for enhancing LLM reasoning. Under the reverse KL objective, the idealized optimum of OPD aligns the student distribution with that of the teacher. When the teacher consistently outperforms the student, this naturally suggests that OPD should yield broad improvements over the pre-OPD student. However, do such improvements extend across the entire range of test-time sampling budgets? In this work, we revisit this expectation through the lens of test-time scaling by varying the sampling budget K and evaluating performance with pass@K. Across multiple settings, we observe two distinct patterns: OPD can improve pass@K at both small and large sampling budgets, but it can also improve small-budget performance while reducing large-budget pass@K. We show one condition that guarantees such a reversal and an idealized reverse KL counterexample where it occurs even when the teacher has higher accuracy on every problem. To choose between two candidate teachers at a target sampling budget, we propose the Teacher Advantage Score at K (TAS@K), which can be computed before OPD training to predict which teacher will lead to a larger improvement in pass@K. Across three domains and thirteen benchmarks, the ordering predicted by TAS@K agrees with the observed pass@K improvements of the resulting OPD models in 83.6\% of experiments, providing a useful signal for teacher selection at the target pass@K.

We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small L^2 error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in epsilon-accuracy can be done in tilde Oleft(d log (1/delta){epsilon}right) steps: 1) the variance-delta Gaussian perturbation of any data distribution; 2) data distributions with 1/delta-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

On-policy distillation is a promising approach for transferring knowledge between language models, where a student learns from dense token-level signals along its own trajectories. This framework typically uses reverse KL divergence, encouraging the student to match the teacher's high-confidence predictions. However, we show that the mode-seeking property of reverse KL reduces generation diversity and yields unstable learning signals when the teacher distribution has high entropy. To address this, we introduce Entropy-Aware On-Policy Distillation. Our key idea is augmenting the standard reverse KL objective with forward KL when teacher entropy is high, capturing the full range of plausible outputs while retaining precise imitation elsewhere. It balances mode-seeking precision with mode-covering robustness without sacrificing on-policy training efficiency. Experiments show that our method maintains generation diversity (sustained token-level entropy) and improves student-teacher alignment (lower forward KL on high-entropy tokens). Across six math reasoning benchmarks, this yields Pass@8 accuracy gains of +1.37 for Qwen3-0.6B-Base, +2.39 for Qwen3-1.7B-Base, and +5.05 for Qwen3-4B-Base compared to baseline on-policy distillation methods. These results demonstrate that accounting for teacher uncertainty is essential for maintaining diversity and achieving effective knowledge transfer.

On-policy distillation (OPD) accelerates post-training by providing dense token-level supervision from a frozen teacher on the student's own rollouts. Vanilla OPD applies this supervision uniformly across prompts, without checking whether the teacher is reliable for each prompt. Because reverse KL is mode-seeking, a confidently wrong teacher can induce a strong yet misleading update. Distributional proxies, such as entropy or teacher-student likelihood agreement, measure uncertainty or agreement but do not directly verify outcome correctness. We introduce Teacher-Gated On-Policy Distillation (TGOPD), built on the principle that teacher reliability should be verified at the prompt level before dense supervision is admitted. TGOPD estimates reliability from a small set of verifier-scored teacher probes and routes each prompt exclusively to dense OPD when the reliability check passes or to verifier-grounded GRPO otherwise. Across 4B and 35B students in mathematics, code, and instruction following, TGOPD outperforms Vanilla OPD in all six single-domain settings and achieves higher seven-benchmark averages at both scales under multi-domain training. By using otherwise-idle teacher capacity for reliability estimation, TGOPD also reduces teacher-side compute waste in asynchronous OPD, increasing teacher-node GPU utilization from 9.8% to 78.9% in the measured 4B single-domain run.

Knowledge distillation is central to LLM post-training, yet its design space remains poorly understood, especially alongside reinforcement learning (RL). We show that the prevailing paradigms, off-policy distillation and on-policy distillation (OPD), implicitly couple two orthogonal choices: prefix source and token-level KL direction. This follows from decomposing sequence-level KL over autoregressive response distributions: forward KL pairs teacher prefixes with token-level forward KL, and reverse KL pairs student prefixes with token-level reverse KL. We argue this coupling is not intrinsic: decoupling the two axes yields four valid objectives. We establish gradient-level identities showing forward KL gives SFT-style cross-entropy matching with teacher soft targets, whereas reverse KL gives an RL-style policy-gradient objective with a dense teacher-student log-ratio reward, connecting them to off-policy SFT, DAgger-style on-policy SFT, offline-RL-style distillation, and OPD. We conduct an extensive controlled study on math reasoning, evaluating the four objectives both as standalone methods and as initializations for subsequent RL. The results reveal three tradeoffs: KL direction induces an accuracy-entropy tradeoff, prefix source a quality-compute tradeoff, and training length an accuracy-stability tradeoff. Motivated by these findings, we propose KL mixing and an entropy-gated length curriculum. KL mixing shows long-sequence distillation requires substantial forward-KL weight to prevent entropy collapse and length inflation without sacrificing accuracy. The entropy-gated length curriculum improves Avg@k and Pass@k by 3.6 and up to 5.8 points, and cuts average response length by roughly 3x versus fixed long-horizon training. Our results provide a framework and practical methods for designing reasoning distillation objectives that balance accuracy, diversity, compute, and RL behavior.

The core learning signal used in language model distillation is the standard Kullback-Leibler (KL) divergence between the student and teacher distributions. Traditional KL divergence tends to be dominated by the next tokens with the highest probabilities, i.e., the teacher's modes, thereby diminishing the influence of less probable yet potentially informative components of the output distribution. We propose a new tail-aware divergence that decouples the contribution of the teacher model's top-K predicted probabilities from that of lower-probability predictions, while maintaining the same computational profile as the KL Divergence. Our decoupled approach reduces the impact of the teacher modes and, consequently, increases the contribution of the tail of the distribution. Experimental results demonstrate that our modified distillation method yields competitive performance in both pre-training and supervised distillation of decoder models across various datasets. Furthermore, the distillation process is efficient and can be performed with a modest academic budget for large datasets, eliminating the need for industry-scale computing.

It is believed that in knowledge distillation (KD), the role of the teacher is to provide an estimate for the unknown Bayes conditional probability distribution (BCPD) to be used in the student training process. Conventionally, this estimate is obtained by training the teacher using maximum log-likelihood (MLL) method. To improve this estimate for KD, in this paper we introduce the concept of conditional mutual information (CMI) into the estimation of BCPD and propose a novel estimator called the maximum CMI (MCMI) method. Specifically, in MCMI estimation, both the log-likelihood and CMI of the teacher are simultaneously maximized when the teacher is trained. Through Eigen-CAM, it is further shown that maximizing the teacher's CMI value allows the teacher to capture more contextual information in an image cluster. Via conducting a thorough set of experiments, we show that by employing a teacher trained via MCMI estimation rather than one trained via MLL estimation in various state-of-the-art KD frameworks, the student's classification accuracy consistently increases, with the gain of up to 3.32\%. This suggests that the teacher's BCPD estimate provided by MCMI method is more accurate than that provided by MLL method. In addition, we show that such improvements in the student's accuracy are more drastic in zero-shot and few-shot settings. Notably, the student's accuracy increases with the gain of up to 5.72\% when 5\% of the training samples are available to the student (few-shot), and increases from 0\% to as high as 84\% for an omitted class (zero-shot). The code is available at https://github.com/iclr2024mcmi/ICLRMCMI.

Sampling from diffusion models involves a slow iterative process that hinders their practical deployment, especially for interactive applications. To accelerate generation speed, recent approaches distill a multi-step diffusion model into a single-step student generator via variational score distillation, which matches the distribution of samples generated by the student to the teacher's distribution. However, these approaches use the reverse Kullback-Leibler (KL) divergence for distribution matching which is known to be mode seeking. In this paper, we generalize the distribution matching approach using a novel f-divergence minimization framework, termed f-distill, that covers different divergences with different trade-offs in terms of mode coverage and training variance. We derive the gradient of the f-divergence between the teacher and student distributions and show that it is expressed as the product of their score differences and a weighting function determined by their density ratio. This weighting function naturally emphasizes samples with higher density in the teacher distribution, when using a less mode-seeking divergence. We observe that the popular variational score distillation approach using the reverse-KL divergence is a special case within our framework. Empirically, we demonstrate that alternative f-divergences, such as forward-KL and Jensen-Shannon divergences, outperform the current best variational score distillation methods across image generation tasks. In particular, when using Jensen-Shannon divergence, f-distill achieves current state-of-the-art one-step generation performance on ImageNet64 and zero-shot text-to-image generation on MS-COCO. Project page: https://research.nvidia.com/labs/genair/f-distill

A central paradox in fine-tuning Large Language Models (LLMs) with Reinforcement Learning with Verifiable Reward (RLVR) is the frequent degradation of multi-attempt performance (Pass@k) despite improvements in single-attempt accuracy (Pass@1). This is often accompanied by catastrophic forgetting, where models lose previously acquired skills. While various methods have been proposed, the choice and function of the divergence term have been surprisingly unexamined as a proactive solution. We argue that standard RLVR objectives -- both those using the mode-seeking reverse KL-divergence and those forgoing a divergence term entirely -- lack a crucial mechanism for knowledge retention. The reverse-KL actively accelerates this decay by narrowing the policy, while its absence provides no safeguard against the model drifting from its diverse knowledge base. We propose a fundamental shift in perspective: using the divergence term itself as the solution. Our framework, Diversity-Preserving Hybrid RL (DPH-RL), leverages mass-covering f-divergences (like forward-KL and JS-divergence) to function as a rehearsal mechanism. By continuously referencing the initial policy, this approach forces the model to maintain broad solution coverage. Extensive experiments on math and SQL generation demonstrate that DPH-RL not only resolves the Pass@k degradation but improves both Pass@1 and Pass@k in- and out-of-domain. Additionally, DPH-RL is more training-efficient because it computes f-divergence using generator functions, requiring only sampling from the initial policy and no online reference model. Our work highlights a crucial, overlooked axis for improving RLVR, demonstrating that the proper selection of a divergence measure is a powerful tool for building more general and diverse reasoning models.

Standard LLM distillation wastes compute on two fronts: problems the student has already mastered (near-zero gradients) and problems far beyond its reach (incoherent gradients that erode existing capabilities). We show that this waste is not merely intuitive but structurally inevitable: the gradient signal-to-noise ratio in distillation provably vanishes at both pass-rate extremes. This theoretical observation leads to Paced, a framework that concentrates distillation on the zone of proximal development -- the frontier of a student model's competence -- via a principled pass-rate weight w(p) = p^α(1 - p)^β derived from the boundary-vanishing structure of distillation gradients. Key results: (1) Theory: We prove that the Beta kernel w(p) = p^α(1-p)^β is a leading-order weight family arising from the SNR structure of distillation, and that it is minimax-robust -- under bounded multiplicative misspecification, worst-case efficiency loss is only O(δ^2). (2)Distillation: On distillation from a larger teacher to a smaller student model with forward KL, Paced achieves significant gain over the base model, while keeping benchmark forgetting at a low level. (3)Self-distillation: On instruction-tuned models with reverse KL, gains are exceeding baselines as well. (4)Two-stage synergy: A forward-KL-then-reverse-KL schedule yields the strongest results in our setting, reaching substantial improvements on standard reasoning benchmarks -- supporting a mode-coverage-then-consolidation interpretation of the distillation process. All configurations require only student rollouts to estimate pass rates, need no architectural changes, and are compatible with any KL direction.

For a parametric model of distributions, the closest distribution in the model to the true distribution located outside the model is considered. Measuring the closeness between two distributions with the Kullback-Leibler (K-L) divergence, the closest distribution is called the "information projection." The estimation risk of the maximum likelihood estimator (MLE) is defined as the expectation of K-L divergence between the information projection and the predictive distribution with plugged-in MLE. Here, the asymptotic expansion of the risk is derived up to n^{-2}-order, and the sufficient condition on the risk for the Bayes error rate between the true distribution and the information projection to be lower than a specified value is investigated. Combining these results, the "p-n criterion" is proposed, which determines whether the MLE is sufficiently close to the information projection for the given model and sample. In particular, the criterion for an exponential family model is relatively simple and can be used for a complex model with no explicit form of normalizing constant. This criterion can constitute a solution to the sample size or model acceptance problem. Use of the p-n criteria is demonstrated for two practical datasets. The relationship between the results and information criteria is also studied.

The increasing capabilities of large language models (LLMs) raise opportunities for artificial general intelligence but concurrently amplify safety concerns, such as potential misuse of AI systems, necessitating effective AI alignment. Reinforcement Learning from Human Feedback (RLHF) has emerged as a promising pathway towards AI alignment but brings forth challenges due to its complexity and dependence on a separate reward model. Direct Preference Optimization (DPO) has been proposed as an alternative, and it remains equivalent to RLHF under the reverse KL regularization constraint. This paper presents f-DPO, a generalized approach to DPO by incorporating diverse divergence constraints. We show that under certain f-divergences, including Jensen-Shannon divergence, forward KL divergences and alpha-divergences, the complex relationship between the reward and optimal policy can also be simplified by addressing the Karush-Kuhn-Tucker conditions. This eliminates the need for estimating the normalizing constant in the Bradley-Terry model and enables a tractable mapping between the reward function and the optimal policy. Our approach optimizes LLMs to align with human preferences in a more efficient and supervised manner under a broad set of divergence constraints. Empirically, adopting these divergences ensures a balance between alignment performance and generation diversity. Importantly, f-DPO outperforms PPO-based methods in divergence efficiency, and divergence constraints directly influence expected calibration error (ECE).

We point out a few pitfalls in implementing gradient estimation for KL divergence in RL training for LLM, as seen in a number of open source projects and papers. The first major pitfall is to differentiate through the KL estimate as loss functions to minimize KL divergence. We show that such implementations are generally incorrect and do not produce the desired KL gradient. Secondly, we show that some implementations do not account for the sequential nature of the estimation problem and produce a partial gradient at best. We demonstrate the impact of such issues with illustrative tabular and LLM experiments, and show the correct way to implement the KL gradient.

Recent advances in large language model (LLM) post-training have leveraged two distinct paradigms to enhance reasoning capabilities: reinforcement learning (RL) and knowledge distillation (KD). While RL enables the emergence of complex reasoning behaviors, it often suffers from low sample efficiency when the initial policy struggles to explore high-reward trajectories. Conversely, KD improves learning efficiency via mimicking the teacher model but tends to generalize poorly to out-of-domain scenarios. In this work, we present KDRL, a unified post-training framework that jointly optimizes a reasoning model through teacher supervision (KD) and self-exploration (RL). Specifically, KDRL leverages policy gradient optimization to simultaneously minimize the reverse Kullback-Leibler divergence (RKL) between the student and teacher distributions while maximizing the expected rule-based rewards. We first formulate a unified objective that integrates GRPO and KD, and systematically explore how different KL approximations, KL coefficients, and reward-guided KD strategies affect the overall post-training dynamics and performance. Empirical results on multiple reasoning benchmarks demonstrate that KDRL outperforms GRPO and various KD baselines while achieving a favorable balance between performance and reasoning token efficiency. These findings indicate that integrating KD and RL serves as an effective and efficient strategy to train reasoning LLMs.

Compression reports summarize how far a compressed language model moved from the dense one, usually by a KL divergence; a deployment that relies on the dense model's outputs needs to know how many of its decisions changed. We show that total variation, not KL, answers this directly. Across 802 compressed and perturbed copies of 19 open models on five corpora and nine mechanically unrelated perturbation families, the rate at which the arg-max token changes (the flip rate) tracks total variation at a ratio with median 1.05, with no fitted constant. KL converts into flips only through its square root and a factor that varies fourfold across models and corpora, because KL averages over tokens before the root is taken; first-order statistics averaged per token, such as Hellinger distance, avoid this, but reports rarely give them. As a result, of two compressors reported on different models and corpora whose flip rates differ by at least 10%, KL assigns the smaller divergence to the one that changes more decisions in 11% of cases, total variation in 1%. Two pre-registered tests mark the limits: on a held-out code corpus the ratio held for all eight models while three predictions about KL each failed for half of them or more, and on three new models with real kernels it stayed in its band for 37 of 38 checkpoints but fell below one on code for two models. In vLLM speculative decoding, total variation measured under teacher forcing predicts greedy draft acceptance with a mean relative error of 1.1--2.4%, without the task-specific calibration that KL needs.

One-step and few-step text-to-image generation must accurately render the details specified in the prompt despite limited opportunities for refinement. Distribution matching distillation (DMD) uses score differences to bring the generated distribution closer to a reference distribution. Following prior work, we estimate this difference with internal velocity branches, avoiding a separate fake-score network. We ask whether an explicit moment constraint improves this learned score supervision. A decomposition of the reverse-KL field into a Gaussian moment component and a non-Gaussian residual identifies statistics that can be constrained explicitly. We introduce Intrinsic Distribution Matching (IDM), which retains the full DMD correction and matches feature means and per-coordinate standard deviations using the generator's exponential-moving-average (EMA) copy, requiring no external feature encoder. Experiments use full-parameter and low-rank adaptation (LoRA) tuning on Z-Image-Turbo 6B and Qwen-Image 20B. At two network evaluations, IDM reaches 94.25% and 90.20% on Chinese and English LongText-Bench, the highest scores among the compared methods at this budget. Controlled ablations show gains from the added moment loss, while generator EMA features perform competitively with an external DINOv2 encoder across the five reported metrics. Code is available at https://github.com/inclusionAI/IDM.

On-policy distillation (OPD) has emerged as an effective paradigm for transferring knowledge between language models, where a student is trained to align its next-token distribution with the teacher's along its own trajectories. To provide dense supervision at tractable cost, many works minimize the reverse Kullback-Leibler (KL) divergence between the student and teacher's normalized distributions over the teacher's top-k tokens. However, this normalized objective discards the information about tail probability: the total probability outside the teacher's top-k tokens. As a result, the optimization can steadily increase the student's tail probability and entropy, empirically degrading downstream accuracy. To address this issue, we propose Tail-Aware Top-k OPD (TA-OPD), a novel distillation method that restores the missing tail probability signal. In particular, TA-OPD minimizes the reverse KL divergence over the top-k tokens plus a tail token that carries the tail probability. In effect, TA-OPD better aligns the student's next-token distribution with the teacher's, preventing the increase in tail probability and entropy caused by top-k normalization. Extensive experiments demonstrate the superiority of TA-OPD, improving Avg@8 by up to 8.05 points on common benchmarks. Our code is available at https://github.com/HuipengHuang/TA-OPD.

In the post-training of large language models (LLMs), Reinforcement Learning from Human Feedback (RLHF) is an effective approach to achieve generation aligned with human preferences. Direct Preference Optimization (DPO) allows for policy training with a simple binary cross-entropy loss without a reward model. The objective of DPO is regularized by reverse KL divergence that encourages mode-seeking fitting to the reference policy. Nonetheless, we indicate that minimizing reverse KL divergence could fail to capture a mode of the reference distribution, which may hurt the policy's performance. Based on this observation, we propose a simple modification to DPO, H-DPO, which allows for control over the entropy of the resulting policy, enhancing the distribution's sharpness and thereby enabling mode-seeking fitting more effectively. In our experiments, we show that H-DPO outperformed DPO across various tasks, demonstrating superior results in pass@k evaluations for mathematical tasks. Moreover, H-DPO is simple to implement, requiring only minor modifications to the loss calculation of DPO, which makes it highly practical and promising for wide-ranging applications in the training of LLMs.

On-policy learning has been argued to reduce catastrophic forgetting, produce sparser parameter updates, and improve generalisation. However, existing comparisons between supervised fine-tuning and reinforcement learning vary many factors simultaneously, making the contribution of rollout policy difficult to isolate. We study the effect of rollout policy in a controlled strong-to-weak distillation setting, by independently varying rollout policy, token-level KL direction, and learning rate across the Llama3 and Qwen2.5 model families and reasoning tasks spanning scientific, medical, and arithmetic domains. Our analysis reveals a nuanced picture of distillation dynamics in which rollout policy does not necessarily play a central role. Instead, token-level KL direction more clearly shapes task performance and output coverage, while learning rate governs forgetting and update sparsity. Analysis of KL gradients and experiments along a continuous student-teacher rollout-policy spectrum explain this pattern: forward KL is remarkably robust to rollout policy, with its performance stable and strong despite changes to the rollout policy, whereas reverse KL is substantially more sensitive and favours student-generated rollouts. On-policy data nevertheless improves generalisation to harder variants of the Countdown arithmetic task under both KL directions, although this advantage does not reliably persist after subsequent RLVR. Our broader conclusions remain robust to removing gradient clipping, using sampled KL estimators, and training on tasks requiring longer reasoning chains. Overall, our results challenge the view that on-policy rollouts are inherently preferable and show that their value depends critically on the objective, evaluation setting, and optimisation hyperparameters.

Sparse on-policy distillation (OPD) allocates teacher supervision to a small subset of tokens in student-generated trajectories. However, useful teacher guidance can yield a noisy update when its gradient is estimated from a sampled next token. We study this estimation problem at a fixed prefix in information geometry and propose an information-efficiency ratio (IER) based on a signal-to-noise decomposition. IER characterizes relative gradient estimation error under an optimal scalar baseline. A candidate-set approximation enables token selection based on IER and its combination with existing usefulness scores, while retaining the sampled reverse-KL training objective. On mathematical and medical reasoning tasks, adding IER improves existing selectors in multiple settings, with sparse configurations matching or exceeding full OPD without token selection at small token budgets of 0.1\%--1\%. These results support accounting for both usefulness and gradient-estimation reliability when allocating sparse supervision. Our code is available at https://github.com/BruceSheng1202/IER-OPD.

Knowledge distillation (KD) is a powerful paradigm for compressing large language models (LLMs), whose effectiveness depends on intertwined choices of divergence direction, optimization strategy, and data regime. We break down the design of existing KD methods and present a unified view that establishes connections between them, reformulating KD as a reweighted log-likelihood objective at the token level. We further propose Hybrid Policy Distillation (HPD), which integrates the complementary advantages of forward and reverse KL to balance mode coverage and mode-seeking, and combines off-policy data with lightweight, approximate on-policy sampling. We validate HPD on long-generation math reasoning as well as short-generation dialogue and code tasks, demonstrating improved optimization stability, computational efficiency, and final performance across diverse model families and scales. The code related to this work is available at https://github.com/zwhong714/Hybrid-Policy-Distillation.

On-Policy Distillation (OPD) has emerged as a dominant post-training paradigm for large language models, especially for reasoning domains. However, OPD remains unstable in practice due to the high gradient variance of its single-sample Monte Carlo estimator, and recipes for stable training are still immature. We propose vOPD (On-Policy Distillation with a control variate baseline), which casts OPD as policy-gradient RL and stabilizes it by introducing a control variate baseline-canonically a value function -- from the RL literature. We show that the OPD value function admits a closed form as the per-token negative reverse KL divergence between the student and the teacher, available directly from the already-computed forward pass with no additional critic or inference. Existing stabilization methods either compute the full token-level reverse KL over the entire vocabulary, adding significant overhead, or restrict it to a top-k support, biasing the objective. vOPD instead preserves the lightweight single-sample estimator, subtracting the value function as a detached baseline to keep the gradient unbiased while reducing variance. Furthermore, we show that a top-k approximation of the baseline further lowers cost without compromising performance. Across mathematical and scientific reasoning benchmarks, vOPD consistently outperforms vanilla OPD and matches the most expensive full-vocabulary baseline, offering an efficient stabilization of On-Policy Distillation through principled RL variance reduction.

Reinforcement learning with outcome-based objectives such as GRPO enables LLM-based agents to solve complex, long-horizon tasks, yet the reusable exploration patterns embedded in interaction trajectories are largely discarded after a single policy update. Existing experience-augmented approaches retrieve historical guidance at inference time, but they apply experiences without accounting for the policy's evolving capability and create persistent dependencies on external retrieval. We propose EDGE (Experience-Distillation for Guided Exploration), a framework that treats retrieved experiences as temporary training-time scaffolds and progressively internalizes their benefits into the parametric policy. Concretely, EDGE partitions each rollout group into experience-conditioned and experience-free trajectories to estimate and admit only positive marginal gains without extra sampling, then distills the induced behavior into the base policy via a reverse-KL objective on its own empirical support. A co-evolutionary experience bank further synthesizes guidance from emerging failure modes and prunes obsolete entries as the policy evolves. Across embodied, web, and search-based QA tasks, EDGE improves over strong RL baselines by up to 12.5 points and remains effective without inference-time scaffolds or a proprietary reflector. The code is available at https://github.com/xvolcano02/EDGE.

Recent progress has shown promise in distilling multi-step video diffusion models into efficient few-step students. Among them, Distribution Matching Distillation (DMD) and its successor DMD2 achieved strong generation quality and fast convergence. However, due to the nature of the reverse Kullback--Leibler (KL) objective, these methods exhibit two persistent failure modes: a substantial drop in sample diversity, and visibly over-saturated outputs that deviate from real-video appearance. In this work, we propose Data-Forcing Distillation (DFD), a simple post-training framework that restores diversity and fidelity in DMD with only a single-line of code change. At its core is the teacher score discrepancy to guide the student toward the real-data distribution, pulling it to missing modes (mitigating mode collapse) and away from problematic modes absent in real data (avoiding over-saturation). We provide an in-depth theoretical analysis of our framework and validate our approach on text-to-video, image-to-video, and autoregressive video generation. With only 100--300 steps of finetuning, DFD effectively restores diversity and fidelity on both Wan2.1-1.3B and Cosmos-Predict2.5-2B model, resolving the over-saturation artifacts with significantly better video dynamics and appearance, and even outperforms the teacher model.

Diffusion language models (dLLMs) are an emerging alternative to autoregressive (AR) generators, but aligning them to human preferences is challenging because sequence log-likelihoods are intractable and pairwise preference data are costly to collect. We introduce ELBO-KTO, which combines an ELBO surrogate for diffusion log-likelihoods with a prospect-theoretic, unpaired preference objective (Kahneman Tversky Optimization, KTO). We analyze the bias and variance induced by the ELBO substitution and employ variance-reduction practices that stabilize gradients during training. Applied to LLaDA-8B-Instruct, ELBO-KTO yields 65.9% and 62.3% adjusted win rates on kto-mix-14k and UltraFeedback-Binary, respectively, versus the base model under an automatic LLM judge. Across downstream tasks, including GSM8K, MMLU, and additional reasoning/knowledge benchmarks, ELBO-KTO trained on UltraFeedback-Binary performs on par with or better than the base model under identical decoding. This establishes unpaired preference optimization as a viable alternative to pairwise alignment in diffusion LLMs.

We introduce Reverse Derivative Ascent: a categorical analogue of gradient based methods for machine learning. Our algorithm is defined at the level of so-called reverse differential categories. It can be used to learn the parameters of models which are expressed as morphisms of such categories. Our motivating example is boolean circuits: we show how our algorithm can be applied to such circuits by using the theory of reverse differential categories. Note our methodology allows us to learn the parameters of boolean circuits directly, in contrast to existing binarised neural network approaches. Moreover, we demonstrate its empirical value by giving experimental results on benchmark machine learning datasets.

The kernel thinning (KT) algorithm of Dwivedi and Mackey (2021) compresses a probability distribution more effectively than independent sampling by targeting a reproducing kernel Hilbert space (RKHS) and leveraging a less smooth square-root kernel. Here we provide four improvements. First, we show that KT applied directly to the target RKHS yields tighter, dimension-free guarantees for any kernel, any distribution, and any fixed function in the RKHS. Second, we show that, for analytic kernels like Gaussian, inverse multiquadric, and sinc, target KT admits maximum mean discrepancy (MMD) guarantees comparable to or better than those of square-root KT without making explicit use of a square-root kernel. Third, we prove that KT with a fractional power kernel yields better-than-Monte-Carlo MMD guarantees for non-smooth kernels, like Laplace and Mat\'ern, that do not have square-roots. Fourth, we establish that KT applied to a sum of the target and power kernels (a procedure we call KT+) simultaneously inherits the improved MMD guarantees of power KT and the tighter individual function guarantees of target KT. In our experiments with target KT and KT+, we witness significant improvements in integration error even in 100 dimensions and when compressing challenging differential equation posteriors.

With the passage of the Right to Be Forgotten (RTBF) regulations and the scaling up of language model training datasets, research on model unlearning in large language models (LLMs) has become more crucial. Before the era of LLMs, machine unlearning research focused mainly on classification tasks in models with small parameters. In these tasks, the content to be forgotten or retained is clear and straightforward. However, as parameter sizes have grown and tasks have become more complex, balancing forget quality and model utility has become more challenging, especially in scenarios involving personal data instead of classification results. Existing methods based on gradient ascent and its variants often struggle with this balance, leading to unintended information loss or partial forgetting. To address this challenge, we propose RKLD, a novel Reverse KL-Divergence-based Knowledge Distillation unlearning algorithm for LLMs targeting the unlearning of personal information. Through RKLD, we achieve significant forget quality and effectively maintain the model utility in our experiments.

Multi-Token Prediction (MTP) is now a common ingredient in language-model pre-training, as it improves downstream performance and enables faster inference through speculative decoding. We identify a flaw in standard MTP training, in which a residual component of the MTP gradient interferes with both next-token and multi-token learning, increasingly so with scale. Specifically, the MTP cross-entropy gradient can be decomposed exactly into a KL self-distillation term that pulls each MTP head toward the model's own next-token prediction (NTP) distribution and improves speculative decoding, and a residual that has a low signal-to-noise ratio and buries the useful KL signal. We introduce the KL gradient-recovery trick: when two distributions share a linear output head, the gradient of their KL divergence can be recovered exactly by combining their cross-entropy gradients through straight-through gradient routing. Unlike minimizing the KL directly, this trick requires no vocabulary-sized logit matrices and is compatible with standard fused cross-entropy kernels. This enables Soft Multi-Token Prediction (Soft-MTP), a one-line code change to MTP that removes the noisy residual and backpropagates only the KL self-distillation signal by reusing the existing NTP and MTP cross-entropy computations. Soft-MTP supports both DeepSeek- and Meta-style MTP, adds no parameters, and incurs no measurable overhead. From 167M to 1.2B parameters, Soft-MTP improves over standard MTP with lower NTP perplexity, faster speculative decoding and higher zero-shot accuracy at every scale, lower MTP perplexity from 689M onward, and stronger math and code reasoning after fine-tuning at 1.2B. Soft-MTP also matches, if not exceeds, direct KL divergence minimization on most metrics. Soft-MTP is, in effect, a free lunch for MTP optimization. Our code can be found at https://github.com/JaydenTeoh/Soft-MTP.

Asynchronous reinforcement learning (RL) for large language model (LLM) agents trains one policy on trajectories generated by another: rollouts come from stale checkpoints, and the inference engine's probabilities differ from the trainer's even at identical parameters. Standard remedies either clip importance ratios, which biases the update, or, as in GRPO, sample a group of responses per prompt, which is costly when episodes are long. We propose KL-Regularized Policy Optimization (KLPO), a framework that anchors the KL regularizer at the sampler. The regularized improvement step then has a closed-form Gibbs solution, and KLPO fits its log-ratio optimality condition by least squares on the sampler's own trajectories, so the sampler probability enters through a log-ratio and no importance weights are needed. Profiling out the regression intercept replaces the intractable log-partition function with the signal's sampler mean plus a sampler-to-trainer KL divergence. For token-level policy mirror descent targets, we show that the resulting gradient can be computed from terminal returns without a critic, via sampler-centered scores or a single trajectory residual, even under stochastic tool outputs. We further prove that independent Monte Carlo estimates of the KL term keep these gradients unbiased, derive the exact KL gap of cheaper top-K and binary approximations, and show that SPPO, GPO, REBEL, and BPO arise as special cases of KLPO. The result is a critic-free update that uses one rollout per prompt and requires neither a learned normalizer nor a group of responses.

Large language models (LLMs) have a surprising failure: when trained on "A has a feature B", they do not generalize to "B is a feature of A", which is termed the Reversal Curse. Even when training with trillions of tokens this issue still appears due to Zipf's law - hence even if we train on the entire internet. This work proposes an alternative training scheme, called reverse training, whereby all words are used twice, doubling the amount of available tokens. The LLM is trained in both forward and reverse directions by reversing the training strings while preserving (i.e., not reversing) chosen substrings, such as entities. We show that data-matched reverse-trained models provide superior performance to standard models on standard tasks, and compute-matched reverse-trained models provide far superior performance on reversal tasks, helping resolve the reversal curse issue.

Knowledge distillation (KD) is widely used for compressing a teacher model to a smaller student model, reducing its inference cost and memory footprint while preserving model capabilities. However, current KD methods for auto-regressive sequence models (e.g., large language models) suffer from missing a standardized objective function. Moreover, the recent use of student-generated outputs to address training-inference mismatches has significantly escalated computational costs. To tackle these issues, we introduce DistiLLM, a more effective and efficient KD framework for auto-regressive language models. DistiLLM comprises two components: (1) a novel skew Kullback-Leibler divergence loss, where we unveil and leverage its theoretical properties, and (2) an adaptive off-policy approach designed to enhance the efficiency in utilizing student-generated outputs. Extensive experiments, including instruction-following tasks, demonstrate the effectiveness of DistiLLM in building high-performing student models while achieving up to 4.3times speedup compared to recent KD methods.

Autoregressive language models are trained exclusively left-to-right. We explore the complementary factorization, training right-to-left at scale, and ask what reasoning patterns emerge when a model conditions on future context to predict the past. We train LEDOM, an open-source purely reverse autoregressive language model (2B/7B parameters, 435B tokens), and find it develops capabilities distinct from forward models, including abductive inference, question synthesis, and natural resolution of the reversal curse. We then explore one application of the reverse model: combining forward likelihood P(y mid x) with reverse posterior P(x mid y) through noisy channel duality. We propose Reverse Reward, which reranks forward outputs using reverse posterior estimates, and prove that bidirectional scoring penalizes hallucinated reasoning chains whose backward reconstruction degrades. Reverse Reward yields gains of up to 6.6\% on AIME 2024 and 15\% on AMC 2023 across multiple strong baselines. We release all models, code, and data here.

Top-k decoding is a widely used method for sampling from LLMs: at each token, only the largest k next-token-probabilities are kept, and the next token is sampled after re-normalizing them to sum to unity. Top-k and other sampling methods are motivated by the intuition that true next-token distributions are sparse, and the noisy LLM probabilities need to be truncated. However, to our knowledge, a precise theoretical motivation for the use of top-k decoding is missing. In this work, we develop a theoretical framework that both explains and generalizes top-k decoding. We view decoding at a fixed token as the recovery of a sparse probability distribution. We consider Bregman decoders obtained by minimizing a separable Bregman divergence (for both the primal and dual cases) with a sparsity-inducing ell_0 regularization. Despite the combinatorial nature of the objective, we show how to optimize it efficiently for a large class of divergences. We show that the optimal decoding strategies are greedy, and further that the loss function is discretely convex in k, so that binary search provably and efficiently finds the optimal k. We show that top-k decoding arises as a special case for the KL divergence, and identify new decoding strategies that have distinct behaviors (e.g., non-linearly up-weighting larger probabilities after re-normalization).

The input side of the Information Bottleneck may contain task-irrelevant variation that still costs rate. We identify this cost exactly. Let T be a source and C a relevance variable. Suppose the deterministic statistic Z = phi(T) satisfies C-Z-T. For any encoder p(X|T), its conditional average over the fibres of phi preserves I(X;C) and lowers the rate by I(X;T|Z). The reverse pullback preserves both coordinates. These maps establish equality of the relevance-rate curves and Lagrangian infima on standard Borel spaces for every tradeoff parameter. They also characterise all attained optima. Every full-source minimiser factors through Z, and every reduced minimiser pulls back to T. When C is finite and distortion is logarithmic loss, replacement of T^n by Z^n also preserves the optimal remote distortion at every blocklength and message budget. The operational rate-distortion functions are therefore equal.

Large-scale kernel ridge regression (KRR) is limited by the need to store a large kernel matrix K_t. To avoid storing the entire matrix K_t, Nystrom methods subsample a subset of columns of the kernel matrix, and efficiently find an approximate KRR solution on the reconstructed matrix. The chosen subsampling distribution in turn affects the statistical and computational tradeoffs. For KRR problems, recent works show that a sampling distribution proportional to the ridge leverage scores (RLSs) provides strong reconstruction guarantees for the approximation. While exact RLSs are as difficult to compute as a KRR solution, we may be able to approximate them well enough. In this paper, we study KRR problems in a sequential setting and introduce the INK-ESTIMATE algorithm, that incrementally computes the RLSs estimates. INK-ESTIMATE maintains a small sketch of K_t, that at each step is used to compute an intermediate estimate of the RLSs. First, our sketch update does not require access to previously seen columns, and therefore a single pass over the kernel matrix is sufficient. Second, the algorithm requires a fixed, small space budget to run dependent only on the effective dimension of the kernel matrix. Finally, our sketch provides strong approximation guarantees on the distance between the true kernel matrix and its approximation, and on the statistical risk of the approximate KRR solution at any time, because all our guarantees hold at any intermediate step.

On-policy distillation (OPD) is appealing for large language model (LLM) post-training because it evaluates teacher feedback on student-generated rollouts rather than fixed teacher traces. In long-horizon settings, however, the common sampled-token variant is fragile: it reduces distribution matching to a one-token signal and becomes increasingly unreliable as rollouts drift away from prefixes the teacher commonly visits. We revisit OPD from the estimator and implementation sides. Theoretically, token-level OPD is biased relative to sequence-level reverse-KL, but it has a much tighter worst-case variance bound; our toy study shows the same tradeoff empirically, with stronger future-reward coupling producing higher gradient variance and less stable learning. Empirically, we identify three failure modes of sampled-token OPD: an imbalanced one-token signal, unreliable teacher guidance on student-generated prefixes, and distortions caused by tokenizer or special-token mismatch. We address these issues with teacher top-K local support matching, implemented as truncated reverse-KL with top-p rollout sampling and special-token masking. Across single-task math reasoning and multi-task agentic-plus-math training, this objective yields more stable optimization and better downstream performance than sampled-token OPD.

This paper studies the theoretical framework of the alignment process of generative models with Reinforcement Learning from Human Feedback (RLHF). We consider a standard mathematical formulation, the reverse-KL regularized contextual bandit for RLHF. Despite its widespread practical application, a rigorous theoretical analysis of this formulation remains open. We investigate its behavior in three distinct settings -- offline, online, and hybrid -- and propose efficient algorithms with finite-sample theoretical guarantees. Moving towards practical applications, our framework, with a robust approximation of the information-theoretical policy improvement oracle, naturally gives rise to several novel RLHF algorithms. This includes an iterative version of the Direct Preference Optimization (DPO) algorithm for online settings, and a multi-step rejection sampling strategy for offline scenarios. Our empirical evaluations on real-world alignment experiment of large language model demonstrate that these proposed methods significantly surpass existing strong baselines, such as DPO and Rejection Sampling Optimization (RSO), showcasing the connections between solid theoretical foundations and their powerful practical implementations.

In the field of large language models (LLMs), Knowledge Distillation (KD) is a critical technique for transferring capabilities from teacher models to student models. However, existing KD methods face limitations and challenges in distillation of LLMs, including efficiency and insufficient measurement capabilities of traditional KL divergence. It is shown that LLMs can serve as an implicit reward function, which we define as a supplement to KL divergence. In this work, we propose Direct Preference Knowledge Distillation (DPKD) for LLMs. DPKD utilizes distribution divergence to represent the preference loss and implicit reward function. We re-formulate KD of LLMs into two stages: first optimizing and objective consisting of implicit reward and reverse KL divergence and then improving the preference probability of teacher outputs over student outputs. We conducted experiments and analysis on various datasets with LLM parameters ranging from 120M to 13B and demonstrate the broad applicability and effectiveness of our DPKD approach. Meanwhile, we prove the value and effectiveness of the introduced implicit reward and output preference in KD through experiments and theoretical analysis. The DPKD method outperforms the baseline method in both output response precision and exact match percentage. Code and data are available at https://aka.ms/dpkd.

Discrete diffusion models for categorical generation are defined by a corruption kernel, which determines the intermediate state space and the associated reverse prediction problem. We study uniform discrete diffusion and ask whether its training objective and reverse transitions can be enriched without changing the underlying categorical corruption process. We introduce Simplax, an exact Dirichlet--categorical augmentation that couples each corrupted categorical state with an auxiliary simplex-valued variable while preserving the original uniform diffusion process as its categorical marginal. This augmentation yields a tractable Rao--Blackwellized reverse-bridge objective and a corresponding stochastic reverse sampler, while retaining the corrupted categorical state as the denoiser input. Empirically, Simplax improves the generative perplexity--entropy tradeoff on unconditional OpenWebText generation. On Sudoku, a model trained exclusively on 30-clue puzzles achieves the highest accuracy among the compared methods across all evaluated clue densities, including the minimum uniquely solvable 17-clue regime, and also achieves the highest validity in unconditional generation.

The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.

On-policy distillation (OPD) reduces train-test state mismatch by training a student on its own generated trajectories, but weak students may visit teacher-misaligned prefixes where supervision is less representative. We introduce SAKI (Supervision Allocation with KL-constrained Interpolation), which combines a KL-constrained teacher-guided rollout with maximal coupling and reuses realized accept/correction events to route token-level supervision. Accepted positions retain sampled-token reverse-KL supervision, while correction positions receive direct supervision on the teacher's highest-probability token. Under maximal coupling, the correction probability is exactly TV(p_t, q_t), so the same trust-region radius controls rollout deviation and upper-bounds intervention and specialized-supervision frequency. We further implement an engine-resident speculative verifier that preserves the exact-q trajectory distribution and coupling semantics while improving matched-workload rollout throughput by 4.22x. Across seven mathematical reasoning benchmarks, SAKI improves the matched teacher-guided baseline in Mean@8 and Pass@8 for both 1.7B and 0.6B students. Placement controls and fixed-prefix analysis further support correction-triggered routing as a conflict-adaptive supervision signal.

Despite the widespread empirical success of ResNet, the generalization properties of deep ResNet are rarely explored beyond the lazy training regime. In this work, we investigate scaled ResNet in the limit of infinitely deep and wide neural networks, of which the gradient flow is described by a partial differential equation in the large-neural network limit, i.e., the mean-field regime. To derive the generalization bounds under this setting, our analysis necessitates a shift from the conventional time-invariant Gram matrix employed in the lazy training regime to a time-variant, distribution-dependent version. To this end, we provide a global lower bound on the minimum eigenvalue of the Gram matrix under the mean-field regime. Besides, for the traceability of the dynamic of Kullback-Leibler (KL) divergence, we establish the linear convergence of the empirical error and estimate the upper bound of the KL divergence over parameters distribution. Finally, we build the uniform convergence for generalization bound via Rademacher complexity. Our results offer new insights into the generalization ability of deep ResNet beyond the lazy training regime and contribute to advancing the understanding of the fundamental properties of deep neural networks.

Recently, a series of papers proposed deep learning-based approaches to sample from unnormalized target densities using controlled diffusion processes. In this work, we identify these approaches as special cases of the Schr\"odinger bridge problem, seeking the most likely stochastic evolution between a given prior distribution and the specified target. We further generalize this framework by introducing a variational formulation based on divergences between path space measures of time-reversed diffusion processes. This abstract perspective leads to practical losses that can be optimized by gradient-based algorithms and includes previous objectives as special cases. At the same time, it allows us to consider divergences other than the reverse Kullback-Leibler divergence that is known to suffer from mode collapse. In particular, we propose the so-called log-variance loss, which exhibits favorable numerical properties and leads to significantly improved performance across all considered approaches.

Knowledge distillation (KD) has gained a lot of attention in the field of model compression for edge devices thanks to its effectiveness in compressing large powerful networks into smaller lower-capacity models. Online distillation, in which both the teacher and the student are learning collaboratively, has also gained much interest due to its ability to improve on the performance of the networks involved. The Kullback-Leibler (KL) divergence ensures the proper knowledge transfer between the teacher and student. However, most online KD techniques present some bottlenecks under the network capacity gap. By cooperatively and simultaneously training, the models the KL distance becomes incapable of properly minimizing the teacher's and student's distributions. Alongside accuracy, critical edge device applications are in need of well-calibrated compact networks. Confidence calibration provides a sensible way of getting trustworthy predictions. We propose BD-KD: Balancing of Divergences for online Knowledge Distillation. We show that adaptively balancing between the reverse and forward divergences shifts the focus of the training strategy to the compact student network without limiting the teacher network's learning process. We demonstrate that, by performing this balancing design at the level of the student distillation loss, we improve upon both performance accuracy and calibration of the compact student network. We conducted extensive experiments using a variety of network architectures and show improvements on multiple datasets including CIFAR-10, CIFAR-100, Tiny-ImageNet, and ImageNet. We illustrate the effectiveness of our approach through comprehensive comparisons and ablations with current state-of-the-art online and offline KD techniques.

The LogSumExp function, dual to the Kullback-Leibler (KL) divergence, plays a central role in many important optimization problems, including entropy-regularized optimal transport (OT) and distributionally robust optimization (DRO). In practice, when the number of exponential terms inside the logarithm is large or infinite, optimization becomes challenging since computing the gradient requires differentiating every term. We propose a novel convexity- and smoothness-preserving approximation to LogSumExp that can be efficiently optimized using stochastic gradient methods. This approximation is rooted in a sound modification of the KL divergence in the dual, resulting in a new f-divergence called the Safe KL divergence. Our experiments and theoretical analysis of the LogSumExp-based stochastic optimization, arising in DRO and continuous OT, demonstrate the advantages of our approach over existing baselines.

Estimating truncated density models is difficult, as these models have intractable normalising constants and hard to satisfy boundary conditions. Score matching can be adapted to solve the truncated density estimation problem, but requires a continuous weighting function which takes zero at the boundary and is positive elsewhere. Evaluation of such a weighting function (and its gradient) often requires a closed-form expression of the truncation boundary and finding a solution to a complicated optimisation problem. In this paper, we propose approximate Stein classes, which in turn leads to a relaxed Stein identity for truncated density estimation. We develop a novel discrepancy measure, truncated kernelised Stein discrepancy (TKSD), which does not require fixing a weighting function in advance, and can be evaluated using only samples on the boundary. We estimate a truncated density model by minimising the Lagrangian dual of TKSD. Finally, experiments show the accuracy of our method to be an improvement over previous works even without the explicit functional form of the boundary.

Trajectory Inference (TI) seeks to recover latent dynamical processes from snapshot data, where only independent samples from time-indexed marginals are observed. In applications such as single-cell genomics, destructive measurements make path-space laws non-identifiable from finitely many marginals, leaving held-out marginal prediction as the dominant but limited evaluation protocol. We introduce a general framework for estimating the Kullback-Leibler divergence (KL) divergence between probability measures on function space, yielding a tractable, data-driven estimator that is scalable to realistic snapshot datasets. We validate the accuracy of our estimator on a benchmark suite, where the estimated functional KL closely matches the analytic KL. Applying this framework to synthetic and real scRNA-seq datasets, we show that current evaluation metrics often give inconsistent assessments, whereas path-space KL enables a coherent comparison of trajectory inference methods and exposes discrepancies in inferred dynamics, especially in regions with sparse or missing data. These results support functional KL as a principled criterion for evaluating trajectory inference under partial observability.

Autoregressive (AR) diffusion offers a promising framework for generating videos of theoretically infinite length. However, a major challenge is maintaining temporal continuity while preventing the progressive quality degradation caused by error accumulation. To ensure continuity, existing methods typically condition on highly denoised contexts; yet, this practice propagates prediction errors with high certainty, thereby exacerbating degradation. In this paper, we argue that a highly clean context is unnecessary. Drawing inspiration from bidirectional diffusion models, which denoise frames at a shared noise level while maintaining coherence, we propose that conditioning on context at the same noise level as the current block provides sufficient signal for temporal consistency while effectively mitigating error propagation. Building on this insight, we propose HiAR, a hierarchical denoising framework that reverses the conventional generation order: instead of completing each block sequentially, it performs causal generation across all blocks at every denoising step, so that each block is always conditioned on context at the same noise level. This hierarchy naturally admits pipelined parallel inference, yielding a 1.8 wall-clock speedup in our 4-step setting. We further observe that self-rollout distillation under this paradigm amplifies a low-motion shortcut inherent to the mode-seeking reverse-KL objective. To counteract this, we introduce a forward-KL regulariser in bidirectional-attention mode, which preserves motion diversity for causal inference without interfering with the distillation loss. On VBench (20s generation), HiAR achieves the best overall score and the lowest temporal drift among all compared methods.

On-policy distillation transfers reasoning capabilities by training a student model on its own generated trajectories using token-level feedback from a teacher. However, we identify a critical bottleneck, Supervision Fidelity Decay (SFD): as student-generated prefixes lengthen, the teacher's next-token distribution becomes less confident and less discriminative. Consequently, the teacher-dependent corrective signal in reverse-KL distillation weakens, causing student drift to compound across long reasoning chains. To mitigate SFD, we introduce Lookahead Group Reward (\ours{)}. Building on the insight that next-step teacher confidence reflects the discriminative strength of future reverse-KL supervision, evaluates the student's top-K candidate tokens by the teacher confidence they induce at the subsequent step and assigns a group-normalized reward. To maintain computational efficiency, we further design an entropy-triggered tree-attention mechanism. Across six math and code benchmarks, improves mean@8 by 2.57 points over OPD for a 7B student, with gains increasing in longer-generation and reaching +4.92 points on AIME-26 at 39k tokens.

Moment restrictions and their conditional counterparts emerge in many areas of machine learning and statistics ranging from causal inference to reinforcement learning. Estimators for these tasks, generally called methods of moments, include the prominent generalized method of moments (GMM) which has recently gained attention in causal inference. GMM is a special case of the broader family of empirical likelihood estimators which are based on approximating a population distribution by means of minimizing a varphi-divergence to an empirical distribution. However, the use of varphi-divergences effectively limits the candidate distributions to reweightings of the data samples. We lift this long-standing limitation and provide a method of moments that goes beyond data reweighting. This is achieved by defining an empirical likelihood estimator based on maximum mean discrepancy which we term the kernel method of moments (KMM). We provide a variant of our estimator for conditional moment restrictions and show that it is asymptotically first-order optimal for such problems. Finally, we show that our method achieves competitive performance on several conditional moment restriction tasks.

What does a discrete diffusion model actually learn? The literature trains denoisers, bridge plug-ins, and concrete scores, each deriving its own ELBO objective under different conventions. This fragmentation makes the object being learned unclear and ELBO values difficult to interpret and compare. All three methods learn the same object in different coordinates: the exact time-reversal of the noising process. We first prove the Oracle Distance identity: for any noising process, the negative ELBO is not just a bound on the likelihood; it equals the data entropy plus the path KL divergence from the reversed noising process to the learned one. Next, for token-factorizing noise, we show that denoiser, bridge plug-in, and concrete score are three coordinates of this reverse process, derive exact conversions between them, and identify the probability distribution learned by each. In particular, contrary to its standard interpretation, the bridge plug-in learns not a denoiser but the cavity distribution: the clean token given the noisy sequence excluding its own position. We confirm the theory by training all three coordinates with masked, uniform, and GIDD kernels on OpenWebText under a shared recipe, and by training uniform-diffusion solvers on four logic tasks. The learned coordinates agree after conversion, while confusing the cavity with the denoiser can severely degrade sampling quality. Together with exact boundary-term formulas, closed-form NELBO calibrations at initialization, and a clock interpretation of time schedules, these results provide a conceptual framework and a mechanical recipe for deriving, implementing, and comparing discrete diffusion processes.

A simple and effective method for the alignment of generative models is the best-of-n policy, where n samples are drawn from a base policy, and ranked based on a reward function, and the highest ranking one is selected. A commonly used analytical expression in the literature claims that the KL divergence between the best-of-n policy and the base policy is equal to log (n) - (n-1)/n. We disprove the validity of this claim, and show that it is an upper bound on the actual KL divergence. We also explore the tightness of this upper bound in different regimes. Finally, we propose a new estimator for the KL divergence and empirically show that it provides a tight approximation through a few examples.

Diffusion large language models (dLLMs) offer a promising alternative to autoregressive decoding by iteratively refining masked sequences, enabling parallel token updates and bidirectional conditioning. Their practical efficiency, however, is limited by sampling procedures that execute a fixed number of reverse denoising steps selected before decoding, spending computation on already-stable positions and sometimes committing unstable ones too early. We present LESS, a training-free, model-agnostic adaptive sampler that treats token commitment as an online stopping problem. LESS implements mutual-stability sampling through a joint stability rule that makes a masked position eligible for unmasking only when its top-1 prediction has high confidence, its top-1 token persists across recent reverse steps, and its predictive distribution is stable under top-K inter-step Jensen--Shannon divergence. We evaluate LESS on Dream-7B, LLaDA-8B, and LLaDA-1.5-8B, covering full-sequence diffusion and semi-autoregressive blockwise sampling regimes, across seven benchmarks spanning general knowledge, math, and code. LESS improves average accuracy over strong training-free adaptive samplers while using 72.1% fewer reverse steps than fixed-budget decoding. Since each reverse step requires a Transformer forward pass, these step-count reductions translate into fewer forward evaluations, lower measured wall-clock latency, and lower estimated inference compute.

Reinforcement Learning with Verified Reward (RLVR) has emerged as a critical paradigm for advancing the reasoning capabilities of Large Language Models (LLMs). Most existing RLVR methods, such as GRPO and its variants, ensure stable updates by constraining policy divergence through clipping likelihood ratios. This paper introduces a unified clipping framework that characterizes existing methods via a general notion of policy divergence, encompassing both likelihood ratios and Kullback-Leibler (KL) divergences and extending to alternative measures. The framework provides a principled foundation for systematically analyzing how different policy divergence measures affect exploration and performance. We further identify the KL3 estimator, a variance-reduced Monte Carlo estimator of the KL divergence, as a key policy divergence constraint. We theoretically demonstrate that the KL3-based constraint is mathematically equivalent to an asymmetric ratio-based clipping that reallocates probability mass toward high-confidence actions, promoting stronger exploration while retaining the simplicity of GRPO-style methods. Empirical results on mathematical reasoning benchmarks demonstrate that incorporating the KL3 estimator into GRPO improves both training stability and final performance, highlighting the importance of principled policy divergence constraints in policy optimization.

Is it possible to perform linear regression on datasets whose labels are shuffled with respect to the inputs? We explore this question by proposing several estimators that recover the weights of a noisy linear model from labels that are shuffled by an unknown permutation. We show that the analog of the classical least-squares estimator produces inconsistent estimates in this setting, and introduce an estimator based on the self-moments of the input features and labels. We study the regimes in which each estimator excels, and generalize the estimators to the setting where partial ordering information is available in the form of experiments replicated independently. The result is a framework that enables robust inference, as we demonstrate by experiments on both synthetic and standard datasets, where we are able to recover approximate weights using only shuffled labels. Our work demonstrates that linear regression in the absence of complete ordering information is possible and can be of practical interest, particularly in experiments that characterize populations of particles, such as flow cytometry.

Knowledge distillation (KD) is widely used for compressing a teacher model to reduce its inference cost and memory footprint, by training a smaller student model. However, current KD methods for auto-regressive sequence models suffer from distribution mismatch between output sequences seen during training and those generated by the student during inference. To address this issue, we introduce Generalized Knowledge Distillation (GKD). Instead of solely relying on a fixed set of output sequences, GKD trains the student on its self-generated output sequences by leveraging feedback from the teacher on such sequences. Unlike supervised KD approaches, GKD also offers the flexibility to employ alternative loss functions between the student and teacher, which can be useful when the student lacks the expressivity to mimic the teacher's distribution. Furthermore, GKD facilitates the seamless integration of distillation with RL fine-tuning (RLHF). We demonstrate the efficacy of GKD for distilling auto-regressive language models on summarization, translation, and arithmetic reasoning tasks, and task-agnostic distillation for instruction-tuning.

The problem of estimating an unknown discrete distribution from its samples is a fundamental tenet of statistical learning. Over the past decade, it attracted significant research effort and has been solved for a variety of divergence measures. Surprisingly, an equally important problem, estimating an unknown Markov chain from its samples, is still far from understood. We consider two problems related to the min-max risk (expected loss) of estimating an unknown k-state Markov chain from its n sequential samples: predicting the conditional distribution of the next sample with respect to the KL-divergence, and estimating the transition matrix with respect to a natural loss induced by KL or a more general f-divergence measure. For the first measure, we determine the min-max prediction risk to within a linear factor in the alphabet size, showing it is Omega(kloglog n / n) and O(k^2loglog n / n). For the second, if the transition probabilities can be arbitrarily small, then only trivial uniform risk upper bounds can be derived. We therefore consider transition probabilities that are bounded away from zero, and resolve the problem for essentially all sufficiently smooth f-divergences, including KL-, L_2-, Chi-squared, Hellinger, and Alpha-divergences.

Diffusion large language models (dLLMs) are a promising alternative to autoregressive generation. However, reasoning-oriented post-training for dLLMs remains challenging. Supervised fine-tuning (SFT) for dLLMs requires dense but often off-policy masked states, while reinforcement learning (RL) relies on sparse rewards or value modeling. This paper proposes trace-based on-policy distillation (TOPD), a teacher-supervised framework that transfers reasoning ability to a target dLLM without reward estimation. The key idea is to supervise a dLLM on its own denoising trajectory, focusing on the trace-aligned token decisions that form the final response. Specifically, TOPD samples on-policy diffusion trajectories from the target dLLM, obtains teacher token distributions from a teacher model on the corresponding partially denoised states, and updates the target dLLM with a token-level Reverse Kullback-Leibler (Reverse-KL) objective. This design preserves dense teacher supervision while aligning training with the model's own denoising states. On mathematical reasoning benchmarks, TOPD enables SDAR-4B-Chat to match the MATH500 accuracy of its RL-trained counterpart TraDo-4B-Instruct, with gains of +5.7 under static evaluation and +4.5 under dynamic evaluation. Compared with the RL-trained counterpart, TOPD achieves this with 4times fewer rollout rounds, corresponding to an estimated 96.0times to-accuracy model-compute speedup.

Causal effects are usually studied in terms of the means of counterfactual distributions, which may be insufficient in many scenarios. Given a class of densities known up to normalizing constants, we propose to model counterfactual distributions by minimizing kernel Stein discrepancies in a doubly robust manner. This enables the estimation of counterfactuals over large classes of distributions while exploiting the desired double robustness. We present a theoretical analysis of the proposed estimator, providing sufficient conditions for consistency and asymptotic normality, as well as an examination of its empirical performance.

Variational inference (VI) seeks to approximate a target distribution pi by an element of a tractable family of distributions. Of key interest in statistics and machine learning is Gaussian VI, which approximates pi by minimizing the Kullback-Leibler (KL) divergence to pi over the space of Gaussians. In this work, we develop the (Stochastic) Forward-Backward Gaussian Variational Inference (FB-GVI) algorithm to solve Gaussian VI. Our approach exploits the composite structure of the KL divergence, which can be written as the sum of a smooth term (the potential) and a non-smooth term (the entropy) over the Bures-Wasserstein (BW) space of Gaussians endowed with the Wasserstein distance. For our proposed algorithm, we obtain state-of-the-art convergence guarantees when pi is log-smooth and log-concave, as well as the first convergence guarantees to first-order stationary solutions when pi is only log-smooth.

Large language models (LLMs) offer impressive performance but are impractical for resource-constrained deployment due to high latency and energy consumption. Knowledge distillation (KD) addresses this by transferring knowledge from a large teacher to a smaller student model. However, conventional KD, notably approaches like Forward KL (FKL) and Reverse KL (RKL), apply uniform divergence loss across the entire vocabulary, neglecting token-level prediction discrepancies. By investigating these representative divergences via gradient analysis, we reveal that FKL boosts underestimated tokens, while RKL suppresses overestimated ones, showing their complementary roles. Based on this observation, we propose Token-wise Distillation (ToDi), a novel method that adaptively combines FKL and RKL per token using a sigmoid-based weighting function derived from the teacher-student probability log-ratio. ToDi dynamically emphasizes the appropriate divergence for each token, enabling precise distribution alignment. We demonstrate that ToDi consistently outperforms recent distillation baselines using uniform or less granular strategies across instruction-following benchmarks. Extensive ablation studies and efficiency analysis further validate ToDi's effectiveness and practicality.

As an adaptive method, Shampoo employs a structured second-moment estimation, and its effectiveness has attracted growing attention. Prior work has primarily analyzed its estimation scheme through the Frobenius norm. Motivated by the natural connection between the second moment and a covariance matrix, we propose studying Shampoo's estimation as covariance estimation through the lens of Kullback-Leibler (KL) minimization. This alternative perspective reveals a previously hidden limitation, motivating improvements to Shampoo's design. Building on this insight, we develop a practical estimation scheme, termed KL-Shampoo, that eliminates Shampoo's reliance on Adam for stabilization, thereby removing the additional memory overhead introduced by Adam. Preliminary results show that KL-Shampoo improves Shampoo's performance, enabling it to stabilize without Adam and even outperform its Adam-stabilized variant, SOAP, in neural network pretraining.

Recent Reinforcement Learning (RL) algorithms making use of Kullback-Leibler (KL) regularization as a core component have shown outstanding performance. Yet, only little is understood theoretically about why KL regularization helps, so far. We study KL regularization within an approximate value iteration scheme and show that it implicitly averages q-values. Leveraging this insight, we provide a very strong performance bound, the very first to combine two desirable aspects: a linear dependency to the horizon (instead of quadratic) and an error propagation term involving an averaging effect of the estimation errors (instead of an accumulation effect). We also study the more general case of an additional entropy regularizer. The resulting abstract scheme encompasses many existing RL algorithms. Some of our assumptions do not hold with neural networks, so we complement this theoretical analysis with an extensive empirical study.

We characterize the statistical efficiency of knowledge transfer through n samples from a teacher to a probabilistic student classifier with input space mathcal S over labels mathcal A. We show that privileged information at three progressive levels accelerates the transfer. At the first level, only samples with hard labels are known, via which the maximum likelihood estimator attains the minimax rate {|{mathcal S||{mathcal A}|}/{n}}. The second level has the teacher probabilities of sampled labels available in addition, which turns out to boost the convergence rate lower bound to {{|{mathcal S}||{mathcal A}|}/{n}}. However, under this second data acquisition protocol, minimizing a naive adaptation of the cross-entropy loss results in an asymptotically biased student. We overcome this limitation and achieve the fundamental limit by using a novel empirical variant of the squared error logit loss. The third level further equips the student with the soft labels (complete logits) on {mathcal A} given every sampled input, thereby provably enables the student to enjoy a rate {|{mathcal S}|}/{n} free of |{mathcal A}|. We find any Kullback-Leibler divergence minimizer to be optimal in the last case. Numerical simulations distinguish the four learners and corroborate our theory.

Unlike the predictable scaling laws in natural language processing and computer vision, protein language models (PLMs) scale poorly: for many tasks, models within the same family plateau or even decrease in performance, with mid-sized models often outperforming the largest in the family. We introduce Reverse Distillation, a principled framework that decomposes large PLM representations into orthogonal subspaces guided by smaller models of the same family. The resulting embeddings have a nested, Matryoshka-style structure: the first k dimensions of a larger model's embedding are exactly the representation from the smaller model. This ensures that larger reverse-distilled models consistently outperform smaller ones. A motivating intuition is that smaller models, constrained by capacity, preferentially encode broadly-shared protein features. Reverse distillation isolates these shared features and orthogonally extracts additional contributions from larger models, preventing interference between the two. On ProteinGym benchmarks, reverse-distilled ESM-2 variants outperform their respective baselines at the same embedding dimensionality, with the reverse-distilled 15 billion parameter model achieving the strongest performance. Our framework is generalizable to any model family where scaling challenges persist. Code and trained models are available at https://github.com/rohitsinghlab/plm_reverse_distillation.

Knowledge distillation is a model compression technique in which a compact "student" network is trained to replicate the predictive behavior of a larger "teacher" network. In logit-based knowledge distillation, it has become the de facto approach to augment cross-entropy with a distillation term. Typically, this term is either a KL divergence that matches marginal probabilities or a correlation-based loss that captures intra- and inter-class relationships. In every case, it acts as an additional term to cross-entropy. This term has its own weight, which must be carefully tuned. In this paper, we adopt a choice-theoretic perspective and recast knowledge distillation under the Plackett-Luce model by interpreting teacher logits as "worth" scores. We introduce "Plackett-Luce Distillation (PLD)", a weighted list-wise ranking loss. In PLD, the teacher model transfers knowledge of its full ranking of classes, weighting each ranked choice by its own confidence. PLD directly optimizes a single "teacher-optimal" ranking. The true label is placed first, followed by the remaining classes in descending teacher confidence. This process yields a convex and translation-invariant surrogate that subsumes weighted cross-entropy. Empirically, across CIFAR-100, ImageNet-1K, and MS-COCO, PLD achieves consistent gains across diverse architectures and distillation objectives, including divergence-based, correlation-based, and feature-based methods, in both homogeneous and heterogeneous teacher-student pairs.

The estimation of repeatedly nested expectations is a challenging task that arises in many real-world systems. However, existing methods generally suffer from high computational costs when the number of nestings becomes large. Fix any non-negative integer D for the total number of nestings. Standard Monte Carlo methods typically cost at least O(varepsilon^{-(2+D)}) and sometimes O(varepsilon^{-2(1+D)}) to obtain an estimator up to varepsilon-error. More advanced methods, such as multilevel Monte Carlo, currently only exist for D = 1. In this paper, we propose a novel Monte Carlo estimator called READ, which stands for "Recursive Estimator for Arbitrary Depth.'' Our estimator has an optimal computational cost of O(varepsilon^{-2}) for every fixed D under suitable assumptions, and a nearly optimal computational cost of O(varepsilon^{-2(1 + delta)}) for any 0 < delta < frac12 under much more general assumptions. Our estimator is also unbiased, which makes it easy to parallelize. The key ingredients in our construction are an observation of the problem's recursive structure and the recursive use of the randomized multilevel Monte Carlo method.

Reinforcement learning with verifiable rewards (RLVR) has recently enhanced the reasoning capabilities of large language models (LLMs), particularly for mathematical problem solving. However, a fundamental limitation remains: as the sampling budget increases, the advantage of RLVR-trained models over their pretrained bases often diminishes or even vanishes, revealing a strong dependence on the base model's restricted search space. We attribute this phenomenon to the widespread use of the reverse Kullback-Leibler (KL) divergence regularizer, whose mode-seeking behavior keeps the policy trapped inside the base model's support region and hampers wider exploration. To address this issue, we propose RAPO (Rewards-Aware Policy Optimization), an algorithm to promote broader yet focused exploration. Our method (i) utilizes the forward KL penalty to replace the reverse KL penalty for out-of-distribution exploration, and (ii) reweights the reference policy to facilitate adaptive in-distribution exploration. We train Qwen2.5-3B and 7B models with RAPO on the 8K SimpleRL-Zero dataset, without supervised fine-tuning, and evaluate them on AIME2024 and AIME2025. Results show that RAPO consistently improves problem-solving performance. Notably, RAPO enables models to surpass the base model's performance ceiling and solves previously intractable problems, advancing the frontier of RLVR for challenging reasoning tasks.

Numerous capability and safety techniques of Large Language Models (LLMs), including RLHF, automated red-teaming, prompt engineering, and infilling, can be cast as sampling from an unnormalized target distribution defined by a given reward or potential function over the full sequence. In this work, we leverage the rich toolkit of Sequential Monte Carlo (SMC) for these probabilistic inference problems. In particular, we use learned twist functions to estimate the expected future value of the potential at each timestep, which enables us to focus inference-time computation on promising partial sequences. We propose a novel contrastive method for learning the twist functions, and establish connections with the rich literature of soft reinforcement learning. As a complementary application of our twisted SMC framework, we present methods for evaluating the accuracy of language model inference techniques using novel bidirectional SMC bounds on the log partition function. These bounds can be used to estimate the KL divergence between the inference and target distributions in both directions. We apply our inference evaluation techniques to show that twisted SMC is effective for sampling undesirable outputs from a pretrained model (a useful component of harmlessness training and automated red-teaming), generating reviews with varied sentiment, and performing infilling tasks.

In this paper, we introduce a novel approach for bounding the cumulant generating function (CGF) of a Dirichlet process (DP) X sim DP(αν_0), using superadditivity. In particular, our key technical contribution is the demonstration of the superadditivity of αmapsto log E_{X sim DP(αν_0)}[exp( E_X[αf])], where E_X[f] = int f dX. This result, combined with Fekete's lemma and Varadhan's integral lemma, converts the known asymptotic large deviation principle into a practical upper bound on the CGF logE_{Xsim DP(αν_0)}{exp(E_{X}{[f]})} for any α> 0. The bound is given by the convex conjugate of the scaled reversed Kullback-Leibler divergence αKL(ν_0Vert cdot). This new bound provides particularly effective confidence regions for sums of independent DPs, making it applicable across various fields.

Kernelized Stein discrepancy (KSD) is a score-based discrepancy widely used in goodness-of-fit tests. It can be applied even when the target distribution has an unknown normalising factor, such as in Bayesian analysis. We show theoretically and empirically that the KSD test can suffer from low power when the target and the alternative distributions have the same well-separated modes but differ in mixing proportions. We propose to perturb the observed sample via Markov transition kernels, with respect to which the target distribution is invariant. This allows us to then employ the KSD test on the perturbed sample. We provide numerical evidence that with suitably chosen transition kernels the proposed approach can lead to substantially higher power than the KSD test.

We propose a Monte Carlo sampler from the reverse diffusion process. Unlike the practice of diffusion models, where the intermediary updates -- the score functions -- are learned with a neural network, we transform the score matching problem into a mean estimation one. By estimating the means of the regularized posterior distributions, we derive a novel Monte Carlo sampling algorithm called reverse diffusion Monte Carlo (rdMC), which is distinct from the Markov chain Monte Carlo (MCMC) methods. We determine the sample size from the error tolerance and the properties of the posterior distribution to yield an algorithm that can approximately sample the target distribution with any desired accuracy. Additionally, we demonstrate and prove under suitable conditions that sampling with rdMC can be significantly faster than that with MCMC. For multi-modal target distributions such as those in Gaussian mixture models, rdMC greatly improves over the Langevin-style MCMC sampling methods both theoretically and in practice. The proposed rdMC method offers a new perspective and solution beyond classical MCMC algorithms for the challenging complex distributions.

Since the advent of knowledge distillation, KL divergence has been the standard loss in distillation. Recently, on-policy distillation (OPD) has emerged as an efficient post-training paradigm for LLMs. As a distillation method, OPD naturally inherits KL divergence as its standard loss. However, in this work, we find that KL divergence may not be necessary for OPD. We show that simply preserving the update direction is sufficient for effective OPD. As long as the update direction is toward the teacher, OPD works. More precisely, it is not the direction of every token, but the direction of a small subset of tokens where the teacher and student disagree strongly. We first show that simply assigning a reward of (+1) to tokens where the teacher probability is higher than the student probability and (-1) where it is lower, which merely encourages updates toward the teacher, reproduces almost the same training mode as OPD with reverse KL. We further show that only the direction of a small subset of tokens with large teacher-student disagreement is critical, and training works as long as their update direction is toward the teacher, even if other tokens are pulled away from the teacher. And as an application of these findings, we introduce Consensus Multi-Teacher On-Policy Distillation (C-MOPD) to improve Multi-Teacher On-Policy Distillation (MOPD). Unlike MOPD, which routes each sample to a single teacher and may cause capability conflicts across domains, C-MOPD lets every sample be supervised by all teachers. Experiments show that C-MOPD consistently outperforms MOPD on both math and code benchmarks. Our code is available at https://github.com/LeapLabTHU/KL-Free-OPD.

One of the major limits of kernel ridge regression (KRR) is that storing and manipulating the kernel matrix K_n for n samples requires O(n^2) space, which rapidly becomes unfeasible for large n. Nystrom approximations reduce the space complexity to O(nm) by sampling m columns from K_n. Uniform sampling preserves KRR accuracy (up to epsilon) only when m is proportional to the maximum degree of freedom of K_n, which may require O(n) columns for datasets with high coherence. Sampling columns according to their ridge leverage scores (RLS) gives accurate Nystrom approximations with m proportional to the effective dimension, but computing exact RLS also requires O(n^2) space. (Calandriello et al. 2016) propose INK-Estimate, an algorithm that processes the dataset incrementally and updates RLS, effective dimension, and Nystrom approximations on-the-fly. Its space complexity scales with the effective dimension but introduces a dependency on the largest eigenvalue of K_n, which in the worst case is O(n). In this paper we introduce SQUEAK, a new algorithm that builds on INK-Estimate but uses unnormalized RLS. As a consequence, the algorithm is simpler, does not need to estimate the effective dimension for normalization, and achieves a space complexity that is only a constant factor worse than exact RLS sampling.

Diffusion models are powerful generative models that simulate the reverse of diffusion processes using score functions to synthesize data from noise. The sampling process of diffusion models can be interpreted as solving the reverse stochastic differential equation (SDE) or the ordinary differential equation (ODE) of the diffusion process, which often requires up to thousands of discretization steps to generate a single image. This has sparked a great interest in developing efficient integration techniques for reverse-S/ODEs. Here, we propose an orthogonal approach to accelerating score-based sampling: Denoising MCMC (DMCMC). DMCMC first uses MCMC to produce samples in the product space of data and variance (or diffusion time). Then, a reverse-S/ODE integrator is used to denoise the MCMC samples. Since MCMC traverses close to the data manifold, the computation cost of producing a clean sample for DMCMC is much less than that of producing a clean sample from noise. To verify the proposed concept, we show that Denoising Langevin Gibbs (DLG), an instance of DMCMC, successfully accelerates all six reverse-S/ODE integrators considered in this work on the tasks of CIFAR10 and CelebA-HQ-256 image generation. Notably, combined with integrators of Karras et al. (2022) and pre-trained score models of Song et al. (2021b), DLG achieves SOTA results. In the limited number of score function evaluation (NFE) settings on CIFAR10, we have 3.86 FID with approx 10 NFE and 2.63 FID with approx 20 NFE. On CelebA-HQ-256, we have 6.99 FID with approx 160 NFE, which beats the current best record of Kim et al. (2022) among score-based models, 7.16 FID with 4000 NFE. Code: https://github.com/1202kbs/DMCMC

Mixture of experts (MoE) has a well-principled finite mixture model construction for prediction, allowing the gating network (mixture weights) to learn from the predictors (explanatory variables) together with the experts' network (mixture component densities). We investigate the estimation properties of MoEs in a high-dimensional setting, where the number of predictors is much larger than the sample size, for which the literature lacks computational and especially theoretical results. We consider the class of finite MoE models with softmax gating functions and Gaussian regression experts, and focus on the theoretical properties of their l_1-regularized estimation via the Lasso. We provide a lower bound on the regularization parameter of the Lasso penalty that ensures an l_1-oracle inequality is satisfied by the Lasso estimator according to the Kullback--Leibler loss. We further state an l_1-ball oracle inequality for the l_1-penalized maximum likelihood estimator from the model selection.

Instruction following (IF) is a critical capability for large language models (LLMs). However, handling complex instructions with multiple constraints remains challenging. Previous methods typically select preference pairs based on the number of constraints they satisfy, introducing noise where chosen examples may fail to follow some constraints and rejected examples may excel in certain respects over the chosen ones. To address the challenge of aligning with multiple preferences, we propose a simple yet effective method called Reverse Preference Optimization (RPO). It mitigates noise in preference pairs by dynamically reversing the constraints within the instruction to ensure the chosen response is perfect, alleviating the burden of extensive sampling and filtering to collect perfect responses. Besides, reversal also enlarges the gap between chosen and rejected responses, thereby clarifying the optimization direction and making it more robust to noise. We evaluate RPO on two multi-turn IF benchmarks, Sysbench and Multi-IF, demonstrating average improvements over the DPO baseline of 4.6 and 2.5 points (on Llama-3.1 8B), respectively. Moreover, RPO scales effectively across model sizes (8B to 70B parameters), with the 70B RPO model surpassing GPT-4o.

Policy gradient algorithms have been successfully applied to enhance the reasoning capabilities of large language models (LLMs). Despite the widespread use of Kullback-Leibler (KL) regularization in policy gradient algorithms to stabilize training, the systematic exploration of how different KL divergence formulations can be estimated and integrated into surrogate loss functions for online reinforcement learning (RL) presents a nuanced and systematically explorable design space. In this paper, we propose regularized policy gradient (RPG), a systematic framework for deriving and analyzing KL-regularized policy gradient methods in the online RL setting. We derive policy gradients and corresponding surrogate loss functions for objectives regularized by both forward and reverse KL divergences, considering both normalized and unnormalized policy distributions. Furthermore, we present derivations for fully differentiable loss functions as well as REINFORCE-style gradient estimators, accommodating diverse algorithmic needs. We conduct extensive experiments on RL for LLM reasoning using these methods, showing improved or competitive results in terms of training stability and performance compared to strong baselines such as GRPO, REINFORCE++, and DAPO. The code is available at https://github.com/complex-reasoning/RPG.

Recently, diffusion probabilistic models (DPMs) have achieved promising results in diverse generative tasks. A typical DPM framework includes a forward process that gradually diffuses the data distribution and a reverse process that recovers the data distribution from time-dependent data scores. In this work, we observe that the stochastic reverse process of data scores is a martingale, from which concentration bounds and the optional stopping theorem for data scores can be derived. Then, we discover a simple way for calibrating an arbitrary pretrained DPM, with which the score matching loss can be reduced and the lower bounds of model likelihood can consequently be increased. We provide general calibration guidelines under various model parametrizations. Our calibration method is performed only once and the resulting models can be used repeatedly for sampling. We conduct experiments on multiple datasets to empirically validate our proposal. Our code is at https://github.com/thudzj/Calibrated-DPMs.

We provide a theoretical framework for Reinforcement Learning with Human Feedback (RLHF). Our analysis shows that when the true reward function is linear, the widely used maximum likelihood estimator (MLE) converges under both the Bradley-Terry-Luce (BTL) model and the Plackett-Luce (PL) model. However, we show that when training a policy based on the learned reward model, MLE fails while a pessimistic MLE provides policies with improved performance under certain coverage assumptions. Additionally, we demonstrate that under the PL model, the true MLE and an alternative MLE that splits the K-wise comparison into pairwise comparisons both converge. Moreover, the true MLE is asymptotically more efficient. Our results validate the empirical success of existing RLHF algorithms in InstructGPT and provide new insights for algorithm design. Furthermore, our results unify the problem of RLHF and max-entropy Inverse Reinforcement Learning (IRL), and provide the first sample complexity bound for max-entropy IRL.

Model overconfidence and poor calibration are common in machine learning and difficult to account for when applying standard empirical risk minimization. In this work, we propose a novel method to alleviate these problems that we call odd-k-out learning (OKO), which minimizes the cross-entropy error for sets rather than for single examples. This naturally allows the model to capture correlations across data examples and achieves both better accuracy and calibration, especially in limited training data and class-imbalanced regimes. Perhaps surprisingly, OKO often yields better calibration even when training with hard labels and dropping any additional calibration parameter tuning, such as temperature scaling. We demonstrate this in extensive experimental analyses and provide a mathematical theory to interpret our findings. We emphasize that OKO is a general framework that can be easily adapted to many settings and a trained model can be applied to single examples at inference time, without significant run-time overhead or architecture changes.

Cross-tokenizer knowledge distillation allows a student model to learn from teachers with incompatible vocabularies. Prior work operates on hidden states or logits; the latter is preferred as a drop-in replacement requiring no auxiliary components. Logit-based methods either use only the correct-token probability, missing the full 'dark knowledge' in the teacher's distribution, or operate on the full output distribution, relying on strict token partitioning and/or unprincipled heuristic ranking. We identify two key shortcomings of full-distribution, logit-based methods: (i) an uncommon-token failure, where critical tokens fall into the unmatched subset (e.g., Llama's 1100 multi-digit numerals under digit-splitting Qwen supervision) and are suppressed during training, reducing GSM8k from 12.89 to 2.56 compared to same-tokenizer KD from a weaker teacher; and (ii) over-conservative matching, where strict 1-to-1 matching excludes near-equivalent tokens across surface forms. These failures require distinct remedies: eliminating the partition when critical tokens are misaligned, and refining it when alignment is reliable. We propose X-Token, an approach with two complementary loss formulations targeting these issues. P-KL removes partitioning and aligns the student's distribution with the teacher's via a sparse projection matrix W (initialized from tokenizer-level string rules) to address the uncommon-token failure. H-KL retains the hybrid form while relaxing matching to align each student token with its top-ranked teacher mapping under W. Both objectives share W and extend naturally to multiple teachers. Empirically, on Llama-3.2-1B, X-Token outperforms the current state of the art GOLD by +3.82 average points with a Qwen3-4B teacher and by +0.5 with a Phi-4-Mini teacher. Further, a two-teacher setup (Phi-4-mini + Llama-3B) improves over single-teacher distillation by +1.3 points.

Knowledge distillation plays a key role in compressing the Large Language Models (LLMs), which boosts a small-size student model under large teacher models' guidance. However, existing LLM distillation methods overly rely on student-generated outputs, which may introduce generation errors and misguide the distillation process. Moreover, the distillation loss functions introduced in previous art struggle to align the most informative part due to the complex distribution of LLMs' outputs. To address these problems, we propose a multi-granularity semantic revision method for LLM distillation. At the sequence level, we propose a sequence correction and re-generation (SCRG) strategy. SCRG first calculates the semantic cognitive difference between the teacher and student to detect the error token, then corrects it with the teacher-generated one, and re-generates the sequence to reduce generation errors and enhance generation diversity. At the token level, we design a distribution adaptive clipping Kullback-Leibler (DAC-KL) loss as the distillation objective function. DAC-KL loss exploits a learnable sub-network to adaptively extract semantically dense areas from the teacher's output, avoiding the interference of redundant information in the distillation process. Finally, at the span level, we leverage the span priors of a sequence to compute the probability correlations within spans, and constrain the teacher and student's probability correlations to be consistent, further enhancing the transfer of semantic information. Extensive experiments across different model families with parameters ranging from 0.1B to 13B demonstrate the superiority of our method compared to existing methods.

Recent advances in machine learning have significantly improved prediction accuracy in various applications. However, ensuring the calibration of probabilistic predictions remains a significant challenge. Despite efforts to enhance model calibration, the rigorous statistical evaluation of model calibration remains less explored. In this work, we develop confidence intervals the ell_2 Expected Calibration Error (ECE). We consider top-1-to-k calibration, which includes both the popular notion of confidence calibration as well as full calibration. For a debiased estimator of the ECE, we show asymptotic normality, but with different convergence rates and asymptotic variances for calibrated and miscalibrated models. We develop methods to construct asymptotically valid confidence intervals for the ECE, accounting for this behavior as well as non-negativity. Our theoretical findings are supported through extensive experiments, showing that our methods produce valid confidence intervals with shorter lengths compared to those obtained by resampling-based methods.

We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples (x,y) from an unknown distribution on R^n times { pm 1}, whose marginal distribution on x is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with 0-1 loss OPT+epsilon, where OPT is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.

On-policy self-distillation (OPSD) enables large language models (LLMs) to learn from privileged information (PI) without a separate teacher model. However, prior work largely treats PI as a fixed design choice, leaving the effects of its content and construction poorly understood. This challenge is particularly important in multi-turn agentic settings, where trajectories contain actions, observations, and reasoning that can be transformed into PI in many ways. We systematically evaluate 16 PI configurations across three multi-turn agentic benchmarks and three student models that vary in scale and reasoning mode. We find that PI effectiveness depends on its content, generation source, and the student model. Information-dense trajectories and PI produced by more capable generators generally yield larger gains, whereas thinking models benefit less consistently. To assess PI before training, we compare PI utility with two KL-based metrics: sampled-token reverse KL (D) and error-step KL contrast (Δd^{-}_{err}). The KL-based metrics track post-training performance more consistently than PI utility, which can be distorted by information leakage. Empirically, selecting each task's PI by Δd^{-}_{err} also improves post-training performance over random selection, most when the PI candidates differ widely. Finally, effective PI remains beneficial when combined with GRPO, allowing the combined objective to outperform a strong GRPO baseline. Overall, our results provide practical guidance for constructing and evaluating PI in OPSD training.

Many policy optimization approaches in reinforcement learning incorporate a Kullback-Leilbler (KL) divergence to the previous policy, to prevent the policy from changing too quickly. This idea was initially proposed in a seminal paper on Conservative Policy Iteration, with approximations given by algorithms like TRPO and Munchausen Value Iteration (MVI). We continue this line of work by investigating a generalized KL divergence -- called the Tsallis KL divergence -- which use the q-logarithm in the definition. The approach is a strict generalization, as q = 1 corresponds to the standard KL divergence; q > 1 provides a range of new options. We characterize the types of policies learned under the Tsallis KL, and motivate when q >1 could be beneficial. To obtain a practical algorithm that incorporates Tsallis KL regularization, we extend MVI, which is one of the simplest approaches to incorporate KL regularization. We show that this generalized MVI(q) obtains significant improvements over the standard MVI(q = 1) across 35 Atari games.

Speculative decoding accelerates autoregressive large language model (LLM) inference by using a lightweight draft model to propose candidate tokens that are then verified in parallel by the target model. The speedup is significantly determined by the acceptance rate, yet standard training minimizes Kullback-Leibler (KL) divergence as a proxy objective. While KL divergence and acceptance rate share the same global optimum, small draft models, having limited capacity, typically converge to suboptimal solutions where minimizing KL does not guarantee maximizing acceptance rate. To address this issue, we propose LK losses, special training objectives that directly target acceptance rate. Comprehensive experiments across four draft architectures and six target models, ranging from 8B to 685B parameters, demonstrate consistent improvements in acceptance metrics across all configurations compared to the standard KL-based training. We evaluate our approach on general, coding and math domains and report gains of up to 8-10% in average acceptance length. LK losses are easy to implement, introduce no computational overhead and can be directly integrated into any existing speculator training framework, making them a compelling alternative to the existing draft training objectives.

Reinforcement learning for large language models faces a fundamental trade-off between sample efficiency and asymptotic performance: strictly on-policy methods discard trajectories after a single update, while off-policy reuse introduces distribution mismatch that existing trust-region techniques mitigate primarily by enforcing conservative optimization, often leaving rich training signals underutilized. To investigate this, we perform extensive off-policy updates on fixed data. Our experiments reveal that aggressive multi-step optimization brings rapid initial gains, but excessive updates cause trajectory probabilities to deviate and entropy to collapse, with performance plateauing early. Tightening KL constraints merely lowers the ceiling without resolving the degradation. This motivates Extreme Region Policy Distillation (ERPD), a two-stage framework that decouples sample efficiency from KL efficiency. The first stage performs weakly constrained off-policy optimization on fixed data to maximally extract training signals. The resulting policy provides token-level supervision. In the second stage, we distill these signals into the base policy under trust-region constraints, filtering harmful drift while preserving useful signals. The distilled policy achieves comparable or better performance with substantially smaller KL divergence, indicating that much of the first-stage divergence was spent on unnecessary drift rather than genuine improvement. Crucially, ERPD accommodates both strong and weak teachers: when aggressive optimization yields no stronger policy, even degenerate teachers provide effective supervision via alternative signal construction strategies. We validate ERPD on mathematical reasoning, showing gains for strong base models where on-policy training plateaus, and reliable improvements with weak teachers.

Distribution Matching Distillation (DMD) is a promising score distillation technique that compresses pre-trained teacher diffusion models into efficient one-step or multi-step student generators. Nevertheless, its reliance on the reverse Kullback-Leibler (KL) divergence minimization potentially induces mode collapse (or mode-seeking) in certain applications. To circumvent this inherent drawback, we propose Adversarial Distribution Matching (ADM), a novel framework that leverages diffusion-based discriminators to align the latent predictions between real and fake score estimators for score distillation in an adversarial manner. In the context of extremely challenging one-step distillation, we further improve the pre-trained generator by adversarial distillation with hybrid discriminators in both latent and pixel spaces. Different from the mean squared error used in DMD2 pre-training, our method incorporates the distributional loss on ODE pairs collected from the teacher model, and thus providing a better initialization for score distillation fine-tuning in the next stage. By combining the adversarial distillation pre-training with ADM fine-tuning into a unified pipeline termed DMDX, our proposed method achieves superior one-step performance on SDXL compared to DMD2 while consuming less GPU time. Additional experiments that apply multi-step ADM distillation on SD3-Medium, SD3.5-Large, and CogVideoX set a new benchmark towards efficient image and video synthesis.

Reinforcement learning (RL) can be used to improve the policy (denoiser) of diffusion large language models (dLLMs), while being hindered by the intractability of the policy likelihood. A dominant and efficient family of methods replaces the likelihood in standard RL with its evidence lower bound (ELBO), estimated from randomly masked sequences. Despite being well aligned with pre-training, these approaches introduce bias through training--inference mismatch by using the ELBO as a likelihood surrogate, which can degrade performance. In this work, we propose Guided Denoiser Self-Distillation (GDSD) to directly distill the denoiser of dLLMs from an advantage-guided self-teacher, derived from the closed-form optimum of reverse-KL regularized RL. GDSD matches the dLLM's denoiser logits to the teacher's via a normalization-free objective, which reduces RL to likelihood-free self-distillation and thus bypasses the TIM biases. Recent ELBO-based methods emerge as instances of applying different distillation divergences, but with diagnosable pathologies that GDSD avoids. On planning, math, and coding benchmarks with LLaDA-8B and Dream-7B, GDSD consistently outperforms prior state-of-the-art ELBO-based methods with a more stable training reward dynamics, achieving test-accuracy improvements of up to +19.6%. These results suggest that direct denoiser self-distillation, without relying on an ELBO likelihood surrogate, can provide a more stable and effective RL procedure for dLLMs. Code is available at https://github.com/GaryBall/GDSD.

The integration of discrete algorithmic components in deep learning architectures has numerous applications. Recently, Implicit Maximum Likelihood Estimation (IMLE, Niepert, Minervini, and Franceschi 2021), a class of gradient estimators for discrete exponential family distributions, was proposed by combining implicit differentiation through perturbation with the path-wise gradient estimator. However, due to the finite difference approximation of the gradients, it is especially sensitive to the choice of the finite difference step size, which needs to be specified by the user. In this work, we present Adaptive IMLE (AIMLE), the first adaptive gradient estimator for complex discrete distributions: it adaptively identifies the target distribution for IMLE by trading off the density of gradient information with the degree of bias in the gradient estimates. We empirically evaluate our estimator on synthetic examples, as well as on Learning to Explain, Discrete Variational Auto-Encoders, and Neural Relational Inference tasks. In our experiments, we show that our adaptive gradient estimator can produce faithful estimates while requiring orders of magnitude fewer samples than other gradient estimators.

Predicting the remaining useful life (RUL) of ball bearings is an active area of research, where novel machine learning techniques are continuously being applied to predict degradation trends and anticipate failures before they occur. However, few studies have explicitly addressed the challenge of handling censored data, where information about a specific event (\eg mechanical failure) is incomplete or only partially observed. To address this issue, we introduce a novel and flexible method for early fault detection using Kullback-Leibler (KL) divergence and RUL estimation using survival analysis that naturally supports censored data. We demonstrate our approach in the XJTU-SY dataset using a 5-fold cross-validation strategy across three different operating conditions. When predicting the time to failure for bearings under the highest load (C1, 12.0 kN and 2100 RPM) with 25% random censoring, our approach achieves a mean absolute error (MAE) of 14.7 minutes (95% CI = 13.6-15.8) using a linear CoxPH model, and an MAE of 12.6 minutes (95% CI = 11.8-13.4) using a nonlinear Random Survival Forests model, compared to an MAE of 18.5 minutes (95% CI = 17.4-19.6) using a linear LASSO model that does not support censoring. Moreover, our approach achieves a mean cumulative relative accuracy (CRA) of 0.7586 over 5 bearings under the highest load, which improves over several state-of-the-art baselines. Our work highlights the importance of considering censored data as part of the model design when building predictive models for early fault detection and RUL estimation.

Attention distillation, which trains one attention distribution to match another by minimizing their Kullback-Leibler (KL) divergence, is widely used in knowledge distillation, model compression, continual learning, and sparse-attention LLM training. However, existing approaches materialize both attention distributions before computing the KL reduction, incurring O(N_QN_K) memory and IO costs that become prohibitive at long context lengths. We present StreamKL, the first fused GPU primitive for attention KL divergence that eliminates this quadratic materialization. StreamKL derives a novel online formulation for the coupled two-distribution KL reduction, enabling a single one-pass forward kernel that streams query-key tiles through on-chip SRAM. For the backward pass, StreamKL recomputes attention probabilities tile-by-tile, avoiding storage of quadratic intermediates. We further design and implement efficient GPU kernels with dedicated optimizations. Experiments show StreamKL delivers up to 43times and 14times speedups over baseline methods in the forward and backward passes, respectively. Most importantly, StreamKL reduces the extra HBM footprint of attention distillation from O(N_QN_K) to O(1), enabling long-context distillation on a single GPU.

Optimizers that exploit the matrix structure of gradients are central to modern LLM pre-training, with two distinct frontiers: explicit Kronecker-factored preconditioning -- most recently KL-Shampoo, which estimates the preconditioner via KL divergence minimization -- and orthogonalization of the gradient momentum, exemplified by Muon and analyzed as steepest descent under the spectral norm. The two routes are typically developed in isolation. We make a structural observation about KL-Shampoo's Kronecker preconditioners: their eigenvalue spectra exhibit a spike-and-flat shape -- a few dominant eigenvalues followed by an approximately uniform tail -- across layers and training stages, holding exactly under a rank-ρ signal-plus-noise gradient model. We exploit this structure by restricting one of KL-Shampoo's Kronecker factors to a parametric family aligned with the spike-and-flat shape: full spectral structure on a tracked r-dimensional subspace, single shared eigenvalue across the remaining n-r directions. On these directions, we apply orthogonalization. An identity shows that this orthogonalization recovers the algebraic form of full KL-Shampoo's preconditioner. On four pre-training scales (GPT-2 124M / 350M, LLaMA 134M / 450M), Pro-KLShampoo consistently outperforms KL-Shampoo at every subspace rank we test in validation loss, peak per-GPU memory, and wallclock time to reach each loss level.

On-Policy Distillation (OPD) trains a student on its own generations using teacher feedback. With different tokenizers, comparing teacher and student predictions requires alignment at both sequence and vocabulary levels. In this paper, we examine whether expanding this alignment coverage improves learning. Across three heterogeneous teacher--student pairs on mathematical reasoning and code generation, strict 1:1 groups already cover most student-generated tokens despite substantial vocabulary mismatch. On responses sampled from the students before distillation, the shared vocabulary retains nearly all teacher and student probability mass at strictly aligned positions on average. Restricting reverse KL to a student-selected top-16 subset of the shared vocabulary at each strict position achieves accuracy comparable to full shared-vocabulary OPD, outperforming the evaluated cross-tokenizer baselines. Adding mean squared error supervision on span log-probabilities in mismatch groups gives complete supervision coverage, yet reduces accuracy. At checkpoints from training with only the strict loss, the span gradients show weak or negative directional agreement with the strict gradients and grow in magnitude relative to them. These diagnostics may help explain the accuracy drop from adding span supervision. Our findings motivate a shift from maximizing alignment coverage to prioritizing supervision reliability: compact supervision at strict positions can be more effective than broader coverage that introduces weakly aligned or conflicting training signals.

Diffusion large language models (dLLMs) accelerate language generation by predicting multiple masks in a single forward pass. However, existing dLLMs can suffer from unreliable predictions in early denoising stages under aggressive parallelism strategies, leading to errors that can propagate to later stages. To tackle this issue, we present Consistency Forcing (CForce) for dLLMs, a distillation method to force the mask predictions of early stages to align with those of later stages. CForce trains the model on pre-collected self-rollout trajectories, thereby improving training-inference alignment. We introduce Confidence Adaptive KL Divergence as a distillation objective to conjoin the merits of forward and reverse KL. We further provide a theoretical analysis for the consistency objective to explain why CForce can approximately minimize the prediction error of early stages. Critically, the same formulation applies to both mask-to-token decoding and edit-capable decoding; in the edit-capable case, later token-to-token refinements provide additional supervision for earlier masked-state predictions. Experiments on non-edit and edit-capable LLaDA models show improved speed-quality trade-offs, especially under high-parallelism decoding budgets. Code is available at: https://github.com/inclusionAI/dFactory.

Large Language Models (LLMs) are typically trained to predict in the forward direction of time. However, recent works have shown that prompting these models to look back and critique their own generations can produce useful feedback. Motivated by this, we explore the question of whether LLMs can be empowered to think (predict and score) backwards to provide unsupervised feedback that complements forward LLMs. Towards this, we introduce Time Reversed Language Models (TRLMs), which can score and generate queries when conditioned on responses, effectively functioning in the reverse direction of time. Further, to effectively infer in the response to query direction, we pre-train and fine-tune a language model (TRLM-Ba) in the reverse token order from scratch. We show empirically (and theoretically in a stylized setting) that time-reversed models can indeed complement forward model predictions when used to score the query given response for re-ranking multiple forward generations. We obtain up to 5\% improvement on the widely used AlpacaEval Leaderboard over the competent baseline of best-of-N re-ranking using self log-perplexity scores. We further show that TRLM scoring outperforms conventional forward scoring of response given query, resulting in significant gains in applications such as citation generation and passage retrieval. We next leverage the generative ability of TRLM to augment or provide unsupervised feedback to input safety filters of LLMs, demonstrating a drastic reduction in false negative rate with negligible impact on false positive rates against several attacks published on the popular JailbreakBench leaderboard.

We propose a goodness-of-fit measure for probability densities modeling observations with varying dimensionality, such as text documents of differing lengths or variable-length sequences. The proposed measure is an instance of the kernel Stein discrepancy (KSD), which has been used to construct goodness-of-fit tests for unnormalized densities. The KSD is defined by its Stein operator: current operators used in testing apply to fixed-dimensional spaces. As our main contribution, we extend the KSD to the variable-dimension setting by identifying appropriate Stein operators, and propose a novel KSD goodness-of-fit test. As with the previous variants, the proposed KSD does not require the density to be normalized, allowing the evaluation of a large class of models. Our test is shown to perform well in practice on discrete sequential data benchmarks.

We propose a direct estimation method for R\'{e}nyi and f-divergence measures based on a new graph theoretical interpretation. Suppose that we are given two sample sets X and Y, respectively with N and M samples, where eta:=M/N is a constant value. Considering the k-nearest neighbor (k-NN) graph of Y in the joint data set (X,Y), we show that the average powered ratio of the number of X points to the number of Y points among all k-NN points is proportional to R\'{e}nyi divergence of X and Y densities. A similar method can also be used to estimate f-divergence measures. We derive bias and variance rates, and show that for the class of gamma-H\"{o}lder smooth functions, the estimator achieves the MSE rate of O(N^{-2gamma/(gamma+d)}). Furthermore, by using a weighted ensemble estimation technique, for density functions with continuous and bounded derivatives of up to the order d, and some extra conditions at the support set boundary, we derive an ensemble estimator that achieves the parametric MSE rate of O(1/N). Our estimators are more computationally tractable than other competing estimators, which makes them appealing in many practical applications.

Reverse-mode differentiation is used for optimization, but it introduces references, which break the purity of the underlying programs, making them notoriously harder to optimize. We present a reverse-mode differentiation on a purely functional language with array operations. It is the first one to deliver a provably efficient, purely functional, and denotationally correct reverse-mode differentiation. We show that our transformation is semantically correct and verifies the cheap gradient principle. Inspired by PROPs and compilation to categories, we introduce a novel intermediate representation that we call 'unary form'. Our reverse-mode transformation is factored as a compilation scheme through this intermediate representation. We obtain provably efficient gradients by performing general partial evaluation optimizations after our reverse-mode transformation, as opposed to manually derived ones. For simple first-order programs, the obtained output programs resemble static-single-assignment (SSA) code. We emphasize the modularity of our approach and show how our language can easily be enriched with more optimized primitives, as required for some speed-ups in practice.

Efficiently and flexibly estimating treatment effect heterogeneity is an important task in a wide variety of settings ranging from medicine to marketing, and there are a considerable number of promising conditional average treatment effect estimators currently available. These, however, typically rely on the assumption that the measured covariates are enough to justify conditional exchangeability. We propose the P-learner, motivated by the R- and DR-learner, a tailored two-stage loss function for learning heterogeneous treatment effects in settings where exchangeability given observed covariates is an implausible assumption, and we wish to rely on proxy variables for causal inference. Our proposed estimator can be implemented by off-the-shelf loss-minimizing machine learning methods, which in the case of kernel regression satisfies an oracle bound on the estimated error as long as the nuisance components are estimated reasonably well.

We study a practical question: can a small correction module fix errors in a frozen language model's outputs without degrading its base capabilities? We propose CRN v2, a lightweight logit-level correction module (~34M trainable parameters, 0.73% of the 4.65B text module) that sits atop a fully frozen Gemma 4 E2B model. The base model is never updated; only the correction module learns, via supervised fine-tuning followed by reference-free DPO on 83,400 error-correction pairs. On a 60-question domain exam (CEHRI: Certified Human-Robot Intelligence, covering facts, arithmetic, and implicit-goal reasoning), CRN v2 corrects 53.3% of base-model errors (reworded variant: 43.3%) while showing no degradation on tested capability benchmarks (MMLU/BoolQ N=200; car-wash N=8). A LoRA baseline at the matched CRN v1 budget (6.6M params, rank 19) achieves 83.3% correction but suffers 30-75% capability loss on the same benchmarks -- the correction-capability tradeoff. An ablation shows that the KL preservation term (lambda=0.1) is critical: lowering it to 0.01 degrades correction to 35.0%. A hidden-state injection variant at earlier layers (1.6M params, SFT-only) reaches 50.0%/55.8% but does not exceed logit correction; shallower injection (layer 4) drops to 30.0%/28.3%; multi-depth logit correction (~35M) reaches only 40%; and longer training (5,000 SFT + 2,000 DPO) stays at 53.3% -- none of the alternative configurations we tested exceeded the rank-128 logit result, consistent with a best-achieved result of ~53% rather than a floor. This is a study of a design principle (frozen base + logit correction + KL anchoring), not a claim of architectural novelty. All code, main-result weights, and evaluation scripts are released (deep variant as code only -- no trained deep checkpoints).

Off-Policy Evaluation (OPE) aims to assess the effectiveness of counterfactual policies using only offline logged data and is often used to identify the top-k promising policies for deployment in online A/B tests. Existing evaluation metrics for OPE estimators primarily focus on the "accuracy" of OPE or that of downstream policy selection, neglecting risk-return tradeoff in the subsequent online policy deployment. To address this issue, we draw inspiration from portfolio evaluation in finance and develop a new metric, called SharpeRatio@k, which measures the risk-return tradeoff of policy portfolios formed by an OPE estimator under varying online evaluation budgets (k). We validate our metric in two example scenarios, demonstrating its ability to effectively distinguish between low-risk and high-risk estimators and to accurately identify the most efficient one. Efficiency of an estimator is characterized by its capability to form the most advantageous policy portfolios, maximizing returns while minimizing risks during online deployment, a nuance that existing metrics typically overlook. To facilitate a quick, accurate, and consistent evaluation of OPE via SharpeRatio@k, we have also integrated this metric into an open-source software, SCOPE-RL (https://github.com/hakuhodo-technologies/scope-rl). Employing SharpeRatio@k and SCOPE-RL, we conduct comprehensive benchmarking experiments on various estimators and RL tasks, focusing on their risk-return tradeoff. These experiments offer several interesting directions and suggestions for future OPE research.

Combining discrete probability distributions and combinatorial optimization problems with neural network components has numerous applications but poses several challenges. We propose Implicit Maximum Likelihood Estimation (I-MLE), a framework for end-to-end learning of models combining discrete exponential family distributions and differentiable neural components. I-MLE is widely applicable as it only requires the ability to compute the most probable states and does not rely on smooth relaxations. The framework encompasses several approaches such as perturbation-based implicit differentiation and recent methods to differentiate through black-box combinatorial solvers. We introduce a novel class of noise distributions for approximating marginals via perturb-and-MAP. Moreover, we show that I-MLE simplifies to maximum likelihood estimation when used in some recently studied learning settings that involve combinatorial solvers. Experiments on several datasets suggest that I-MLE is competitive with and often outperforms existing approaches which rely on problem-specific relaxations.

Reinforcement learning (RL) post-training has recently driven major gains in long chain-of-thought reasoning large language models (LLMs), but the high inference cost of such models motivates distillation into smaller students. Most existing knowledge distillation (KD) methods are designed for supervised fine-tuning (SFT), relying on fixed teacher traces or teacher-student Kullback-Leibler (KL) divergence-based regularization. When combined with RL, these approaches often suffer from distribution mismatch and objective interference: teacher supervision may not align with the student's evolving rollout distribution, and the KL regularizer can compete with reward maximization and require careful loss balancing. To address these issues, we propose RL-aware distillation (RLAD), which performs selective imitation during RL -- guiding the student toward the teacher only when it improves the current policy update. Our core component, Trust Region Ratio Distillation (TRRD), replaces the teacher-student KL regularizer with a PPO/GRPO-style likelihood-ratio objective anchored to a teacher--old-policy mixture, yielding advantage-aware, trust-region-bounded distillation on student rollouts and naturally balancing exploration, exploitation, and imitation. Across diverse logic reasoning and math benchmarks, RLAD consistently outperforms offline distillation, standard GRPO, and KL-based on-policy teacher-student knowledge distillation.

Autoregressive language models are trained by minimizing the cross-entropy of the model distribution Q relative to the data distribution P -- that is, minimizing the forward cross-entropy, which is equivalent to maximum likelihood estimation (MLE). We have observed that models trained in this way may "over-generalize", in the sense that they produce non-human-like text. Moreover, we believe that reverse cross-entropy, i.e., the cross-entropy of P relative to Q, is a better reflection of how a human would evaluate text generated by a model. Hence, we propose learning with MixCE, an objective that mixes the forward and reverse cross-entropies. We evaluate models trained with this objective on synthetic data settings (where P is known) and real data, and show that the resulting models yield better generated text without complex decoding strategies. Our code and models are publicly available at https://github.com/bloomberg/mixce-acl2023

In the misspecified kernel ridge regression problem, researchers usually assume the underground true function f_{rho}^{*} in [H]^{s}, a less-smooth interpolation space of a reproducing kernel Hilbert space (RKHS) H for some sin (0,1). The existing minimax optimal results require |f_{rho}^{*}|_{L^{infty}}<infty which implicitly requires s > alpha_{0} where alpha_{0}in (0,1) is the embedding index, a constant depending on H. Whether the KRR is optimal for all sin (0,1) is an outstanding problem lasting for years. In this paper, we show that KRR is minimax optimal for any sin (0,1) when the H is a Sobolev RKHS.