Power Distribution Bridges Sampling, Self-Reward RL, and Self-Distillation
Authors: Akiyoshi Tomihari, Issei Sato
Organizations: Department of Computer Science, The University of Tokyo
Abstract
Recent analyses question whether reinforcement learning (RL) is responsible for strong reasoning in large language models (LLMs). At the same time, distillation and inference-time sampling, including power sampling, have emerged as effective ways to improve LLM performance. However, the relationship among RL, distillation, and sampling remains unclear. In this study, we focus on the power distribution, the target distribution of power sampling, and show that the power distribution bridges sampling, self-reward KL-regularized RL, and self-distillation. From the sampling perspective, we show that inexpensive local approximations cannot reproduce sequence-level power without information about possible suffixes. From the RL perspective, the power distribution is the closed-form optimizer of KL-regularized RL when the model's sequence-level log-probabilities are used as the reward. This identification leads to power self-distillation, an offline distillation surrogate that shares the same target distribution and amortizes the cost of power sampling into supervised training on teacher samples. We further show that power self-distillation can achieve self-reward sharpening, while improvement in a downstream true reward is governed by the covariance between true reward and self-reward under the power distribution. Experiments on reasoning tasks support our analysis: power sampling raises self-reward, true-reward gains depend on alignment with self-reward, and power self-distillation can match or exceed the performance of power sampling at much lower inference cost.
Large language model (LLM) post-training is essential for improving reasoning, adaptation, and alignment. Existing methods mainly follow two paradigms: reinforcement learning (RL) and on-policy distillation (OPD). However, RL relies on coarse-grained outcome supervision, resulting in difficult credit assignment and limited capability to acquire new knowledge. OPD, meanwhile, unconditionally matches teacher logits through KL divergence, which creates a dilemma: similar teachers provide little new knowledge, while substantially different teachers often yield ineffective guidance, largely restricting OPD to within-family distillation. We propose Distilled Reinforcement Learning (Distilled RL), which integrates teacher supervision into the RL objective to provide fine-grained guidance, selectively transfer new knowledge and avoid unconditional imitation. Distilled RL contains three components: reverse importance sampling with clipping, negative sample reset, and sequence-level geometric normalization. Through a concise and interpretable case study, we demonstrate that Distilled RL can effectively transfer previously unavailable knowledge from a teacher model to a student model. Extensive experiments across both within-family and cross-family distillation settings show that Distilled RL substantially outperforms standard RL and OPD in terms of both pass@1 and pass@k. Our code is available at https://github.com/597358816/Distilled-RL.
Reinforcement learning from verifiable rewards assigns a single scalar to each rollout, leaving token-level credit assignment underspecified in long reasoning traces. On-policy self-distillation addresses this by letting the same model act as a teacher conditioned on privileged information, producing a dense per-token signal. But the common choice of a ground-truth answer is only an endpoint cue: on terse-answer tasks, the teacher falls silent at the intermediate positions where path-level guidance matters most. We propose Hindsight Self-Distillation (HSD), which conditions the teacher on a successful peer rollout drawn from the current training group. Such a peer is an exact sample from the success-conditioned policy, requiring no additional sampled rollouts. By providing a full successful continuation rather than only the final answer, the resulting credit signal concentrates at the divergence position between a failed rollout and a successful peer. Across Qwen3-8B and Qwen3-32B on math and code benchmarks, HSD obtains the best result against GRPO variants and on-policy distillation baselines, with the largest gains on terse-answer tasks such as AIME.
On-Policy Self-Distillation (OPSD) has emerged as a popular paradigm for large language model (LLM) self-improvement, allowing models to act as their own teachers by leveraging privileged information such as ground-truth solutions. However, recent findings indicate that OPSD can severely degrade the performance of LLMs on complex reasoning tasks: By forcing the student to imitate an artificially confident reasoning trace conditioned on privileged information, OPSD inadvertently suppresses expressions of uncertainty and penalizes the exploratory, self-corrective behaviors required to solve challenging problems. To address this, we introduce Negative Self-Distillation (NSD), a new framework that optimizes LLMs by diverging from flawed reasoning rather than imitating privileged solutions. Instead of relying on ground-truth answers or external supervision, NSD uses the model itself to generate a question-specific negative condition (eg, acting as a ``careless reasoner'') and pushes the student's distribution away from this self-generated negative teacher. Naively applying unlearning objectives to achieve this divergence is problematic, as flawed reasoning tokens are confounded with basic linguistic tokens; indiscriminately penalizing both risks catastrophically degrading the model's foundational language capabilities. We resolve this by designing a dynamic gating mechanism that automatically identifies and isolates reasoning-critical tokens, ensuring gradient updates target only behavioral flaws while preserving the model's linguistic priors. Empirically, NSD consistently outperforms OPSD and other label-free, self-bootstrapping reinforcement learning (RL) baselines.