cs.LGMay 4, 2026

Reference-Sampled Boltzmann Projection for KL-Regularized RLVR: Target-Matched Weighted SFT, Finite One-Shot Gaps, and Policy Mirror Descent

Authors: Yao Shu, Chenxing Wei, Hongbin Lin, Shuang Qiu, Hui Xiong

Abstract

Online reinforcement learning with verifiable rewards (RLVR) turns checkable outcomes into a scalable training signal, but it keeps rollout generation, verifier scoring, and reference-policy evaluations on the optimization path. Static weighted supervised fine-tuning (SFT) on precomputed rollouts seems to remove this bottleneck, yet a weighted likelihood is not specified by rewards alone: its sampler and weights induce the policy being fit. This paper identifies the reference-sampled weighted-SFT objective whose induced policy equals the fixed-reference KL-regularized RLVR optimizer. The optimizer is the standard Boltzmann target policy, obtained by exponentially tilting the reference policy by verifier reward. Matching a weighted-SFT induced policy to this target forces density-ratio weights; in the reference-sampled subclass, this reduces uniquely, up to prompt scaling, to the prompt-normalized Boltzmann weight exp⁡(r(x,y)/β)/Z(x)\exp(r(x,y)/β)/Z(x). BOLT, a Boltzmann-Targeted SFT procedure, is the empirical estimator of this projection. The finite one-shot analysis separates the exact stored-support price βlog⁡(1/π∗(SN∣x))β\log(1/π^*(S_N\mid x)) from partition estimation, effective-sample-size variance, generalization, optimization, and approximation errors. This decomposition explains why extra SFT epochs cannot repair missing reference-policy coverage and exposes the temperature--coverage--variance frontier. When coverage needs adaptive sampling, refreshed Boltzmann projections become KL policy mirror descent; finite inner solves enter as additive drift from the exact mirror step. Single-run Qwen experiments provide projection evidence for the target-matched weight, one-shot saturation, refreshed-sampler gains, and optimization-time savings, within the stated single-run scope.

Explore similar work

CardsList
Jul 28, 2026cs.LG

Early Verdicts, Better Budgets: Sequential Adaptive Rollout Allocation for Compute-Efficient RLVR

Reinforcement learning with verifiable rewards (RLVR) is bottlenecked by rollout generation, yet many sampled prompts produce saturated groups (all responses correct or all incorrect) whose zero reward variance yields no policy-gradient signal. Existing remedies either oversample a larger candidate pool and discard saturated prompts (dynamic sampling), paying heavy extra rollouts, or predict prompt difficulty before sampling, which is fragile under a shifting policy. We observe that a group's effectiveness is usually decided early, within the first few of its rollouts, so spending a full group on an already-decided prompt is wasteful. We cast per-step rollout collection as a budget-constrained sequential allocation (optimal stopping) problem and introduce SARA (Sequential Adaptive Rollout Allocation). SARA maintains a Beta posterior over each prompt's success rate, evaluates a closed-form predictor of group effectiveness, and applies a two-threshold, SPRT-style rule that commits effective groups, abandons saturated ones after a short probe, and reallocates the freed budget to fresh prompts, without any extra prediction rollouts. We prove abandonment reliability, expected rollout savings, fixed-budget yield dominance, and a link between effective-group yield and the GRPO gradient norm. On mathematical reasoning and planning with 1.5B/3B models on a single GPU, SARA matches DPS (both below the DS oracle) while using 22% fewer rollouts than DS; composing SARA with DPS yields the best accuracy, slightly above DS, at 67% fewer rollouts (near-uniform cost).
Pixel Nomand, Elena Voss, Marcus Hale +1
Verifiable RewardsBudgetsPolicy GradientVerdict
AskPDF
May 19, 2026cs.LG

When to Stop Reusing: Dynamic Gradient Gating for Sample-Efficient RLVR

Reinforcement Learning with Verifiable Rewards (RLVR) has become the dominant paradigm for advanced reasoning in Large Language Models (LLMs), but rollout samples are expensive to obtain, making sample efficiency a critical bottleneck. A natural remedy is to reuse each rollout batch for multiple gradient updates, a standard practice in classical RL. Yet in RLVR, this amplifies policy shift, leading to severe performance degradation. Detecting the onset of degradation early enough to stop reuse remains an open and challenging problem. We close this gap by identifying the \textit{Disproportionate Weight Divergence (DWD)} phenomenon: performance degradation is synchronized with a sharp surge in the \texttt{lm_head} weight change, while intermediate layers remain stable. Empirically, we verify that DWD emerges consistently across diverse LLMs and tasks. Theoretically, we prove that (i) harmful gradients concentrate at the \texttt{lm_head} while intermediate layers are structurally attenuated, and (ii) the \texttt{lm_head} gradient norm lower-bounds the policy divergence. These results establish the \texttt{lm_head} gradient norm as a principled, real-time signal of catastrophic policy shift. Guided by this insight, we propose \textit{Dynamic Gradient Gating (DGG)}, a lightweight intervention that monitors the \texttt{lm_head} gradient norm in real time and intercepts harmful gradients before they corrupt the optimizer. DGG consistently matches or exceeds the standard single-use baseline, achieving up to 2.93×2.93\times sample efficiency and 2.14×2.14\times wall-clock speedup across math, ALFWorld, WebShop, and search-augmented QA tasks.
Yuchun Miao, Sen Zhang, Yuqi Zhang +4
Verifiable RewardsGradientReuse
AskPDF
May 14, 2026cs.LG

Boosting Reinforcement Learning with Verifiable Rewards via Randomly Selected Few-Shot Guidance

Reinforcement Learning with Verifiable Rewards (RLVR) has achieved great success in developing Large Language Models (LLMs) with chain-of-thought rollouts for many tasks such as math and coding. Nevertheless, RLVR struggles with sample efficiency on difficult problems where correct rollouts are hard to generate. Prior works propose to address this issue via demonstration-guided RLVR, i.e., to conduct Supervised FineTuning (SFT) when RL fails; however, SFT often requires a lot of data, which can be expensive to acquire. In this paper, we propose FEST, a FEw-ShoT demonstration-guided RLVR algorithm. It attains compelling results with only 128 demonstrations randomly selected from an SFT dataset. We find that three components are vital for the success: supervised signal, on-policy signal, and decaying weights on the few-shot SFT dataset to prevent overfitting from multiple-epoch training. On several benchmarks, FEST outperforms baselines with magnitudes less SFT data, even matching their performance with full dataset.
Kai Yan, Alexander G. Schwing, Yu-Xiong Wang
Verifiable RewardsReinforcement Fine-TuningSafe Reinforcement LearningSupervised Fine-TuningGuidance
AskPDF