# Discounted Beta-Bernoulli Reward Estimation for Sample-Efficient Reinforcement Learning with Verifiable Rewards

arXiv: https://arxiv.org/abs/2603.18444 · HTML: https://mlsanity.com/paper/2603.18444/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 14 of its 19 references that are in the map.
- First posted on arXiv 19 Mar 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 14 of 14)

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Challenging BIG-Bench Tasks and Whether Chain-of-Thought Can Solve Them](https://mlsanity.com/paper/2210.09261/) (2022)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Multi-Step Reasoning with Large Language Models, a Survey](https://mlsanity.com/paper/2407.11511/) (2024)
- [Act Only When It Pays: Efficient Reinforcement Learning for LLM Reasoning via Selective Rollouts](https://mlsanity.com/paper/2506.02177/) (2025)
- [RLPR: Extrapolating RLVR to General Domains without Verifiers](https://mlsanity.com/paper/2506.18254/) (2025)
- [No Prompt Left Behind: Exploiting Zero-Variance Prompts in LLM Reinforcement Learning via Entropy-Guided Advantage Shaping](https://mlsanity.com/paper/2509.21880/) (2025)
- [Unlocking Exploration in RLVR: Uncertainty-aware Advantage Shaping for Deeper Reasoning](https://mlsanity.com/paper/2510.10649/) (2025)
- [Improving Sampling Efficiency in RLVR through Adaptive Rollout and Response Reuse](https://mlsanity.com/paper/2509.25808/) (2025)

## Built on by (most cited first, 2 of 2)

- [Learning Beyond What You Sample: Off-Policy-Aware Cross-Model Trajectory Exchange for RLVR](https://mlsanity.com/paper/2609.37868/) (2026)
- [AECG: Asymmetric Experience Consolidation and Governance In Multi-Agent Systems](https://mlsanity.com/paper/2610.05176/) (2026)

## Abstract (arXiv)

Reinforcement learning with verifiable rewards (RLVR) has emerged as an effective post-training paradigm for improving the reasoning capabilities of large language models. However, existing group-based RLVR methods often suffer from severe sample inefficiency. This inefficiency stems from reliance on point estimation of rewards from a small number of rollouts, leading to high estimation variance, variance collapse, and ineffective utilization of generated responses. In this work, we reformulate RLVR from a statistical estimation perspective by modeling rewards as samples drawn from a policy-induced distribution and casting advantage computation as the problem of estimating the reward distribution from finite data. Building on this view, we propose Discounted Beta-Bernoulli (DBB) reward estimation, which leverages historical reward statistics for the non-stationary distribution. Although biased, the resulting estimator exhibits reduced and stable variance, theoretically avoids estimated variance collapse, and achieves lower mean squared error than standard point estimation. Extensive experiments across six in-distribution and three out-of-distribution reasoning benchmarks demonstrate that GRPO with DBB consistently outperforms naive GRPO, achieving average Acc@8 improvements of 3.22/2.42 points in-distribution and 12.49/6.92 points out-of-distribution on the 1.7B and 8B models, respectively, without additional computational cost or memory usage.