Beyond Pass@1: Self-Play with Variational Problem Synthesis Sustains RLVR

TL;DR

This paper introduces SvS, a self-play strategy for RLVR that synthesizes variational problems to sustain policy diversity and significantly improve reasoning performance on multiple benchmarks.

Contribution

The paper proposes an online self-play with variational problem synthesis (SvS) method that maintains policy entropy and enhances Pass@k performance in RLVR training.

Findings

Achieves 18.3% and 22.8% absolute gains in Pass@32 on AIME24 and AIME25 benchmarks.

Maintains policy entropy and diversity during training, preventing entropy collapse.

Demonstrates robustness across 12 reasoning benchmarks and various model sizes.

Abstract

Reinforcement Learning with Verifiable Rewards (RLVR) has recently emerged as a key paradigm for post-training Large Language Models (LLMs), particularly for complex reasoning tasks. However, vanilla RLVR training has been shown to improve Pass@1 performance at the expense of policy entropy, leading to reduced generation diversity and limiting the Pass@k performance, which typically represents the upper bound of LLM reasoning capability. In this paper, we systematically analyze the policy's generation diversity from the perspective of training problems and find that augmenting and updating training problems helps mitigate entropy collapse during training. Based on these observations, we propose an online Self-play with Variational problem Synthesis (SvS) strategy for RLVR training, which uses the policy's correct solutions to synthesize variational problems while ensuring their reference answers remain identical to the originals. This self-improving strategy effectively maintains policy entropy during training and substantially improves Pass@k compared with standard RLVR, sustaining prolonged improvements and achieving absolute gains of 18.3% and 22.8% in Pass@32 performance on the competition-level AIME24 and AIME25 benchmarks, as well as on code generation tasks. Experiments on 12 reasoning benchmarks across varying model sizes from 3B to 32B consistently demonstrate the generalizability and robustness of SvS.