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. Experiments on 12 reasoning benchmarks across varying model
sizes from 3B to 32B consistently demonstrate the generalizability and
robustness of SvS.