From $P(y|x)$ to $P(y)$: Investigating Reinforcement Learning in Pre-train Space

TL;DR

This paper introduces PreRL and DSRL, novel reinforcement learning methods that optimize reasoning in pre-train space, significantly improving reasoning capabilities and exploration in large language models.

Contribution

It proposes a new RL approach in pre-train space with negative sample reinforcement and a dual space strategy, enhancing reasoning and exploration in language models.

Findings

NSR-PreRL increases reasoning transition and reflection by over 14 and 6 times.

DSRL outperforms strong baselines in reasoning tasks.

Pre-train space pruning effectively guides policy toward correct reasoning subspace.

Abstract

While reinforcement learning with verifiable rewards (RLVR) significantly enhances LLM reasoning by optimizing the conditional distribution P(y|x), its potential is fundamentally bounded by the base model's existing output distribution. Optimizing the marginal distribution P(y) in the Pre-train Space addresses this bottleneck by encoding reasoning ability and preserving broad exploration capacity. Yet, conventional pre-training relies on static corpora for passive learning, leading to a distribution shift that hinders targeted reasoning enhancement. In this paper, we introduce PreRL (Pre-train Space RL), which applies reward-driven online updates directly to P(y). We theoretically and empirically validate the strong gradient alignment between log P(y) and log P(y|x), establishing PreRL as a viable surrogate for standard RL. Furthermore, we uncover a critical mechanism: Negative Sample Reinforcement (NSR) within PreRL serves as an exceptionally effective driver for reasoning. NSR-PreRL rapidly prunes incorrect reasoning spaces while stimulating endogenous reflective behaviors, increasing transition and reflection thoughts by 14.89x and 6.54x, respectively. Leveraging these insights, we propose Dual Space RL (DSRL), a Policy Reincarnation strategy that initializes models with NSR-PreRL to expand the reasoning horizon before transitioning to standard RL for fine-grained optimization. Extensive experiments demonstrate that DSRL consistently outperforms strong baselines, proving that pre-train space pruning effectively steers the policy toward a refined correct reasoning subspace.