Sparse but Critical: A Token-Level Analysis of Distributional Shifts in RLVR Fine-Tuning of LLMs

TL;DR

This paper systematically analyzes how reinforcement learning with verifiable rewards (RLVR) causes sparse, targeted token-level distributional shifts in large language models, and how these shifts relate to improved reasoning performance.

Contribution

It provides a detailed empirical study of token-level distributional effects of RLVR on LLMs, revealing the sparsity and structure of these shifts and their functional importance.

Findings

RL fine-tuning induces sparse, targeted token distribution changes.

Small interventions with RL tokens can recover performance gains.

Divergence-weighted advantage signals can improve RLVR outcomes.

Abstract

Reinforcement learning with verifiable rewards (RLVR) has significantly improved reasoning in large language models (LLMs), yet the token-level mechanisms underlying these improvements remain unclear. We present a systematic empirical study of RLVR's distributional effects organized around three main analyses: (1) token-level characterization of distributional shifts between base and RL models, (2) the impact of token-level distributional shifts on sequence-level reasoning performance through cross-sampling interventions, and (3) fine-grained mechanics of these shifts at the token level. We find that RL fine-tuning induces highly sparse and targeted changes, with only a small fraction of token distributions exhibiting meaningful divergence between the base and RL policies. We further characterize the structure and evolution of these shifts through analyses of token entropy, positional concentration, and reallocation of probability mass. To assess the functional importance of these sparse changes, we conduct cross-sampling experiments that selectively swap token choices between the base and RL models with varying intervention budgets. We show that inserting only a small fraction of RL-sampled tokens into base generations progressively recovers RL performance gains, while injecting a similarly small number of base token choices into otherwise RL-generated sequences collapses performance to base levels, isolating a small set of token-level decisions directly responsible for RLVR's performance gains. Finally, we explore divergence-weighted variants of the advantage signal as a diagnostic intervention, finding that they can yield improvements over baselines. Together, our results shed light on the distributional changes induced by RLVR and provide a fine-grained, token-level lens for understanding RLVR fine-tuning as a targeted refinement process.