Semantic-Space Exploration and Exploitation in RLVR for LLM Reasoning

Reinforcement Learning with Verifiable Rewards (RLVR) for LLM reasoning is often framed as balancing exploration and exploitation in action space, typically operationalized with token-level proxies (e.g., output entropy or confidence). We argue that this apparent trade-off is largely a measurement artifact: token-level statistics reflect next-token uncertainty rather than how reasoning progresses over multi-token semantic structures. We therefore study exploration and exploitation in the hidden-state space of response trajectories. We use Effective Rank (ER) to quantify representational exploration and introduce its temporal derivatives, Effective Rank Velocity (ERV) and Effective Rank Acceleration (ERA), to characterize exploitative refinement dynamics. Empirically and theoretically, ER and ERV exhibit near-zero correlation in semantic space, suggesting the two capacities can be improved simultaneously. Motivated by this, we propose Velocity-Exploiting Rank Learning (VERL), which shapes the RL advantage with an auxiliary signal derived from ER/ERV and uses the more stable ERA as a meta-control variable to adaptively balance the incentives. Across multiple base models, RL algorithms, and reasoning benchmarks, VERL yields consistent improvements, including large gains on challenging tasks (e.g., 21.4\% in Gaokao 2024).