Pass@k Training for Adaptively Balancing Exploration and Exploitation of Large Reasoning Models

TL;DR

This paper introduces Pass@k Training, a novel approach that uses Pass@k as a reward in reinforcement learning for large language models, improving exploration and balancing it with exploitation.

Contribution

It presents the first use of Pass@k as a reward in RLVR, derives an analytical advantage function, and demonstrates how exploration and exploitation can mutually enhance each other.

Findings

Pass@k Training improves exploration ability of large models.

Analytical advantage derivation leads to efficient training.

Exploration and exploitation can mutually reinforce each other.

Abstract

Reinforcement learning with verifiable rewards (RLVR), which typically adopts Pass@1 as the reward, has faced the issues in balancing exploration and exploitation, causing policies to prefer conservative actions, converging to a local optimum. Identifying an appropriate reward metric is therefore crucial. Regarding the prior work, although Pass@k has been used in evaluation, its connection to LLM exploration ability in RLVR remains largely overlooked. To investigate this, we first use Pass@k as the reward to train the policy model (i.e., $\textbf{Pass@k Training}$), and observe the improvement on its exploration ability. Next, we derive an analytical solution for the advantage of Pass@k Training, leading to an efficient and effective process. Building on this, our analysis reveals that exploration and exploitation are not inherently conflicting objectives, while they can mutually enhance each other. Moreover, Pass@k Training with analytical derivation essentially involves directly designing the advantage function. Inspired by this, we preliminarily explore the advantage design for RLVR, showing promising results and highlighting a potential future direction.