Optimizing Anytime Reasoning via Budget Relative Policy Optimization
Authors: name: Penghui Qi, name: Zichen Liu, name: Tianyu Pang, name: Chao Du, name: Wee Sun Lee, name: Min Lin

Scaling test-time compute is crucial for enhancing the reasoning capabilities
of large language models (LLMs). Existing approaches typically employ
reinforcement learning (RL) to maximize a verifiable reward obtained at the end
of reasoning traces. However, such methods optimize only the final performance
under a large and fixed token budget, which hinders efficiency in both training
and deployment. In this work, we present a novel framework, AnytimeReasoner, to
optimize anytime reasoning performance, which aims to improve token efficiency
and the flexibility of reasoning under varying token budget constraints. To
achieve this, we truncate the complete thinking process to fit within sampled
token budgets from a prior distribution, compelling the model to summarize the
optimal answer for each truncated thinking for verification. This introduces
verifiable dense rewards into the reasoning process, facilitating more
effective credit assignment in RL optimization. We then optimize the thinking
and summary policies in a decoupled manner to maximize the cumulative reward.
Additionally, we introduce a novel variance reduction technique, Budget
Relative Policy Optimization (BRPO), to enhance the robustness and efficiency
of the learning process when reinforcing the thinking policy. Empirical results
in mathematical reasoning tasks demonstrate that our method consistently
outperforms GRPO across all thinking budgets under various prior distributions,
enhancing both training and token efficiency.