Assumption 2. The following conditions are assumed throughout: A; (32) (iv) r has polynomial growth in x and a, i.e., there exists a constant C > 0 and µ 1 such that To do so, let's assume Theorem 6. Assume that for a policy π and for every x, Assumption 3. Assume the following conditions hold: Lemma 9. Let π, ˆ π be two feedback policies. We need a lemma for the perturbation bounds. Here we present a detailed version of the CPPO algorithm. D.3 below, which clearly illustrates the advantage of square-root KL divergence.

Through numerical experiments, we demonstrate the effectiveness and advantages of our approach.

This paper introduces Completion Pruning Policy Optimization (CPPO) to accelerate the training of reasoning models based on Group Relative Policy Optimization (GRPO). GRPO, while effective, incurs high training costs due to the need for sampling multiple completions for each question. Our experiment and theoretical analysis reveals that the number of completions impacts model accuracy yet increases training time multiplicatively, and not all completions contribute equally to policy training -- their contribution depends on their relative advantage. To address these issues, we propose CPPO, which prunes completions with low absolute advantages, significantly reducing the number needed for gradient calculation and updates. Additionally, we introduce a dynamic completion allocation strategy to maximize GPU utilization by incorporating additional questions, further enhancing training efficiency. Experimental results demonstrate that CPPO achieves up to $8.32\times$ speedup on GSM8K and $3.51\times$ on Math while preserving or even enhancing the accuracy compared to the original GRPO. We release our code at https://github.com/lzhxmu/CPPO.

Though deep reinforcement learning (DRL) has obtained substantial success, it may encounter catastrophic failures due to the intrinsic uncertainty of both transition and observation. Most of the existing methods for safe reinforcement learning can only handle transition disturbance or observation disturbance since these two kinds of disturbance affect different parts of the agent; besides, the popular worst-case return may lead to overly pessimistic policies. To address these issues, we first theoretically prove that the performance degradation under transition disturbance and observation disturbance depends on a novel metric of Value Function Range (VFR), which corresponds to the gap in the value function between the best state and the worst state. Based on the analysis, we adopt conditional value-at-risk (CVaR) as an assessment of risk and propose a novel reinforcement learning algorithm of CVaR-Proximal-Policy-Optimization (CPPO) which formalizes the risk-sensitive constrained optimization problem by keeping its CVaR under a given threshold. Experimental results show that CPPO achieves a higher cumulative reward and is more robust against both observation and transition disturbances on a series of continuous control tasks in MuJoCo.