Reinforcement Learning with Verifiable Rewards: GRPO's Effective Loss, Dynamics, and Success Amplification

Abstract: Group Relative Policy Optimization (GRPO) was introduced recently and used successfully to train DeepSeek-R1 models for promoting reasoning capabilities of LLMs using verifiable or binary rewards. We show in this paper that GRPO with verifiable rewards can be written as a Kullback--Leibler (KL) regularized contrastive loss, where the contrastive samples are synthetic data sampled from the old policy. The optimal GRPO policy $\pi_{n}$ can be expressed explicitly in terms of the binary reward, as well as the first- and second-order statistics of the old policy ($\pi_{n-1}$) and the reference policy $\pi_{\text{ref}}$. Iterating this scheme, we obtain a sequence of policies $\pi_{n}$ for which we can quantify the probability of success $p_n$. We show that the probability of success of the policy satisfies a recurrence that converges to a fixed point of a function that depends on the initial probability of success $p_{\text{ref}}$ and the regularization parameter $\beta$ of the $KL$ regularizer. We show that the fixed point $p^*$ is guaranteed to be larger than $p_{\text{ref}}$, thereby demonstrating that GRPO effectively amplifies the probability of success of the policy.