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

Group Relative Policy Optimization (GRPO) was introduced 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 ($\mathsf{KL}$) 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 $\pi_0$. 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 $p_0$ and the regularization parameter $\beta$ of the $\mathsf{KL}$ regularizer. We show that the fixed point $p^*$ is guaranteed to be larger than $p_0$, thereby demonstrating that GRPO effectively amplifies the probability of success of the policy.