Reinforcement Learning (RL) algorithms sample multiple n>1 solution attempts for each problem and reward them independently. This optimizes for pass@1 performance and prioritizes the strength of isolated samples at the expense of the diversity and collective utility of sets of samples. This under-utilizes the sampling capacity, limiting exploration and eventual improvement on harder examples. As a fix, we propose Pass-at-k Policy Optimization (PKPO), a transformation on the final rewards which leads to direct optimization of pass@k performance, thus optimizing for sets of samples that maximize reward when considered jointly. Our contribution is to derive novel low variance unbiased estimators for pass@k and its gradient, in both the binary and continuous reward settings. We show optimization with our estimators reduces to standard RL with rewards that have been jointly transformed by a stable and efficient transformation function. While previous efforts are restricted to k=n, ours is the first to enable robust optimization of pass@k for any arbitrary k <= n. Moreover, instead of trading off pass@1 performance for pass@k gains, our method allows annealing k during training, optimizing both metrics and often achieving strong pass@1 numbers alongside significant pass@k gains. We validate our reward transformations on toy experiments, which reveal the variance reducing properties of our formulations. We also include real-world examples using the open-source LLM, GEMMA-2. We find that our transformation effectively optimizes for the target k. Furthermore, higher k values enable solving more and harder problems, while annealing k boosts both the pass@1 and pass@k . Crucially, for challenging task sets where conventional pass@1 optimization stalls, our pass@k approach unblocks learning, likely due to better exploration by prioritizing joint utility over the utility of individual samples.

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