Pass@K Policy Optimization: Solving Harder Reinforcement Learning Problems
This addresses a bottleneck in reinforcement learning for tasks where diverse sample sets are crucial, such as in language models, by enabling robust optimization for any k, which is incremental but novel in its flexibility and annealing approach.
The paper tackles the problem of reinforcement learning algorithms under-utilizing sampling capacity by optimizing for pass@1 performance, which limits exploration on harder tasks. They propose Pass-at-k Policy Optimization (PKPO), which directly optimizes pass@k performance, enabling solving more and harder problems and achieving significant pass@k gains while often maintaining strong pass@1 numbers.
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.