Title: Knapsack RL: Unlocking Exploration of LLMs via Optimizing Budget Allocation
Primary area: foundation or frontier models, including LLMs
Keywords: Reinforcement Learning, Exploration, Large Language Models, Policy Optimization

Abstract: Large Language Models (LLMs) can self-improve through reinforcement learning, where they generate trajectories to explore and discover better solutions. However, this exploration process is computationally expensive, often forcing current methods to assign limited exploration budgets to each task. This uniform allocation creates problematic edge cases: easy tasks consistently succeed while difficult tasks consistently fail, both producing zero gradients during training updates for the widely used Group Relative Policy Optimization (GRPO). We address this problem from the lens of exploration budget allocation. Viewing each task's exploration as an "item" with a distinct "value" and "cost", we establish a connection to the classical knapsack problem. From this, we derive an optimal assignment rule that transfers exploration budgets from easy tasks to challenging ones. When applied to GRPO, our method increases the effective ratio of non-zero policy gradients by 20–40% during training. As a computational "free lunch", it also enables substantially larger exploration budgets (e.g., 93 rollouts) for especially challenging tasks—budgets that would be computationally prohibitive under uniform allocation. These improvements translate to meaningful gains on mathematical reasoning benchmarks, with average improvements of 2–4 points and peak gains of 9 points on specific tasks. Notably, achieving comparable performance with traditional homogeneous allocation would require about 2x the computational resources.