Title: Beyond Pass@k: Breadth-Depth Metrics for Reasoning Boundaries
Primary area: foundation or frontier models, including LLMs
Keywords: LLMs for reasoning, reinforcement learning with verifiable rewards, evaluation metrics, math datasets

Abstract: Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a powerful paradigm to improve Large Language Models on reasoning tasks such as coding, math or logic. To assess the reasoning boundary (the fraction of problems a model can solve) researchers often report pass@k at large sampling budgets. Recent results reveal a crossover phenomenon: while RLVR models outperform the base model at small k values, the base model usually outperforms them when sampling a very large number of completions. This has been interpreted as evidence that base models have a larger reasoning boundary. We argue that on tasks with discrete answer spaces, such as math with numeric outputs, pass@k at large k reflects the increasingly higher chance of success in the limit of the number of trials rather than genuine reasoning, and can therefore be misleading. We propose cover@tau, which measures the fraction of problems that a model can solve for which at least a tau proportion of completions are correct. Unlike pass@k, cover@tau captures reasoning under an explicit reliability threshold: models that rely on random guessing degrade rapidly as tau increases. We evaluate several RLVR models using cover@tau-based metrics and illustrate how the relative rankings of popular algorithms change compared to pass@1, offering a different perspective on reasoning boundaries.

TL;DR: We introduce a new metric, cover@tau, to measure the reasoning boundary of LLMs at explicit reliability thresholds