Theory paper proves pass@1 gains in RLVR incur a structural cost in pass@k performance

·2026.11.02 13:30

Key point

A new theory paper demonstrates that under strategy specialization, every pass@1 improvement in RLVR strictly lowers success on some prompts, with a tight bound on the pass@k crossover.

Details

Proves a "no free lunch" theorem for RLVR: under exclusive specialization, any update raising pass@1 strictly lowers success on at least one prompt.
Establishes a tight bound for the pass@k crossover using Karlin total positivity, showing the number of crossings is at most the sign variation of the shift in per-prompt success.
Identifies the optimal pass@1 vs pass@k trade-off frontier, noting that forbidding per-prompt loss removes both the crossover and the pass@1 gain.
Provides a practical diagnostic using existing eval logs to determine crossing counts and worst-case pass@k loss, validated on Omni-MATH data showing one crossing between k=64 and k=256.

reinforcement-learning theory llm-evaluation arxiv rlvr pass-at-k

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