The Cost of Mismatch: Noise Amplification in Zeroth-Order Reinforcement Learning

Zeroth-order (ZO) optimization has emerged as a promising approach for memory-efficient large language model (LLM) post-training, yet its reliable application to reinforcement learning with verifiable rewards (RLVR) remains largely unexplored. Motivated by the distinct data regimes of on-policy RL and fixed-data optimization, we develop a generic variance-theoretic framework to investigate what limits the reliability of ZO gradient estimation. Our analysis reveals that estimation reliability is structurally determined by whether the two finite-difference evaluations are performed on matched data. On-policy reinforcement learning (RL) under the mismatched-data regime produces a 1/μ^2 oracle-noise amplification that sets an irreducible noise floor as the perturbation scale $μ$ decreases, whereas Boltzmann Targeted SFT (BOLT) under the matched-data regime eliminates this amplification via noise cancellation. We further derive convergence guarantees with explicit constants that cleanly separate the two regimes. These results naturally lead to ZO-BOLT, which applies ZO optimization to BOLT's RLVR-aligned weighted-SFT objective. Controlled perturbation-scale sweeps and matched-versus-mismatched end-to-end comparisons validate the predicted effect of data matching. Experiments across three model variants and two mathematical reasoning benchmarks further show that ZO-BOLT consistently improves over the pretrained baseline while retaining a near-inference memory footprint. The code is publicly available at https://github.com/lianmin-chen/the_cost_of_mismatch.