Temporal-Difference Calibration of Black-Box Simulators

We develop a derivative-free temporal-difference (TD) framework for calibrating black-box simulators using one-step transitions from a target system and noisy simulator-generated continuation values. The calibration parameter is defined through Bellman orthogonality, recovering the physical parameter under correct simulator specification and yielding a near-optimal $L_2$ approximation of the policy value under misspecification. We propose three TD algorithms based on random-direction finite differences, distinguished by their query costs and reuse of noisy simulator evaluations. We establish almost-sure convergence, last-iterate mean-square error bounds, and central limit theorems for Polyak-Ruppert averages across four query-noise regimes. Our analysis reveals a specification-dependent bias phenomenon: for two algorithms, the finite-difference bias of the TD estimating equation vanishes at the calibration target under correct specification, even when the gradient surrogate remains biased. This permits a fixed perturbation radius and rate exponents approaching $1/2$ in the simulator rollout budget, whereas misspecification generally recovers the classical Kiefer-Wolfowitz rates. We also develop a few-replicate online bootstrap that yields Student-$t$ calibrated confidence intervals under negligible query noise, and identify a covariance mismatch that can invalidate conventional multiplier inference along Markov trajectories. Finally, a second-stage TD recursion corrects the remaining value discrepancy without additional simulator queries.