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Life sciences · Preprint
arXiv · August 11, 2026
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This preprint derives asymptotically valid confidence regions for state-value contrasts in fixed-stepsize temporal-difference learning via a self-normalized Brownian-bridge approach that avoids bandwidth or long-run covariance estimation. The method is theoretically grounded but remains in the domain of proof-of-concept simulation; applicability to real reinforcement learning or clinical decision-support systems is not established.
Theoretical analysis with simulation experiments. Intervention: Constant-stepsize temporal-difference learning with parallel Richardson–Romberg recursions and Brownian-bridge self-normalisation for confidence region construction.
Functional central limit theorem established for constant-stepsize linear TD, retaining multiplicative component from random TD matrix and stationary iterate error Joint functional limit derived for parallel Richardson–Romberg recursions from single Markov trajectory Brownian-bridge self-normalizer yields asymptotically pivotal confidence regions without bandwidth selection or long-run covariance estimation
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A theoretical and computational contribution establishing asymptotic properties of temporal-difference learning inference, with proof-of-concept experiments but no empirical validation in real clinical or operational decision-making settings.
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Constant-stepsize temporal-difference (TD) learning is attractive for policy evaluation, but inference from a single Markov trajectory must account for serial dependence and a stepsize-dependent stationary target. For fixed-stepsize linear TD, we establish a functional central limit theorem whose covariance retains the multiplicative component induced by the random TD matrix and the stationary iterate error. We then derive a joint functional limit for parallel Richardson--Romberg (RR) recursions driven by the same trajectory. A Brownian-bridge self-normalizer yields asymptotically pivotal confidence regions for prespecified state-value contrasts without estimating the long-run covariance or selecting a bandwidth or batch length. For such a contrast, the procedure admits a one-pass implementation whose memory does not grow with the trajectory length. At a fixed stepsize, the inferential center is the RR stationary target. We also study horizon-indexed designs in which the stepsize remains constant within each run and decreases across longer horizons. Under an explicit RR-dependent rate window, the residual RR target shift, multiplicative remainder, and initialization effect are negligible at the root-$n$ scale, yielding inference for the projected Bellman solution. Experiments on FrozenLake and Garnet illustrate stationary-target coverage, RR target correction, and the finite-sample behavior of the horizon-indexed design.
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