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This paper advances the theoretical understanding of Temporal-Difference learning by introducing a stochastic differential equation approximation that accounts for Markovian noise effects often neglected in classical ODE models. By explicitly modeling these fluctuations, the work provides a rigorous explanation for error floors observed during constant-stepsize training regimes. This contribution is significant for researchers developing more robust reinforcement learning algorithms with linear function approximators under realistic sampling conditions.
A Diffusion Approximation for Temporal-Difference Learning with Linear Features under Markovian Noise
arXiv:2606.18183v1 Announce Type: cross Abstract: Temporal difference (TD) learning with linear function approximation is a core method for policy evaluation. Its classical continuous-time description is an ordinary differential equation (ODE), which captures the asymptotic mean dynamics but neglects stochastic fluctuations determining the error floor. We introduce a stochastic differential equation (SDE) approximation for linear TD(0) under Markovian noise. The resulting model distinguishes the contraction dynamics governed by the projected Bellman operator from the influence of Markovian sampling. As a consequence, the model explains the constant-stepsize error floor through the interaction between Markovian long-run covariance and the contraction geometry of the projected Bellman operator.
Abstract:Temporal difference (TD) learning with linear function approximation is a core method for policy evaluation. Its classical continuous-time description is an ordinary differential equation (ODE), which captures the asymptotic mean dynamics but neglects stochastic fluctuations determining the error floor. We introduce a stochastic differential equation (SDE) approximation for linear TD(0) under Markovian noise. The resulting model distinguishes the contraction dynamics governed by the projected Bellman operator from the influence of Markovian sampling. As a consequence, the model explains the constant-stepsize error floor through the interaction between Markovian long-run covariance and the contraction geometry of the projected Bellman operator.
Machine Learning (stat.ML); Machine Learning (cs.LG); Probability (math.PR)
arXiv:2606.18183 [stat.ML]
(or arXiv:2606.18183v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2606.18183 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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