Policy evaluation from a single path: Multi-step methods, mixing and mis-specification

We study non-parametric estimation of the value function of an infinite-horizon γ-discounted Markov reward process (MRP) using observations from a single trajectory. We provide non-asymptotic guarantees for a general family of kernel-based multi-step temporal difference (TD) estimates, including canonical K-step look-ahead TD for K = 1, 2, ldots and the TD(λ) family for λ ∈ [0,1) as special cases. Our bounds capture its dependence on Bellman fluctuations, mixing time of the Markov chain, any mis-specification in the model, as well as the choice of weight function defining the estimator itself, and reveal some delicate interactions between mixing time and model mis-specification. For a given TD method applied to a well-specified model, its statistical error under trajectory data is similar to that of i.i.d. sample transition pairs, whereas under mis-specification, temporal dependence in data inflates the statistical error. However, any such deterioration can be mitigated by increased look-ahead. We complement our upper bounds by proving minimax lower bounds that establish optimality of TD-based methods with appropriately chosen look-ahead and weighting, and reveal some fundamental differences between value function estimation and ordinary non-parametric regression.