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Abstract:
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We study the off-policy evaluation (OPE) problem in an infinite-horizon Markov decision process with continuous states and actions. We recast the Q-function estimation into a special form of the nonparametric instrumental variables (NPIV) estimation problem. We first show that under one mild condition the NPIV formulation of Q-function estimation is well-posed in the sense of L2-measure of ill-posedness with respect to the data generating distribution, bypassing a strong assumption on the discount factor ? imposed in the recent literature for obtaining the L2 convergence rates of various Q-function estimators. Thanks to this new well-posed property, we derive the first minimax lower bounds for the convergence rates of nonparametric estimation of Q-function and its derivatives in both sup-norm and L2-norm, which are shown to be the same as those for the classical nonparametric regression (Stone, 1982). We then propose a sieve two-stage least squares estimator and establish its rate-optimality in both norms under some mild conditions.
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