On Well-posedness and Minimax Optimal Rates of Nonparametric Q-function Estimation in Off-policy Evaluation

TL;DR

This paper establishes the well-posedness and minimax optimal convergence rates for nonparametric Q-function estimation in off-policy evaluation, using a novel NPIV formulation that bypasses previous restrictions.

Contribution

It introduces a new well-posedness result for Q-function estimation via NPIV, derives minimax lower bounds, and proposes a rate-optimal sieve two-stage least squares estimator.

Findings

Well-posedness of Q-function estimation without strong discount factor assumptions.

Minimax lower bounds matching classical nonparametric regression rates.

A sieve two-stage least squares estimator achieving these rates.

Abstract

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 $L^2$-measure of ill-posedness with respect to the data generating distribution, bypassing a strong assumption on the discount factor $\gamma$ imposed in the recent literature for obtaining the $L^2$ 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 $L^2$-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. Our general results on the well-posedness and the minimax lower bounds are of independent interest to study not only other nonparametric estimators for $Q$-function but also efficient estimation on the value of any target policy in off-policy settings.