Fitted Q-Iteration with Continuous Actions and Adaptive Data: Smoothed Coverage and Minimax Rates

Abstract

Fitted Q-iteration (FQI) is well understood when the data come from a fixed distribution and the action set is finite, whereas value-based continuous control collects its data adaptively and maximizes over continuous actions. Both features break the standard analysis: the regression data are dependent, and every density-ratio concentrability coefficient of a deterministic greedy policy is infinite when the exploration has no atoms. Conditioning on the history freezes the regression target, so each round of FQI is a regression with martingale noise, and we obtain localized rates that are fast for classes of polylogarithmic metric entropy. We transfer the regression error to the loss of the greedy policy by smoothing the measures on which this loss is evaluated, at a price that depends on the action dimension and the smoothness of the residuals in the action. Under deterministic dynamics the state is smoothed as well, and the state dimension enters the price. Minimax lower bounds with the same exponents in the sample size show that no learner avoids these prices, for restart data in the case of the state dimension. The coverage conditions do not involve the unknown greedy policies; they hold under exploration floors with restarts and, for action smoothing, bounded transition densities.