Best Policy Identification in Linear MDPs

TL;DR

This paper studies the problem of efficiently identifying the best policy in linear Markov Decision Processes using sample-efficient algorithms, providing theoretical bounds and extending to episodic settings.

Contribution

It derives an instance-specific lower bound and proposes near-optimal algorithms with proven sample complexity bounds for linear MDPs.

Findings

Sample complexity upper bound of ${rac{d}{( ext{gap})^2}}$ times logarithmic factors.

Algorithm matches existing lower bounds in the moderate-confidence regime.

Extension of algorithms to episodic linear MDPs.

Abstract

We investigate the problem of best policy identification in discounted linear Markov Decision Processes in the fixed confidence setting under a generative model. We first derive an instance-specific lower bound on the expected number of samples required to identify an $\varepsilon$-optimal policy with probability $1-\delta$. The lower bound characterizes the optimal sampling rule as the solution of an intricate non-convex optimization program, but can be used as the starting point to devise simple and near-optimal sampling rules and algorithms. We devise such algorithms. One of these exhibits a sample complexity upper bounded by ${\cal O}({\frac{d}{(\varepsilon+\Delta)^2}} (\log(\frac{1}{\delta})+d))$ where $\Delta$ denotes the minimum reward gap of sub-optimal actions and $d$ is the dimension of the feature space. This upper bound holds in the moderate-confidence regime (i.e., for all $\delta$), and matches existing minimax and gap-dependent lower bounds. We extend our algorithm to episodic linear MDPs.