Agnostic Q-learning with Function Approximation in Deterministic Systems: Tight Bounds on Approximation Error and Sample Complexity

The current paper studies the problem of agnostic $Q$-learning with function approximation in deterministic systems where the optimal $Q$-function is approximable by a function in the class $\mathcal{F}$ with approximation error $\delta \ge 0$. We propose a novel recursion-based algorithm and show that if $\delta =O\left(\rho \mathrm{/}\sqrt{{\dim }_E}\right)$, then one can find the optimal policy using $O\left({\dim }_E\right)$ trajectories, where $\rho$ is the gap between the optimal $Q$-value of the best actions and that of the second-best actions and ${\dim }_E$ is the Eluder dimension of $\mathcal{F}$. Our result has two implications:

1. In conjunction with the lower bound in [Du et al., ICLR 2020], our upper bound suggests that the condition $\delta = \widetilde{\Theta}\left(\rho/\sqrt{$\mathrm{d}\mathrm{i}\mathrm{m}$_E}\right)$ is necessary and sufficient for algorithms with polynomial sample complexity.
2. In conjunction with the lower bound in [Wen and Van Roy, NIPS 2013], our upper bound suggests that the sample complexity $\tilde{\mathrm{\Theta }}\left({\mathrm{d}\mathrm{i}\mathrm{m}}_E\right)$ is tight even in the agnostic setting. Therefore, we settle the open problem on agnostic $Q$-learning proposed in [Wen and Van Roy, NIPS 2013]. We further extend our algorithm to the stochastic reward setting and obtain similar results.