Agnostic Q-learning with Function Approximation in Deterministic Systems: Tight Bounds on Approximation Error and Sample Complexity
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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 F with approximation error δ> 0. We propose a novel recursion-based algorithm and show that if δ = O(ρ/√(_E)), then one can find the optimal policy using O(_E) trajectories, where ρ is the gap between the optimal Q-value of the best actions and that of the second-best actions and _E is the Eluder dimension of 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 δ = Θ(ρ/√(dim_E)) 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 Θ(dim_E) 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.
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