Description

Speaker: Lin Chen (UC Berkeley)

Title: Infinite-Horizon Offline Reinforcement Learning with Linear Function Approximation: Curse of Dimensionality and Algorithm

Abstract: In this paper, we investigate the sample complexity of policy evaluation in infinite-horizon offline reinforcement learning (also known as the off-policy evaluation problem) with linear function approximation. We identify a hard regime d\gamma^{2}>1, where d is the dimension of the feature vector and \gamma is the discount rate. In this regime, for any q\in[\gamma^{2},1], we can construct a hard instance such that the smallest eigenvalue of its feature covariance matrix is q/d and it requires \Omega(\frac{d}{\gamma^{2}(q-\gamma^{2})\varepsilon^{2}}\exp(\Theta(d\gamma^{2}))) samples to approximate the value function up to an additive error \varepsilon. Note that the lower bound of the sample complexity is exponential in d. If q=\gamma^{2}, even infinite data cannot suffice. Under the low distribution shift assumption, we show that there is an algorithm that needs at most O(\max\{ \frac{\Vert \theta^{\pi}\Vert _{2}^{4}}{\varepsilon^{4}}\log\frac{d}{\delta},\frac{1}{\varepsilon^{2}}(d+\log\frac{1}{\delta})\} ) samples (\theta^{\pi} is the parameter of the policy in linear function approximation) and guarantees approximation to the value function up to an additive error of \varepsilon with probability at least 1-\delta.

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