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

• URL: http://arxiv.org/abs/2103.09847v1
• Date: Wed, 17 Mar 2021 18:18:57 GMT
• Title: Infinite-Horizon Offline Reinforcement Learning with Linear Function Approximation: Curse of Dimensionality and Algorithm
• Authors: Lin Chen, Bruno Scherrer, Peter L. Bartlett
• Abstract summary: In this paper, we investigate the sample complexity of policy evaluation in offline reinforcement learning. Under the low distribution shift assumption, we show that there is an algorithm that needs at most $Oleft(maxleft fracleftVert thetapirightVert _24varepsilon4logfracddelta,frac1varepsilon2left(d+logfrac1deltaright)right right)$ samples to approximate the
• Score: 46.36534144138337
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\Omega\left(\frac{d}{\gamma^{2}\left(q-\gamma^{2}\right)\varepsilon^{2}}\exp\left(\Theta\left(d\gamma^{2}\right)\right)\right)$ 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 $O\left(\max\left\{ \frac{\left\Vert \theta^{\pi}\right\Vert _{2}^{4}}{\varepsilon^{4}}\log\frac{d}{\delta},\frac{1}{\varepsilon^{2}}\left(d+\log\frac{1}{\delta}\right)\right\} \right)$ 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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