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 $1-\delta$.

本文研究线性函数逼近下无穷时域离线强化学习的策略评估的样本复杂性以及分布漂移假设下的算法，提出了算法的样本复杂性的下界，以及样本复杂性的上界。