Stochastic Halpern iteration in normed spaces and applications to reinforcement learning

We analyze the oracle complexity of the stochastic Halpern iteration with minibatch, where we aim to approximate fixed-points of nonexpansive and contractive operators in a normed finite-dimensional space. We show that if the underlying stochastic oracle has uniformly bounded variance, our method exhibits an overall oracle complexity of $\tilde{O}(\varepsilon^{-5})$, to obtain $\varepsilon$ expected fixed-point residual for nonexpansive operators, improving recent rates established for the stochastic Krasnoselskii-Mann iteration. Also, we establish a lower bound of $\Omega(\varepsilon^{-3})$ which applies to a wide range of algorithms, including all averaged iterations even with minibatching. Using a suitable modification of our approach, we derive a $O(\varepsilon^{-2}(1-\gamma)^{-3})$ complexity bound in the case in which the operator is a $\gamma$-contraction to obtain an approximation of the fixed-point. As an application, we propose new model-free algorithms for average and discounted reward MDPs. For the average reward case, our method applies to weakly communicating MDPs without requiring prior parameter knowledge.