# Fixed-Point Iterations in Markov Decision Processes

Prof. Roberto Cominetti

### Abstract

This short course is an introduction to some novel techniques for studying the convergence of stochastic fixed point iterations — such as the Krasnoselskii-Mann, Halpern, and similar iterative methods — with explicit approximation guarantees in the form of finite-time error bounds and convergence rates. Starting from the base case of deterministic iterations, in which the relevant maps can be computed exactly, we will move to settings where the iterations are affected by numerical inaccuracies or stochastic perturbations. The stochastic case is a natural setting for studying the convergence of reinforcement learning procedures in Markov decision processes (MDP’s). We will survey the recent results for MDP’s with discounted payoffs, and illustrate the full power of the new techniques, which becomes more evident when considering the case of long-term average rewards.

Content

The tentative program for the course is (click on each lecture for the slides):

Lecture 1: MDP’s as fixed point iterations. What are we looking for?

Lecture 2: Optimal transport error bounds for fixed-point iterations.

Lecture 3: Extension to stochastic fixed point iterations.

Lecture 4: Application to reinforcement learning for MDPs.

The error-bound analysis is strongly based on finite-dimensional optimal transport. While the course does not presume familiarity with this topic, some acquaintance with linear and convex programming is desirable. Also, the analysis of stochastic iterations would benefit from prior knowledge about finite-dimensional martingales, although the main results used in the analysis will be recalled as needed.

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