On the Use of Non-Stationary Policies for Infinite-Horizon Discounted Markov Decision Processes

TL;DR

This paper demonstrates that using non-stationary policies in infinite-horizon discounted Markov Decision Processes significantly improves performance bounds and simplifies the computation of near-optimal policies, especially when the discount factor is close to one.

Contribution

It introduces performance bounds for non-stationary policies derived from Value Iteration, showing they outperform stationary policies and simplify approximate policy computation.

Findings

Non-stationary policies reduce performance bounds by a factor related to the discount factor.

Using non-stationary policies simplifies the problem of computing approximately optimal policies.

Performance bounds for non-stationary policies are tighter when the discount factor is close to 1.

Abstract

We consider infinite-horizon $\gamma$-discounted Markov Decision Processes, for which it is known that there exists a stationary optimal policy. We consider the algorithm Value Iteration and the sequence of policies $\pi_1,...,\pi_k$ it implicitely generates until some iteration $k$. We provide performance bounds for non-stationary policies involving the last $m$ generated policies that reduce the state-of-the-art bound for the last stationary policy $\pi_k$ by a factor $\frac{1-\gamma}{1-\gamma^m}$. In particular, the use of non-stationary policies allows to reduce the usual asymptotic performance bounds of Value Iteration with errors bounded by $\epsilon$ at each iteration from $\frac{\gamma}{(1-\gamma)^2}\epsilon$ to $\frac{\gamma}{1-\gamma}\epsilon$, which is significant in the usual situation when $\gamma$ is close to 1. Given Bellman operators that can only be computed with some error $\epsilon$, a surprising consequence of this result is that the problem of "computing an approximately optimal non-stationary policy" is much simpler than that of "computing an approximately optimal stationary policy", and even slightly simpler than that of "approximately computing the value of some fixed policy", since this last problem only has a guarantee of $\frac{1}{1-\gamma}\epsilon$.