Policy Iteration is well suited to optimize PageRank

TL;DR

This paper demonstrates that policy iteration is efficient for a class of Markov Decision Processes related to PageRank optimization, supported by theoretical analysis and numerical evidence, suggesting polynomial complexity in these cases.

Contribution

The paper identifies a class of MDPs based on PageRank optimization where policy iteration is conjectured to be polynomial-time, supported by proofs and numerical experiments.

Findings

Policy iteration is efficient for PageRank-based MDPs.

Adding constraints links PageRank optimization to path length optimization.

Numerical evidence supports polynomial complexity conjecture.

Abstract

The question of knowing whether the policy Iteration algorithm (PI) for solving Markov Decision Processes (MDPs) has exponential or (strongly) polynomial complexity has attracted much attention in the last 50 years. Recently, Fearnley proposed an example on which PI needs an exponential number of iterations to converge. Though, it has been observed that Fearnley's example leaves open the possibility that PI behaves well in many particular cases, such as in problems that involve a fixed discount factor, or that are restricted to deterministic actions. In this paper, we analyze a large class of MDPs and we argue that PI is efficient in that case. The problems in this class are obtained when optimizing the PageRank of a particular node in the Markov chain. They are motivated by several practical applications. We show that adding natural constraints to this PageRank Optimization problem (PRO) makes it equivalent to the problem of optimizing the length of a stochastic path, which is a widely studied family of MDPs. Finally, we conjecture that PI runs in a polynomial number of iterations when applied to PRO. We give numerical arguments as well as the proof of our conjecture in a number of particular cases of practical importance.