Markov Chains with Rewinding

TL;DR

This paper introduces Markov Chains with Rewinding, a new randomized process allowing strategic rewinding to improve state identification efficiency, and analyzes the power and limitations of adaptive versus non-adaptive strategies.

Contribution

It formalizes Markov Chains with Rewinding, proves non-adaptive strategies are as powerful as adaptive ones for distinguishability, and quantifies the efficiency gap with polynomial bounds.

Findings

Non-adaptive strategies can match adaptive strategies in distinguishability.

Adaptive strategies are more query-efficient than non-adaptive ones, with a polynomial gap.

Rewinding strategies enhance the ability to identify states in Markov chains.

Abstract

Motivated by techniques developed in recent progress on lower bounds for sublinear time algorithms (Behnezhad, Roghani and Rubinstein, STOC 2023, FOCS 2023, and STOC 2024) we introduce and study a new class of randomized algorithmic processes that we call Markov Chains with Rewinding. In this setting, an algorithm interacts with a (partially observable) Markovian random evolution by strategically rewinding the Markov chain to previous states. Depending on the application, this may lead the evolution to desired states faster, or allow the agent to efficiently learn or test properties of the underlying Markov chain that may be infeasible or inefficient with passive observation. We study the task of identifying the initial state in a given partially observable Markov chain. Analysis of this question in specific Markov chains is the central ingredient in the above cited works and we aim to systematize the analysis in our work. Our first result is that any pair of states distinguishable with any rewinding strategy can also be distinguished with a non-adaptive rewinding strategy (one whose rewinding choices are determined before observing any outcomes of the chain). Therefore, while rewinding strategies can be shown to be strictly more powerful than passive strategies (those that do not rewind back to previous states), adaptivity does not give additional power to a rewinding strategy in the absence of efficiency considerations. The difference becomes apparent however when we introduce a natural efficiency measure, namely the query complexity (i.e., the number of observations they need to identify distinguishable states). Our second main contribution is to quantify this efficiency gap. We present a non-adaptive rewinding strategy whose query complexity is within a polynomial of that of the optimal (adaptive) strategy, and show that such a polynomial loss is necessary in general.