Markov Chains with Rewinding

2026-02-17

AI Summary

This study addresses the problem of efficiently identifying the initial state in partially observable Markov chains, particularly under passive observation and limited-efficiency constraints. To this end, the authors introduce and systematically analyze, for the first time, a novel model termed “Markov chains with support for backtracking,” which permits algorithms to strategically revert the process to prior states to accelerate learning or decision-making. The key contributions include establishing the equivalence between non-adaptive and adaptive backtracking strategies in terms of state distinguishability, and constructing a non-adaptive strategy whose query complexity exceeds that of the optimal adaptive strategy by only a polynomial factor—a gap proven to be unavoidable. The theoretical analysis integrates probability theory, information theory, and computational complexity, thereby establishing a new analytical framework for reasoning about backtracking mechanisms.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Markov Chains
• Rewinding
• Partial Observability
• State Identification
• Query Complexity

Innovation

• Markov Chains with Rewinding
• non-adaptive strategy
• query complexity
• state distinguishability
• partial observability