Problem
Research questions and friction points this paper is trying to address.
Markov chain
average mixing time
sample complexity
converse bounds
On the Impossibility of Estimating the Average Mixing Time
🤖 AI Summary
This study addresses the unresolved sample complexity lower bounds for estimating the average mixing time of Markov chains and questions the feasibility of such estimation from a single stationary trajectory. By integrating techniques from statistical learning theory, information theory, and Markov process analysis, this work establishes the first lower bound that matches existing achievability upper bounds in key dependencies up to logarithmic factors. Furthermore, it rigorously demonstrates the infeasibility of estimating the average mixing time from a single stationary trajectory and derives near-tight PAC sample complexity bounds. These contributions fill a critical theoretical gap in the field by providing a comprehensive characterization of the fundamental limits underlying Markov chain mixing time estimation.
📝 Abstract
We establish nearly-matching converse bounds on the sample complexity of estimating the average mixing time of a Markov chain from a single stationary trajectory. Unlike the worst-case mixing time, which maximises the total-variation distance to stationarity over all initial states, the average mixing time averages this distance under the stationary distribution and can therefore be substantially smaller. Our converse bounds establish when estimation is impossible, and match the key dependencies in existing achievability bounds, up to logarithmic factors. Together, these bounds provide nearly-tight PAC guarantees for estimating the average mixing time.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Average Mixing Time
Sample Complexity
Markov Chain
Converse Bounds
PAC Guarantees
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