A Non-Asymptotic Analysis of Mismatched Guesswork

TL;DR

This paper provides a finite-sample analysis of mismatched guesswork, quantifying the additional cost when using a guessing strategy optimized for a different distribution, using permutation distance and total variation bounds.

Contribution

It introduces a non-asymptotic framework for mismatched guesswork, connecting it to permutation distance and providing bounds based on distribution differences.

Findings

Bound the guesswork cost using total variation distance

Establish equivalence with Kendall tau permutation distance

Quantify mismatch penalty in finite samples

Abstract

The problem of mismatched guesswork considers the additional cost incurred by using a guessing function which is optimal for a distribution $q$ when the random variable to be guessed is actually distributed according to a different distribution $p$. This problem has been well-studied from an asymptotic perspective, but there has been little work on quantifying the difference in guesswork between optimal and suboptimal strategies for a finite number of symbols. In this non-asymptotic regime, we consider a definition for mismatched guesswork which we show is equivalent to a variant of the Kendall tau permutation distance applied to optimal guessing functions for the mismatched distributions. We use this formulation to bound the cost of guesswork under mismatch given a bound on the total variation distance between the two distributions.