Evaluating Multiple Guesses by an Adversary via a Tunable Loss Function

TL;DR

This paper analyzes an adversary's guessing strategy using a tunable loss function, characterizes optimal strategies, and introduces an information leakage measure in a multiple guesses setting.

Contribution

It provides a complete characterization of optimal adversarial strategies under $ ext{alpha}$-loss and introduces a new information leakage measure for multiple guesses.

Findings

Optimal strategies are characterized for all $ ext{alpha}$ values.

The paper recovers known results for $ ext{alpha}=\infty$.

Conditions are derived where leakage remains unchanged with multiple guesses.

Abstract

We consider a problem of guessing, wherein an adversary is interested in knowing the value of the realization of a discrete random variable $X$ on observing another correlated random variable $Y$. The adversary can make multiple (say, $k$) guesses. The adversary's guessing strategy is assumed to minimize $\alpha$-loss, a class of tunable loss functions parameterized by $\alpha$. It has been shown before that this loss function captures well known loss functions including the exponential loss ($\alpha=1/2$), the log-loss ($\alpha=1$) and the $0$-$1$ loss ($\alpha=\infty$). We completely characterize the optimal adversarial strategy and the resulting expected $\alpha$-loss, thereby recovering known results for $\alpha=\infty$. We define an information leakage measure from the $k$-guesses setup and derive a condition under which the leakage is unchanged from a single guess.