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The Query Complexity of Mastermind with $\ell_p$ Distances by Fernandez, Manuel; Woodruff, David P.; Yasuda, Taisuke is a scholarly article available to read on EtoBox.

What is The Query Complexity of Mastermind with $\ell_p$ Distances about?

Consider a variant of the Mastermind game in which queries are $\ell_p$ distances, rather than the usual Hamming distance. That is, a codemaker chooses a hidden vector $\mathbf{y}\in\{-k,-k+1,\dots,k-1,k\}^n$ and answers to queries of the form $\Vert\mathbf{y}-\mathbf{x}\Vert_p$ where $\mathbf{x}\in\{-k,-k+1,\dots,k-1,k\}^n$. The goal is to minimize the number of queries made in order to correctly guess $\mathbf{y}$. Motivated by this question, in this work, we develop a nonadaptive polynomial time algorithm that works for a natural class of separable distance measures, i.e.\ coordinate-wise sums of functions of the absolute value. This in particular includes distances such as the smooth max (LogSumExp) as well as many widely-studied $M$-estimator losses, such as $\ell_p$ norms, the $\ell_1$-$\ell_2$ loss, the Huber loss, and the Fair estimator loss. When we apply this result to $\ell_p$ queries, we obtain an upper bound of $O\left(\min\left\{n,\frac{n\log k}{\log n}\right\}\right)$ queries for any real $1\leq p<\infty$. We also show matching lower bounds up to constant factors for the $\ell_p$ problem, even for adaptive algorithms for the approximation version of the problem, in w

Author

Fernandez, Manuel; Woodruff, David P.; Yasuda, Taisuke

Published

2019

Language

EN