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Learning Multiple Secrets in Mastermind

Milind Prabhu, David Woodruff

TL;DR

It is shown that any $r$-round adaptive randomized algorithm that learns $H$ with constant probability must make $\exp(\Omega(d^{3^{-(r-1)}}))$ queries even when the input has $\text{poly}(d)$ points; thus, any query algorithm must necessarily use $\Omega(\log \log d)$ rounds of adaptivity.

Abstract

In the Generalized Mastermind problem, there is an unknown subset $H$ of the hypercube $\{0,1\}^d$ containing $n$ points. The goal is to learn $H$ by making a few queries to an oracle, which, given a point $q$ in $\{0,1\}^d$, returns the point in $H$ nearest to $q$. We give a two-round adaptive algorithm for this problem that learns $H$ while making at most $\exp(\tilde{O}(\sqrt{d \log n}))$ queries. Furthermore, we show that any $r$-round adaptive randomized algorithm that learns $H$ with constant probability must make $\exp(Ω(d^{3^{-(r-1)}}))$ queries even when the input has $\text{poly}(d)$ points; thus, any $\text{poly}(d)$ query algorithm must necessarily use $Ω(\log \log d)$ rounds of adaptivity. We give optimal query complexity bounds for the variant of the problem where queries are allowed to be from $\{0,1,2\}^d$. We also study a continuous variant of the problem in which $H$ is a subset of unit vectors in $\mathbb{R}^d$, and one can query unit vectors in $\mathbb{R}^d$. For this setting, we give an $O(n^{d/2})$ query deterministic algorithm to learn the hidden set of points.

Learning Multiple Secrets in Mastermind

TL;DR

It is shown that any -round adaptive randomized algorithm that learns with constant probability must make queries even when the input has points; thus, any query algorithm must necessarily use rounds of adaptivity.

Abstract

In the Generalized Mastermind problem, there is an unknown subset of the hypercube containing points. The goal is to learn by making a few queries to an oracle, which, given a point in , returns the point in nearest to . We give a two-round adaptive algorithm for this problem that learns while making at most queries. Furthermore, we show that any -round adaptive randomized algorithm that learns with constant probability must make queries even when the input has points; thus, any query algorithm must necessarily use rounds of adaptivity. We give optimal query complexity bounds for the variant of the problem where queries are allowed to be from . We also study a continuous variant of the problem in which is a subset of unit vectors in , and one can query unit vectors in . For this setting, we give an query deterministic algorithm to learn the hidden set of points.
Paper Structure (19 sections, 19 theorems, 15 equations, 1 figure, 3 algorithms)

This paper contains 19 sections, 19 theorems, 15 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Mastermind With Multiple Secrets
  3. Motivation
  4. Data Reconstruction from a Nearest Neighbor Data Structure.
  5. Optimization.
  6. Adverserially Robust Learning.
  7. Our Contributions
  8. Related Work
  9. Preliminaries
  10. Notation
  11. Some Useful Concentration Inequalities
  12. Results for Problem 2
  13. A Simple 2-Round Adaptive Algorithm
  14. Overview of \ref{['alg: hamming-nearest-point']}.
  15. A Lower Bound Against r-Round Adaptive Algorithms
  16. ...and 4 more sections

Key Result

Corollary 2.3

Suppose that $Y_1, \ldots, Y_m$ are $m$ independent $\textup{Bin}(d,1/2)$ random variables. Define $Y_{\min}$ to be the minimum of the random variables $Y_i$, i.e., $Y_{\min} = \min\limits_i Y_i$. For $m = 2^{o(d)}$ and $\tfrac{1}{2^m} < \delta < 1$ we have,

Figures (1)

  • Figure 1: The figure illustrates how a hard distribution for $r$ round algorithms can be used to create a hard distribution for $r+1$ round algorithms.

Theorems & Definitions (47)

  • Definition 1.1: $r$-Round Adaptive Algorithm
  • Corollary 2.3
  • Definition 2.4: Hypergeometric Distribution
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Claim 3.3
  • proof
  • Claim 3.4
  • proof
  • ...and 37 more