On the Optimal Boolean Function for Prediction under Quadratic Loss

TL;DR

This paper investigates the optimal Boolean functions for prediction under quadratic loss in noisy channels, revealing that majority functions are optimal in noiseless or weak noise scenarios, while dictators excel under strong noise, and no universal optimal exists.

Contribution

It introduces the analysis of Boolean functions' prediction cost under quadratic loss across different noise levels, extending prior work focused on logarithmic loss.

Findings

Majority functions asymptotically minimize prediction cost in noiseless cases.

Majority outperforms dictators under weak noise conditions.

Dictators outperform majority in strong noise scenarios.

Abstract

Suppose $Y^{n}$ is obtained by observing a uniform Bernoulli random vector $X^{n}$ through a binary symmetric channel. Courtade and Kumar asked how large the mutual information between $Y^{n}$ and a Boolean function $\mathsf{b}(X^{n})$ could be, and conjectured that the maximum is attained by a dictator function. An equivalent formulation of this conjecture is that dictator minimizes the prediction cost in a sequential prediction of $Y^{n}$ under logarithmic loss, given $\mathsf{b}(X^{n})$. In this paper, we study the question of minimizing the sequential prediction cost under a different (proper) loss function - the quadratic loss. In the noiseless case, we show that majority asymptotically minimizes this prediction cost among all Boolean functions. We further show that for weak noise, majority is better than dictator, and that for strong noise dictator outperforms majority. We conjecture that for quadratic loss, there is no single sequence of Boolean functions that is simultaneously (asymptotically) optimal at all noise levels.