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

On the Optimal Boolean Function for Prediction under Quadratic Loss