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• W2952207910 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." @default.
• W2952207910 created "2019-06-27" @default.
• W2952207910 creator A5032708627 @default.
• W2952207910 creator A5076051906 @default.
• W2952207910 date "2016-07-08" @default.
• W2952207910 modified "2023-09-27" @default.
• W2952207910 title "On the Optimal Boolean Function for Prediction under Quadratic Loss" @default.
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