TY - GEN
T1 - On the optimal boolean function for prediction under quadratic loss
AU - Weinberger, Nir
AU - Shayevitz, Ofer
N1 - Publisher Copyright: © 2016 IEEE.
PY - 2016/8/10
Y1 - 2016/8/10
N2 - Suppose Y n is obtained by observing a uniform Bernoulli random vector Xn through a binary symmetric channel. Courtade and Kumar asked how large the mutual information between Y n and a Boolean function b(Xn) could be, and conjectured that the maximum is attained by the dictator function. An equivalent formulation of this conjecture is that dictator minimizes the prediction cost in sequentially predicting Y n under logarithmic loss, given b(Xn). 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 Boolean function that is simultaneously optimal at all noise levels.
AB - Suppose Y n is obtained by observing a uniform Bernoulli random vector Xn through a binary symmetric channel. Courtade and Kumar asked how large the mutual information between Y n and a Boolean function b(Xn) could be, and conjectured that the maximum is attained by the dictator function. An equivalent formulation of this conjecture is that dictator minimizes the prediction cost in sequentially predicting Y n under logarithmic loss, given b(Xn). 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 Boolean function that is simultaneously optimal at all noise levels.
KW - Boolean functions
KW - Pinsker's inequality
KW - logarithmic loss function
KW - quadratic loss function
KW - sequential prediction
UR - http://www.scopus.com/inward/record.url?scp=84985920220&partnerID=8YFLogxK
U2 - 10.1109/ISIT.2016.7541348
DO - 10.1109/ISIT.2016.7541348
M3 - منشور من مؤتمر
T3 - IEEE International Symposium on Information Theory - Proceedings
SP - 495
EP - 499
BT - Proceedings - ISIT 2016; 2016 IEEE International Symposium on Information Theory
T2 - 2016 IEEE International Symposium on Information Theory, ISIT 2016
Y2 - 10 July 2016 through 15 July 2016
ER -