Constructive minimax classification of discrete observations with arbitrary loss function

Abstract

This paper develops a multihypothesis testing framework for calculating numerically the optimal minimax test with discrete observations and an arbitrary loss function. Discrete observations are common in data processing and make tractable the calculation of the minimax test. Each hypothesis is both associated to a parameter defining the distribution of the observations and to an action which describes the decision to take when the hypothesis is true. The loss function measures the gap between the parameters and the actions. The minimax test minimizes the maximum classification risk. It is the solution of a finite linear programming problem which gives the worst case classification risk and the worst case prior distribution. The minimax test equalizes the classification risks whose prior probabilities are strictly positive. The minimax framework is applied to vector channel decoding which consists in classifying some codewords transmitted on a binary asymmetric channel. The Hamming metric is used to measure the number of differences between the emitted codeword and the decoded one.

Box-constrained optimization for minimax supervised learning

“…so that the solution δ C of ( 8) is the Bayes classifier given by ( 10) with the priors (11). The least favorable priors are generally difficult to compute as underlined in [12,[26][27][28]. Subsection 2.2 is devoted to the calculation of the minimum Bayes risk V (π) over the simplex.…”