Box-constrained optimization for minimax supervised learning

In this paper, we present the optimization procedure for computing the discrete boxconstrained minimax classifier introduced in [1, 2]. Our approach processes discrete or beforehand discretized features. A box-constrained region defines some bounds for each class proportion independently. The box-constrained minimax classifier is obtained from the computation of the least favorable prior which maximizes the minimum empirical risk of error over the box-constrained region. After studying the discrete empirical Bayes risk over the probabilistic simplex, we consider a projected subgradient algorithm which computes the prior maximizing this concave multivariate piecewise affine function over a polyhedral domain. The convergence of our algorithm is established.