Learning Games and Rademacher Observations Losses

Abstract

It has recently been shown that supervised learning with the popular logistic loss is equivalent to optimizing the exponential loss over sufficient statistics about the class: Rademacher observations (rados). We first show that this unexpected equivalence can actually be generalized to other example / rado losses, with necessary and sufficient conditions for the equivalence, exemplified on four losses that bear popular names in various fields: exponential (boosting), mean-variance (finance), Linear Hinge (on-line learning), ReLU (deep learning), and unhinged (statistics). Second, we show that the generalization unveils a surprising new connection to regularized learning, and in particular a sufficient condition under which regularizing the loss over examples is equivalent to regularizing the rados (with Minkowski sums) in the equivalent rado loss. This brings simple and powerful rado-based learning algorithms for sparsity-controlling regularization, that we exemplify on a boosting algorithm for the regularized exponential rado-loss, which formally boosts over four types of regularization, including the popular ridge and lasso, and the recently coined slope --- we obtain the first proven boosting algorithm for this last regularization. Through our first contribution on the equivalence of rado and example-based losses, Omega-R.AdaBoost~appears to be an efficient proxy to boost the regularized logistic loss over examples using whichever of the four regularizers. Experiments display that regularization consistently improves performances of rado-based learning, and may challenge or beat the state of the art of example-based learning even when learning over small sets of rados. Finally, we connect regularization to differential privacy, and display how tiny budgets can be afforded on big domains while beating (protected) example-based learning.