  A commonly used learning rule is to approximately minimize the \emph{average}
loss over the training set. Other learning algorithms, such as AdaBoost and
hard-SVM, aim at minimizing the \emph{maximal} loss over the training set. The
average loss is more popular, particularly in deep learning, due to three main
reasons. First, it can be conveniently minimized using online algorithms, that
process few examples at each iteration. Second, it is often argued that there
is no sense to minimize the loss on the training set too much, as it will not
be reflected in the generalization loss. Last, the maximal loss is not robust
to outliers. In this paper we describe and analyze an algorithm that can
convert any online algorithm to a minimizer of the maximal loss. We prove that
in some situations better accuracy on the training set is crucial to obtain
good performance on unseen examples. Last, we propose robust versions of the
approach that can handle outliers.
