A Simple Analysis for Exp-concave Empirical Minimization with Arbitrary Convex Regularizer

In this paper, we present a simple analysis of bf fast rates with it high probability of bf empirical minimization for it stochastic composite optimization over a finite-dimensional bounded convex set with exponential concave loss functions and an arbitrary convex regularization. To the best of our knowledge, this result is the first of its kind. As a byproduct, we can directly obtain the fast rate with it high probability for exponential concave empirical risk minimization with and without any convex regularization, which not only extends existing results of empirical risk minimization but also provides a unified framework for analyzing exponential concave empirical risk minimization with and without it any convex regularization. Our proof is very simple only exploiting the covering number of a finite-dimensional bounded set and a concentration inequality of random vectors.