  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.
