  Although there exist plentiful theories of empirical risk minimization (ERM)
for supervised learning, current theoretical understandings of ERM for a
related problem---stochastic convex optimization (SCO), are limited. In this
work, we strengthen the realm of ERM for SCO by exploiting smoothness and
strong convexity conditions to improve the risk bounds. First, we establish an
$\widetilde{O}(d/n + \sqrt{F_*/n})$ risk bound when the random function is
nonnegative, convex and smooth, and the expected function is Lipschitz
continuous, where $d$ is the dimensionality of the problem, $n$ is the number
of samples, and $F_*$ is the minimal risk. Thus, when $F_*$ is small we obtain
an $\widetilde{O}(d/n)$ risk bound, which is analogous to the
$\widetilde{O}(1/n)$ optimistic rate of ERM for supervised learning. Second, if
the objective function is also $\lambda$-strongly convex, we prove an
$\widetilde{O}(d/n + \kappa F_*/n )$ risk bound where $\kappa$ is the condition
number, and improve it to $O(1/[\lambda n^2] + \kappa F_*/n)$ when
$n=\widetilde{\Omega}(\kappa d)$. As a result, we obtain an $O(\kappa/n^2)$
risk bound under the condition that $n$ is large and $F_*$ is small, which to
the best of our knowledge, is the first $O(1/n^2)$-type of risk bound of ERM.
Third, we stress that the above results are established in a unified framework,
which allows us to derive new risk bounds under weaker conditions, e.g.,
without convexity of the random function and Lipschitz continuity of the
expected function. Finally, we demonstrate that to achieve an $O(1/[\lambda
n^2] + \kappa F_*/n)$ risk bound for supervised learning, the
$\widetilde{\Omega}(\kappa d)$ requirement on $n$ can be replaced with
$\Omega(\kappa^2)$, which is dimensionality-independent.
