  Error bound conditions (EBC) are properties that characterize the growth of
an objective function when a point is moved away from the optimal set. They
have recently received increasing attention in the field of optimization for
developing optimization algorithms with fast convergence. However, the studies
of EBC in statistical learning are hitherto still limited. The main
contributions of this paper are two-fold. First, we develop fast and
intermediate rates of empirical risk minimization (ERM) under EBC for risk
minimization with Lipschitz continuous, and smooth convex random functions.
Second, we establish fast and intermediate rates of an efficient stochastic
approximation (SA) algorithm for risk minimization with Lipschitz continuous
random functions, which requires only one pass of $n$ samples and adapts to
EBC. For both approaches, the convergence rates span a full spectrum between
$\widetilde O(1/\sqrt{n})$ and $\widetilde O(1/n)$ depending on the power
constant in EBC, and could be even faster than $O(1/n)$ in special cases for
ERM. Moreover, these convergence rates are automatically adaptive without using
any knowledge of EBC. Overall, this work not only strengthens the understanding
of ERM for statistical learning but also brings new fast stochastic algorithms
for solving a broad range of statistical learning problems.
