  Stochastic approximation (SA) is a classical approach for stochastic convex
optimization. Previous studies have demonstrated that the convergence rate of
SA can be improved by introducing either smoothness or strong convexity
condition. In this paper, we make use of smoothness and strong convexity
simultaneously to boost the convergence rate. Let $\lambda$ be the modulus of
strong convexity, $\kappa$ be the condition number, $F_*$ be the minimal risk,
and $\alpha>1$ be some small constant. First, we demonstrate that, in
expectation, an $O(1/[\lambda T^\alpha] + \kappa F_*/T)$ risk bound is
attainable when $T = \Omega(\kappa^\alpha)$. Thus, when $F_*$ is small, the
convergence rate could be faster than $O(1/[\lambda T])$ and approaches
$O(1/[\lambda T^\alpha])$ in the ideal case. Second, to further benefit from
small risk, we show that, in expectation, an $O(1/2^{T/\kappa}+F_*)$ risk bound
is achievable. Thus, the excess risk reduces exponentially until reaching
$O(F_*)$, and if $F_*=0$, we obtain a global linear convergence. Finally, we
emphasize that our proof is constructive and each risk bound is equipped with
an efficient stochastic algorithm attaining that bound.
