  We focus on the problem of minimizing a convex function $f$ over a convex set
$S$ given $T$ queries to a stochastic first order oracle. We argue that the
complexity of convex minimization is only determined by the rate of growth of
the function around its minimizer $x^*_{f,S}$, as quantified by a Tsybakov-like
noise condition. Specifically, we prove that if $f$ grows at least as fast as
$\|x-x^*_{f,S}\|^\kappa$ around its minimum, for some $\kappa > 1$, then the
optimal rate of learning $f(x^*_{f,S})$ is
$\Theta(T^{-\frac{\kappa}{2\kappa-2}})$. The classic rate $\Theta(1/\sqrt T)$
for convex functions and $\Theta(1/T)$ for strongly convex functions are
special cases of our result for $\kappa \rightarrow \infty$ and $\kappa=2$, and
even faster rates are attained for $\kappa <2$. We also derive tight bounds for
the complexity of learning $x_{f,S}^*$, where the optimal rate is
$\Theta(T^{-\frac{1}{2\kappa-2}})$. Interestingly, these precise rates for
convex optimization also characterize the complexity of active learning and our
results further strengthen the connections between the two fields, both of
which rely on feedback-driven queries.
