  In stochastic convex optimization the goal is to minimize a convex function
$F(x) \doteq {\mathbf E}_{{\mathbf f}\sim D}[{\mathbf f}(x)]$ over a convex set
$\cal K \subset {\mathbb R}^d$ where $D$ is some unknown distribution and each
$f(\cdot)$ in the support of $D$ is convex over $\cal K$. The optimization is
commonly based on i.i.d.~samples $f^1,f^2,\ldots,f^n$ from $D$. A standard
approach to such problems is empirical risk minimization (ERM) that optimizes
$F_S(x) \doteq \frac{1}{n}\sum_{i\leq n} f^i(x)$. Here we consider the question
of how many samples are necessary for ERM to succeed and the closely related
question of uniform convergence of $F_S$ to $F$ over $\cal K$. We demonstrate
that in the standard $\ell_p/\ell_q$ setting of Lipschitz-bounded functions
over a $\cal K$ of bounded radius, ERM requires sample size that scales
linearly with the dimension $d$. This nearly matches standard upper bounds and
improves on $\Omega(\log d)$ dependence proved for $\ell_2/\ell_2$ setting by
Shalev-Shwartz et al. (2009). In stark contrast, these problems can be solved
using dimension-independent number of samples for $\ell_2/\ell_2$ setting and
$\log d$ dependence for $\ell_1/\ell_\infty$ setting using other approaches. We
further show that our lower bound applies even if the functions in the support
of $D$ are smooth and efficiently computable and even if an $\ell_1$
regularization term is added. Finally, we demonstrate that for a more general
class of bounded-range (but not Lipschitz-bounded) stochastic convex programs
an infinite gap appears already in dimension 2.
