  In this paper, we consider convex stochastic optimization problems arising in
machine learning applications (e.g., risk minimization) and mathematical
statistics (e.g., maximum likelihood estimation). There are two main approaches
to solve such kinds of problems, namely the Stochastic Approximation approach
(online approach) and the Sample Average Approximation approach, also known as
the Monte Carlo approach, (offline approach). In the offline approach, the
problem is replaced by its empirical counterpart (the empirical risk
minimization problem). The natural question is how to define the problem sample
size, i.e., how many realizations should be sampled so that the quite accurate
solution of the empirical problem be the solution of the original problem with
the desired precision. This issue is one of the main issues in modern machine
learning and optimization. In the last decade, a lot of significant advances
were made in these areas to solve convex stochastic optimization problems on
the Euclidean balls (or the whole space). In this work, we are based on these
advances and study the case of arbitrary balls in the $\ell_p$-norms. We also
explore the question of how the parameter $p$ affects the estimates of the
required number of terms as a function of empirical risk.
