  This paper concerns a high-dimensional stochastic programming problem of
minimizing a function of expected cost with a matrix argument. To this problem,
one of the most widely applied solution paradigms is the sample average
approximation (SAA), which uses the average cost over sampled scenarios as a
surrogate to approximate the expected cost. Traditional SAA theories require
the sample size to grow rapidly when the problem dimensionality increases.
Indeed, for a problem of optimizing over a $p$-by-$p$ matrix, the sample
complexity of the SAA is given by $\tilde O(1)\cdot
\frac{p^2}{\epsilon^2}\cdot{polylog}(\frac{1}{\epsilon})$ to achieve an
$\epsilon$-suboptimality gap, for some poly-logarithmic function
${polylog}(\,\cdot\,)$ and some quantity $\tilde O(1)$ independent of
dimensionality $p$ and sample size $n$. In contrast, this paper considers a
regularized SAA (RSAA) with a low-rankness-inducing penalty. We demonstrate
that the sample complexity of RSAA is $\tilde O(1)\cdot
\frac{p}{\epsilon^3}\cdot {polylog}(p,\,\frac{1}{\epsilon})$, which is almost
linear in $p$ and thus indicates a substantially lower dependence on
dimensionality. Therefore, RSAA can be more advantageous than SAA especially
for larger scale and higher dimensional problems. Due to the close
correspondence between stochastic programming and statistical learning, our
results also indicate that high-dimensional low-rank matrix recovery is
possible generally beyond a linear model, even if the common assumption of
restricted strong convexity is completely absent.
