  In this paper, we utilize stochastic optimization to reduce the space
complexity of convex composite optimization with a nuclear norm regularizer,
where the variable is a matrix of size $m \times n$. By constructing a low-rank
estimate of the gradient, we propose an iterative algorithm based on stochastic
proximal gradient descent (SPGD), and take the last iterate of SPGD as the
final solution. The main advantage of the proposed algorithm is that its space
complexity is $O(m+n)$, in contrast, most of previous algorithms have a $O(mn)$
space complexity. Theoretical analysis shows that it achieves $O(\log
T/\sqrt{T})$ and $O(\log T/T)$ convergence rates for general convex functions
and strongly convex functions, respectively.
