Fast Stochastic Algorithms for Low-rank and Nonsmooth Matrix Problems

Composite convex optimization problems which include both a nonsmooth term and a low-rank promoting term have important applications in machine learning and signal processing, such as when one wishes to recover an unknown matrix that is simultaneously low-rank and sparse. However, such problems are highly challenging to solve in large-scale: the low-rank promoting term prohibits efficient implementations of proximal methods for composite optimization and even simple subgradient methods. On the other hand, methods which are tailored for low-rank optimization, such as conditional gradient-type methods, which are often applied to a smooth approximation of the nonsmooth objective, are slow since their runtime scales with both the large Lipshitz parameter of the smoothed gradient vector and with 1/ε. In this paper we develop efficient algorithms for textitstochastic optimization of a strongly-convex objective which includes both a nonsmooth term and a low-rank promoting term. In particular, to the best of our knowledge, we present the first algorithm that enjoys all following critical properties for large-scale problems: i) (nearly) optimal sample complexity, ii) each iteration requires only a single textitlow-rank SVD computation, and iii) overall number of thin-SVD computations scales only with log1/ε (as opposed to textrmpoly(1/ε) in previous methods). We also give an algorithm for the closely-related finite-sum setting. At the heart of our results lie a novel combination of a variance-reduction technique and the use of a textitweak-proximal oracle which is key to obtaining all above three properties simultaneously.