Nonconvex reformulations via low-rank factorization for stochastic convex semidefinite optimization problem have attracted arising attention due to their empirical efficiency and scalability. Compared with the original convex formulations, the nonconvex ones typically involve much fewer variables, allowing them to scale to scenarios with millions of variables. However, it opens a new challenge that under what conditions the nonconvex stochastic algorithms may find the population minimizer within the optimal statistical precision despite their empirical success in applications. In this paper, we provide an answer that the stochastic gradient descent (SGD) method can be adapted to solve the nonconvex reformulation of the original convex problem, with a global linear convergence when using a fixed step size, i.e., converging exponentially fast to the population minimizer within an optimal statistical precision in the restricted strongly convex case. If a diminishing step size is adopted, the bad effect caused by the variance of gradients on the optimization error can be eliminated but the rate is dropped to be sublinear. The core of our treatment relies on a novel second-order descent lemma, which is more general than the existing best result in the literature and improves the analysis on both online and batch algorithms. The developed theoretical results and effectiveness of the suggested SGD are also verified by a series of experiments.

The stochastic gradient (SG) method can quickly solve a problem with a large number of components in the objective, or a stochastic optimization problem, to a moderate accuracy. The block coordinate descent/update (BCD) method, on the other hand, can quickly solve problems with multiple (blocks of) variables. This paper introduces a method that combines the great features of SG and BCD for problems with many components in the objective and with multiple (blocks of) variables. This paper proposes a block SG (BSG) method for both convex and nonconvex programs. BSG generalizes SG by updating all the blocks of variables in the Gauss–Seidel type (updating the current block depends on the previously updated block), in either a fixed or randomly shuffled order. Although BSG has slightly more work at each iteration, it typically outperforms SG because of BSG’s Gauss–Seidel updates and larger step sizes, the latter of which are determined by the smaller per-block Lipschitz constants. The convergence of BSG is established for both convex and nonconvex cases. In the convex case, BSG has the same order of convergence rate as SG. In the nonconvex case, its convergence is established in terms of the expected violation of a first-order optimality condition. In both cases our analysis is nontrivial since the typical unbiasedness assumption no longer holds. BSG is numerically evaluated on the following problems: stochastic least squares and logistic regression, which are convex, and low-rank tensor recovery and bilinear logistic regression, which are nonconvex. On the convex problems, BSG performed significantly better than SG. On the nonconvex problems, BSG significantly outperformed the deterministic BCD method because the latter tends to stagnate early near local minimizers. Overall, BSG inherits the benefits of both SG approximation and block coordinate updates and is especially useful for solving large-scale nonconvex problems.

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Factor models are a class of powerful statistical models that have been widely used to deal with dependent measurements that arise frequently from various applications from genomics and neuroscience to economics and finance. As data are collected at an ever-growing scale, statistical machine learning faces some new challenges: high dimensionality, strong dependence among observed variables, heavy-tailed variables and heterogeneity. High-dimensional robust factor analysis serves as a powerful toolkit to conquer these challenges.