Zhongruo Wang

Zhongruo Wang, Bingyuan Liu, Shixiang Chen et al.

Spectral clustering is one of the fundamental unsupervised learning methods widely used in data analysis. Sparse spectral clustering (SSC) imposes sparsity to the spectral clustering and it improves the interpretability of the model. This paper considers a widely adopted model for SSC, which can be formulated as an optimization problem over the Stiefel manifold with nonsmooth and nonconvex objective. Such an optimization problem is very challenging to solve. Existing methods usually solve its convex relaxation or need to smooth its nonsmooth part using certain smoothing techniques. In this paper, we propose a manifold proximal linear method (ManPL) that solves the original SSC formulation. We also extend the algorithm to solve the multiple-kernel SSC problems, for which an alternating ManPL algorithm is proposed. Convergence and iteration complexity results of the proposed methods are established. We demonstrate the advantage of our proposed methods over existing methods via the single-cell RNA sequencing data analysis.

Zhongruo Wang, Krishnakumar Balasubramanian, Shiqian Ma et al.

In this paper, we study zeroth-order algorithms for minimax optimization problems that are nonconvex in one variable and strongly-concave in the other variable. Such minimax optimization problems have attracted significant attention lately due to their applications in modern machine learning tasks. We first consider a deterministic version of the problem. We design and analyze the Zeroth-Order Gradient Descent Ascent (\texttt{ZO-GDA}) algorithm, and provide improved results compared to existing works, in terms of oracle complexity. We also propose the Zeroth-Order Gradient Descent Multi-Step Ascent (\texttt{ZO-GDMSA}) algorithm that significantly improves the oracle complexity of \texttt{ZO-GDA}. We then consider stochastic versions of \texttt{ZO-GDA} and \texttt{ZO-GDMSA}, to handle stochastic nonconvex minimax problems. For this case, we provide oracle complexity results under two assumptions on the stochastic gradient: (i) the uniformly bounded variance assumption, which is common in traditional stochastic optimization, and (ii) the Strong Growth Condition (SGC), which has been known to be satisfied by modern over-parametrized machine learning models. We establish that under the SGC assumption, the complexities of the stochastic algorithms match that of deterministic algorithms. Numerical experiments are presented to support our theoretical results.