On Constrained Nonconvex Stochastic Optimization: A Case Study for Generalized Eigenvalue Decomposition

We study constrained nonconvex optimization problems in machine learning and signal processing. It is well-known that these problems can be rewritten to a min-max problem in a Lagrangian form. However, due to the lack of convexity, their landscape is not well understood and how to find the stable equilibria of the Lagrangian function is still unknown. To bridge the gap, we study the landscape of the Lagrangian function. Further, we define a special class of Lagrangian functions. They enjoy the following two properties: 1. Equilibria are either stable or unstable (Formal definition in Section 2); 2.Stable equilibria correspond to the global optima of the original problem. We show that a generalized eigenvalue (GEV) problem, including canonical correlation analysis and other problems as special examples, belongs to the class. Specifically, we characterize its stable and unstable equilibria by leveraging an invariant group and symmetric property (more details in Section 3). Motivated by these neat geometric structures, we propose a simple, efficient, and stochastic primal-dual algorithm solving the online GEV problem. Theoretically, under sufficient conditions, we establish an asymptotic rate of convergence and obtain the first sample complexity result for the online GEV problem by diffusion approximations, which are widely used in applied probability. Numerical results are also provided to support our theory.