  We study constrained nonconvex optimization problems in machine learning,
signal processing, and stochastic control. It is well-known that these problems
can be rewritten to a minimax 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 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, 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, we provide
sufficient conditions, based on which we establish an asymptotic convergence
rate and obtain the first sample complexity result for the online GEV problem
by diffusion approximations, which are widely used in applied probability and
stochastic control. Numerical results are provided to support our theory.
