On Landscape of Lagrangian Functions and Stochastic Search for Constrained Nonconvex Optimization

TL;DR

This paper explores the complex landscape of Lagrangian functions in constrained nonconvex optimization, characterizes stable equilibria, and introduces an efficient stochastic primal-dual algorithm with theoretical convergence guarantees for generalized eigenvalue problems.

Contribution

It characterizes the landscape of Lagrangian functions, defines a class with stable equilibria corresponding to global optima, and proposes a novel stochastic primal-dual algorithm with convergence analysis.

Findings

Characterized stable and unstable equilibria of the Lagrangian landscape.

Proposed an efficient stochastic primal-dual algorithm for online GEV problems.

Established the first sample complexity result for the online GEV problem.

Abstract

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.