On exploration of an interior mirror descent flow for stochastic nonconvex constrained problem

TL;DR

This paper introduces a continuous Riemannian subgradient flow for nonsmooth nonconvex constrained optimization, unifying and analyzing the behavior of Hessian barrier and mirror descent methods, and proposing new algorithms with improved convergence properties.

Contribution

It formulates a novel interior mirror descent flow as a differential inclusion, unifies existing methods, and offers new insights and algorithms for nonconvex constrained optimization.

Findings

The continuous flow explains the convergence and spurious stationary points of existing methods.

Conditions are provided to avoid spurious stationary points under strict complementarity.

A random perturbation strategy ensures convergence to approximate stationary points.

Abstract

We study a nonsmooth nonconvex optimization problem defined over nonconvex constraints, where the feasible set is given by the intersection of the closure of an open set and a smooth manifold. By endowing the open set with a Riemannian metric induced by a barrier function, we obtain a Riemannian subgradient flow formulated as a differential inclusion, which remains strictly within the interior of the feasible set. This continuous dynamical system unifies two classes of iterative optimization methods, namely the Hessian barrier method and mirror descent scheme, by revealing that these methods can be interpreted as discrete approximations of the continuous flow. We explore the long-term behavior of the trajectories generated by this dynamical system and show that the existing deficient convergence properties of the Hessian barrier and mirror descent scheme can be unifily and more insightfully interpreted through these of the continuous trajectory. For instance, the notorious spurious stationary points \cite{chen2024spurious} observed in Hessian barrier method and mirror descent scheme are interpreted as stable equilibria of the dynamical system that do not correspond to real stationary points of the original optimization problem. We provide two sufficient condition such that these spurious stationary points can be avoided if the strict complementarity conditions holds. In the absence of these regularity condition, we propose a random perturbation strategy that ensures the trajectory converges (subsequentially) to an approximate stationary point. Building on these insights, we introduce two iterative Riemannian subgradient methods, form of interior point methods, that generalizes the existing Hessian barrier method and mirror descent scheme for solving nonsmooth nonconvex optimization problems.