Nonlinearly preconditioned gradient flows

TL;DR

This paper analyzes the continuous-time dynamics of nonlinearly preconditioned gradient methods, defined by $\dot x = -\nabla \phi^*(\nabla f(x))$, and establishes global well-posedness and Lyapunov-based convergence under mild assumptions. For convex $f$, it derives sublinear convergence via a Lyapunov function and, under an anisotropic gradient-dominance (a generalized PL) condition, exponential convergence in function values. A duality with mirror descent is uncovered, linking the flow to a mirror-flow in dual variables and yielding an infinite-horizon optimal-control formulation with the value function $V(x_0)=D_f(x_0, y^{*})$. The results connect non-Euclidean optimization flows with known continuous-time models, providing a structured, control-theoretic lens for analyzing anisotropically preconditioned gradient methods and their convergence behavior. These insights offer rigorous rate guarantees and a theoretical bridge to mirror-descent-based analyses and non-Euclidean geometry in optimization.

Abstract

We study a continuous-time dynamical system which arises as the limit of a broad class of nonlinearly preconditioned gradient methods. Under mild assumptions, we establish existence of global solutions and derive Lyapunov-based convergence guarantees. For convex costs, we prove a sublinear decay in a geometry induced by some reference function, and under a generalized gradient-dominance condition we obtain exponential convergence. We further uncover a duality connection with mirror descent, and use it to establish that the flow of interest solves an infinite-horizon optimal-control problem of which the value function is the Bregman divergence generated by the cost. These results clarify the structure and optimization behavior of nonlinearly preconditioned gradient flows and connect them to known continuous-time models in non-Euclidean optimization.

Figures (3)

• Figure 1: Preconditioners corresponding to different reference functions.
• Figure 2: Visualization of a mirror descent update.
• Figure 3: Visualization of a nonlinearly preconditioned gradient method update.

Theorems & Definitions (12)

• Remark II.2
• Proposition II.3
• proof
• Theorem II.4
• proof
• Remark II.5
• Proposition II.6
• proof
• Remark II.7
• Remark III.2