A Continuous-Time Perspective on Global Acceleration for Monotone Equation Problems
Tianyi Lin, Michael. I. Jordan
TL;DR
This paper tackles global acceleration for monotone equation problems by introducing accelerated rescaled gradient systems and drawing a deep connection to closed-loop control. It develops a unified framework that yields both high-order and simple first-order discrete-time methods with provable global rates measured in the residue $\|F(x)\|$, specifically $O(k^{-p/2})$ when $F$ is $p$-th order Lipschitz and under strong Lipschitz conditions for first-order schemes. The main theoretical contributions include global existence and uniqueness of continuous trajectories, asymptotic convergence results, a Lyapunov-based nonasymptotic rate, and discrete-time convergence proofs that mirror the continuous-time analysis. Practically, the framework unifies existing high-order ME methods and produces new lightweight first-order schemes, with restarted variants offering local superlinear convergence when $p\ge2$. The work highlights the pivotal role of rescaled gradients in global acceleration for ME problems and points to further connections with Lagrangian/Hamiltonian communities for future research.
Abstract
We propose a new framework to design and analyze accelerated methods that solve general monotone equation (ME) problems $F(x)=0$. Traditional approaches include generalized steepest descent methods and inexact Newton-type methods. If $F$ is uniformly monotone and twice differentiable, these methods achieve local convergence rates while the latter methods are globally convergent thanks to line search and hyperplane projection. However, a global rate is unknown for these methods. The variational inequality methods can be applied to yield a global rate that is expressed in terms of $\|F(x)\|$ but these results are restricted to first-order methods and a Lipschitz continuous operator. It has not been clear how to obtain global acceleration using high-order Lipschitz continuity. This paper takes a continuous-time perspective where accelerated methods are viewed as the discretization of dynamical systems. Our contribution is to propose accelerated rescaled gradient systems and prove that they are equivalent to closed-loop control systems. Based on this connection, we establish the properties of solution trajectories. Moreover, we provide a unified algorithmic framework obtained from discretization of our system, which together with two approximation subroutines yields both existing high-order methods and new first-order methods. We prove that the $p^{th}$-order method achieves a global rate of $O(k^{-p/2})$ in terms of $\|F(x)\|$ if $F$ is $p^{th}$-order Lipschitz continuous and the first-order method achieves the same rate if $F$ is $p^{th}$-order strongly Lipschitz continuous. If $F$ is strongly monotone, the restarted versions achieve local convergence with order $p$ when $p \geq 2$. Our discrete-time analysis is largely motivated by the continuous-time analysis and demonstrates the fundamental role that rescaled gradients play in global acceleration for solving ME problems.