A Continuous-Time Perspective on Global Acceleration for Monotone Equation Problems

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 ≥ 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.