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.

翻译：我们提出了一个新的框架来设计和分析求解一般单调方程（ME）问题 $F(x)=0$ 的加速方法。传统方法包括广义最速下降法和不精确牛顿型方法。若 $F$ 为一致单调且二阶可微，这些方法可实现局部收敛速率，而后一类方法由于线搜索和超平面投影而具有全局收敛性。然而，这些方法的全局收敛速率尚属未知。变分不等式方法可用于推导以 $\|F(x)\|$ 表示的全局速率，但这些结果仅限于一阶方法和 Lipschitz 连续算子。如何利用高阶 Lipschitz 连续性获得全局加速尚不明确。本文采用连续时间视角，将加速方法视为动力系统的离散化。我们的贡献在于提出加速重缩放梯度系统，并证明其等价于闭环控制系统。基于这一联系，我们建立了解轨迹的性质。此外，我们提供了一个通过系统离散化获得的统一算法框架，该框架与两个近似子程序相结合，既可导出已有的高阶方法，也能产生新的一阶方法。我们证明，若 $F$ 是 $p$ 阶 Lipschitz 连续的，则 $p$ 阶方法在 $\|F(x)\|$ 意义下可实现 $O(k^{-p/2})$ 的全局收敛速率；若 $F$ 是 $p$ 阶强 Lipschitz 连续的，则一阶方法也能达到相同速率。若 $F$ 是强单调的，当 $p \geq 2$ 时，其重启版本可实现 $p$ 阶局部收敛。我们的离散时间分析在很大程度上受连续时间分析的启发，并证明了重缩放梯度在求解 ME 问题的全局加速中发挥的根本性作用。