Monotone Inclusions, Acceleration, and Closed-Loop Control

We propose and analyze a new dynamical system with a closed-loop control law in a Hilbert space $\mathcal{H}$, aiming to shed light on the acceleration phenomenon for monotone inclusion problems, which unifies a broad class of optimization, saddle point, and variational inequality (VI) problems under a single framework. Given an operator $A:\mathcal{H}\rightrightarrows \mathcal{H}$ that is maximal monotone, we propose a closed-loop control system that is governed by the operator $I-{(I+\lambda (t)A)}^{-1}$, where a feedback law $\lambda (·)$ is tuned by the resolution of the algebraic equation $\lambda (t)\Vert {(I+\lambda (t)A)}^{-1}x(t)-x(t){\Vert }^{p-1}=\theta$ for some $\theta >0$. Our first contribution is to prove the existence and uniqueness of a global solution via the Cauchy–Lipschitz theorem. We present a simple Lyapunov function for establishing the weak convergence of trajectories via the Opial lemma and strong convergence results under additional conditions. We then prove a global ergodic convergence rate of $O(t^{-(p+1)/2})$ in terms of a gap function and a global pointwise convergence rate of $O(t^{-p/2})$ in terms of a residue function. Local linear convergence is established in terms of a distance function under an error bound condition. Further, we provide an algorithmic framework based on the implicit discretization of our system in a Euclidean setting, generalizing the large-step hybrid proximal extragradient framework. Even though the discrete-time analysis is a simplification and generalization of existing analyses for a bounded domain, it is largely motivated by the aforementioned continuous-time analysis, illustrating the fundamental role that the closed-loop control plays in acceleration in monotone inclusion. A highlight of our analysis is a new result concerning $p\text{th}$-order tensor algorithms for monotone inclusion problems, complementing the recent analysis for saddle point and VI problems.

Funding: This work was supported in part by the Mathematical Data Science Program of the Office of Naval Research [Grant N00014-18-1-2764] and by the Vannevar Bush Faculty Fellowship Program [Grant N00014-21-1-2941].