Capturing Acceleration in Monotone Inclusion: A Closed-Loop Control Perspective

We propose and analyze a new dynamical system with a closed-loop control law in a Hilbert space 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 : H ⇒ H that is maximal monotone, we study a closed-loop control system that is governed by the operator I − (I + λ(t)A) where a feedback law λ(·) is tuned by the resolution of the algebraic equation λ(t)‖(I + λ(t)A)x(t) − x(t)‖p−1 = θ for some θ ∈ (0, 1). Our first contribution in this paper is to prove the existence and uniqueness of a global solution via the Cauchy-Lipschitz theorem. We present a simple Lyapunov function that allows for establishing the weak convergence of trajectories via the Opial lemma and strong convergence results under additional conditions. We establish a global ergodic convergence rate of O(t) in terms of a gap function and a global pointwise convergence rate of O(t) in terms of a residue function. Local linear convergence is established in terms of a distance function under an error bound condition. Our second contribution is to provide an algorithmic framework based on the implicit discretization of our closed-loop control system in a Euclidean setting, generalizing the large-step HPE framework of Monteiro and Svaiter [2012]. Even though the discrete-time analysis is a simplification and generalization of the analysis in Monteiro and Svaiter [2012] for bounded domain, it is 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 collection of new results concerning p-th order tensor algorithms for monotone inclusion problems, which complement the recent analysis in Bullins and Lai [2020] for saddle point and VI problems.