Tikhonov regularization of monotone operator flows not only ensures strong convergence of the trajectories but also speeds up the vanishing of the residuals

Radu Ioan Bot, Dang-Khoa Nguyen

Published 2024-06-02Version 1

In the framework of real Hilbert spaces, we investigate first-order dynamical systems governed by monotone and continuous operators. It has been established that for these systems, only the ergodic trajectory converges to a zero of the operator. A notable example is the counterclockwise $\pi/2$-rotation operator on $\mathbb{R}^2$, which illustrates that general trajectory convergence cannot be expected. However, trajectory convergence is assured for operators with the stronger property of cocoercivity. For this class of operators, the trajectory's velocity and the opertor values along the trajectory converge in norm to zero at a rate of $o(\frac{1}{\sqrt{t}})$ as $t \rightarrow +\infty$. In this paper, we demonstrate that when the monotone operator flow is augmented with a Tikhonov regularization term, the resulting trajectory converges strongly to the element of the set of zeros with minimal norm. In addition, rates of convergence in norm for the trajectory's velocity and the operator along the trajectory can be derived in terms of the regularization function. In some particular cases, these rates of convergence can outperform the ones of the coercive operator flows and can be as fast as $O(\frac{1}{t})$ as $t \rightarrow +\infty$. In this way, we emphasize a surprising acceleration feature of the Tikhonov regularization. Additionally, we explore these properties for monotone operator flows that incorporate time rescaling and an anchor point. For a specific choice of the Tikhonov regularization function, these flows are closely linked to second-order dynamical systems with a vanishing damping term. The convergence and convergence rate results we achieve for these systems complement recent findings for the Fast Optimistic Gradient Descent Ascent (OGDA) dynamics, leading to surprising outcomes.