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

TL;DR

This paper shows that Tikhonov regularization applied to monotone operator flows in Hilbert spaces guarantees strong convergence to minimal norm solutions and accelerates residual decay, with implications for fixed point algorithms.

Contribution

It introduces a novel analysis of Tikhonov regularization effects on monotone flows, revealing acceleration and convergence rate improvements over existing methods.

Findings

Strong convergence to minimal norm solutions is achieved.

Residuals decay at rates up to O(1/t), surpassing some existing flows.

Discrete algorithms like Halpern iteration benefit from the regularization.

Abstract

In the framework of real Hilbert spaces, we investigate first-order dynamical systems governed by monotone and continuous operators. 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 and show that they are closely linked to second-order dynamics 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. When the monotone operator is defined as the identity minus a nonexpansive operator, the monotone equations transform into a fixed point problem. In such cases, explicitly discretizing the system with Tikhonov regularization, enhanced by an anchor point, leads to the Halpern fixed point iteration. We identify two regimes for the regularization sequence which ensure that the generated sequence of iterates converges strongly to the fixed point nearest to the anchor point. Furthermore, we establish a general theoretical framework that provides convergence rates for the vanishing of the discrete velocity and the fixed point residual. For certain regularization sequences, we derive specific convergence rates that align with those observed in continuous time.