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

The paper addresses solving monotone operator equations $M(x)=0$ in real Hilbert spaces by augmenting first-order flows with a time-varying Tikhonov regularization $oldsymbol{\varepsilon}(t)$. It develops a unified theory linking continuous-time flows, time-rescaled and anchor-point variants, and second-order dynamics with vanishing damping, establishing strong convergence to the minimum-norm zero $oldsymbol{\xi_*}$ and explicit rates for $ig\|\dot{x}(t)ig\|$ and $igig\|M(x(t))igig angle$ governed by $oldsymbol{\varepsilon}(t)$ and the integrating factor $oldsymbol{\gamma_{oldsymbol{\varepsilon}}(t)}$, with $O(1/t)$ rates in key cases. The work also connects these continuous dynamics to Halpern-type fixed-point discretizations, proving strong convergence to the fixed point closest to the anchor and providing a broad rate framework for the discrete velocity and residual; it further extends to averaged operators and links to Fast OGDA. Altogether, the results reveal an acceleration effect of Tikhonov regularization in monotone flows and offer practical continuous-discrete methods for strongly convergent fixed-point computations.

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.

Theorems & Definitions (60)

• Remark 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Theorem 2.4: the case $\varepsilon \left( t \right) \coloneq \frac{1}{t}$
• proof
• Remark 2.5
• Proposition 2.6
• proof