[Paper Review] Halpern Iteration for Near-Optimal and Parameter-Free Monotone Inclusion and Strong Solutions to Variational Inequalities

Jelena Diakonikolas|arXiv (Cornell University)|Feb 20, 2020

Optimization and Variational Analysis32 references4 citations

TL;DR

This paper presents near-optimal, parameter-free algorithms for monotone inclusion and variational inequality problems using Halpern iteration, achieving optimal convergence rates up to logarithmic factors. By leveraging connections between nonexpansive maps, monotone operators, and proximal mappings, it establishes tight iteration complexity bounds and proves optimality via reductions and lower bounds.

ABSTRACT

We leverage the connections between nonexpansive maps, monotone Lipschitz operators, and proximal mappings to obtain near-optimal (i.e., optimal up to poly-log factors in terms of iteration complexity) and parameter-free methods for solving monotone inclusion problems. These results immediately translate into near-optimal guarantees for approximating strong solutions to variational inequality problems, approximating convex-concave min-max optimization problems, and minimizing the norm of the gradient in min-max optimization problems. Our analysis is based on a novel and simple potential-based proof of convergence of Halpern iteration, a classical iteration for finding fixed points of nonexpansive maps. Additionally, we provide a series of algorithmic reductions that highlight connections between different problem classes and lead to lower bounds that certify near-optimality of the studied methods.

Motivation & Objective

• To develop parameter-free, near-optimal algorithms for solving monotone inclusion problems with Lipschitz-continuous operators.
• To establish tight iteration complexity bounds for variational inequality and min-max optimization problems via Halpern iteration.
• To demonstrate optimality of the proposed methods through algorithmic reductions and lower bounds in the oracle model.
• To clarify the relationship between monotone inclusion, fixed points of nonexpansive maps, and proximal-point methods.
• To close the gap between upper and lower bounds for monotone inclusion under various operator conditions.

Proposed method

• Uses Halpern iteration as the core algorithmic framework: $\mathbf{u}_{k+1} = \lambda_{k+1}\mathbf{u}_0 + (1 - \lambda_{k+1})T(\mathbf{u}_k)$, where $T$ is nonexpansive.
• Establishes convergence via a novel potential-based proof for Halpern iteration, ensuring convergence to the closest fixed point in $\ell_2$-norm.
• Reduces monotone inclusion problems to fixed-point problems of nonexpansive maps using proximal mappings and operator lifting.
• Applies algorithmic reductions to connect monotone inclusion to variational inequalities, min-max optimization, and gradient norm minimization.
• Derives lower bounds in the operator oracle model to certify near-optimality of the proposed methods under different operator classes.
• Uses restarting and adaptive step-size strategies to achieve optimal complexity in strongly monotone and cocoercive settings.

Experimental results

Research questions

• RQ1 Can Halpern iteration be adapted to yield near-optimal, parameter-free algorithms for monotone inclusion with Lipschitz operators?
• RQ2 What is the optimal iteration complexity for solving monotone inclusion in the $L$-Lipschitz, $m$-strongly monotone, and $\frac{1}{L}$-cocoercive settings?
• RQ3 How do the proposed algorithms compare to existing methods in terms of convergence guarantees and iteration complexity?
• RQ4 Can tight lower bounds be derived to certify the optimality of the proposed methods in the oracle model?
• RQ5 What is the relationship between monotone inclusion, fixed points of nonexpansive maps, and proximal-point methods?

Key findings

• The proposed Halpern-based algorithm achieves $O\left(\frac{LD^2}{\epsilon}\log\left(\frac{LD}{\epsilon}\right)\right)$ iteration complexity for $L$-Lipschitz operators, matching the lower bound up to a logarithmic factor.
• For $\frac{1}{L}$-cocoercive operators, the method attains $O\left(\frac{LD}{\epsilon}\log\left(\frac{LD}{\epsilon}\right)\right)$ complexity, matching the lower bound up to $\log(D/\epsilon)$.
• In the $m$-strongly monotone case, the restarting variant achieves $O\left(\frac{L}{m}\log\left(\frac{LD}{\epsilon}\right)\log\left(\frac{L}{m}\right)\right)$ complexity, matching the lower bound up to logarithmic factors.
• The paper proves that the Halpern iteration converges to the $\ell_2$-closest fixed point of a nonexpansive map under mild conditions on $\lambda_k$.
• Lower bounds show that the proposed methods are optimal up to logarithmic factors in all considered settings.
• The results imply near-optimal guarantees for approximating strong solutions to variational inequalities and minimizing the gradient norm in min-max optimization.