# Halpern Iteration for Near-Optimal and Parameter-Free Monotone Inclusion and Strong Solutions to Variational Inequalities

@article{Diakonikolas2020HalpernIF, title={Halpern Iteration for Near-Optimal and Parameter-Free Monotone Inclusion and Strong Solutions to Variational Inequalities}, author={Jelena Diakonikolas}, journal={ArXiv}, year={2020}, volume={abs/2002.08872} }

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…

## 24 Citations

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A new approach is developed that combines the power of the sum-of-squares programming with the low dimensionality of the update rule of the extragradient method and establishes the monotonicity of a new performance measure – the tangent residual.

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A new type of accelerated algorithms to solve some classes of maximally monotones equations as well as monotone inclusions using a so-called Halpern-type fixed-point iteration to solve convex-concave minimax problems and a new accelerated DR scheme to derive a new variant of the alternating direction method of multipliers (ADMM).

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Abstract Halpern’s iteration method, discovered by Halpern in 1967, is an iterative algorithm for finding fixed points of a nonexpansive mapping in Hilbert and Banach spaces. Since many optimization…

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- Computer ScienceUAI
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The complexity of the NC-SC smooth minimax problems is studied in both general and averaged smooth finite-sum settings, and a generic acceleration scheme is introduced that deploys existing gradient-based methods to solve a sequence of crafted strongly-convexstrongly-concave subproblems.

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- Computer ScienceArXiv
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A novel potential function-based framework to study the convergence of standard methods for making the gradients small in smooth convex optimization and convex-concave min-max optimization and provides a new lower bound for minimizing norm of cocoercive operators that allows us to argue about optimality of methods in the min- max setup.

Convergence of Adaptive Methods for Equilibrium Problems in Hadamard Spaces

- MathematicsIT&I Workshops
- 2020

New adaptive algorithms for pseudomonotone bifunctions of Lipschitz type are proposed and theorems on the weak convergence of sequences generated by the algorithms are proved.

On the Initialization for Convex-Concave Min-max Problems

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It is shown that strict-convexitystrict-concavity is sufficient to get the convergence rate to depend on the initialization and that the so-called “parameter-free” algorithms allow to achieve improved initialization-dependent asymptotic rates without any learning rate to tune.

The Connection Between Nesterov's Accelerated Methods and Halpern Fixed-Point Iterations

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We derive a direct connection between Nesterov’s accelerated first-order algorithm and the Halpern fixed-point iteration scheme for approximating a solution of a co-coercive equation. We show that…

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