# From Halpern's Fixed-Point Iterations to Nesterov's Accelerated Interpretations for Root-Finding Problems

arXiv: https://arxiv.org/abs/2203.04869 · HTML: https://mlsanity.com/paper/2203.04869/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Mar 2022.

## Built on by (most cited first, 20 of 20)

- [Accelerated Single-Call Methods for Constrained Min-Max Optimization](https://mlsanity.com/paper/2210.03096/) (2022)
- [Accelerated Algorithms for Constrained Nonconvex-Nonconcave Min-Max Optimization and Comonotone Inclusion](https://mlsanity.com/paper/2206.05248/) (2022)
- [Towards Weaker Variance Assumptions for Stochastic Optimization](https://mlsanity.com/paper/2504.09951/) (2025)
- [Doubly Optimal No-Regret Learning in Monotone Games](https://mlsanity.com/paper/2301.13120/) (2023)
- [Randomized Block-Coordinate Optimistic Gradient Algorithms for Root-Finding Problems](https://mlsanity.com/paper/2301.03113/) (2023)
- [Sublinear Convergence Rates of Extragradient-Type Methods: A Survey on Classical and Recent Developments](https://mlsanity.com/paper/2303.17192/) (2023)
- [Direct Acceleration of Stochastic Root-Finding Without Variance Reduction and Regularization](https://mlsanity.com/paper/2608.12043/) (2026)
- [Variance-Reduced Fast Krasnoselkii-Mann Methods for Finite-Sum Root-Finding Problems](https://mlsanity.com/paper/2406.02413/) (2024)
- [An Inexact Halpern Iteration with Application to Distributionally Robust Optimization](https://mlsanity.com/paper/2402.06033/) (2024)
- [Primal-dual extrapolation methods for monotone inclusions under local Lipschitz continuity](https://mlsanity.com/paper/2206.00973/) (2022)
- [Gradient Descent-Type Methods: Background and Simple Unified Convergence Analysis](https://mlsanity.com/paper/2212.09413/) (2022)
- [A Class of Accelerated Fixed-Point-Based Methods with Delayed Inexact Oracles and Its Applications](https://mlsanity.com/paper/2512.13547/) (2025)
- [RAMPAGE: RAndomized Mid-Point for debiAsed Gradient Extrapolation](https://mlsanity.com/paper/2603.22155/) (2026)
- [Extragradient-Type Methods with O (1/k) Last-Iterate Convergence Rates for Co-Hypomonotone Inclusions](https://mlsanity.com/paper/2302.04099/) (2023)
- [HPPP: Halpern-type Preconditioned Proximal Point Algorithms and Applications to Image Restoration](https://mlsanity.com/paper/2407.13120/) (2024)
- [Control and Reinforcement Learning through the Lens of Optimization: An Algorithmic Perspective](https://mlsanity.com/paper/2512.07377/) (2025)
- [Accelerating Min-Max Optimization via Power-Law Stepsizes](https://mlsanity.com/paper/2606.01764/) (2026)
- [On Same-Sample and Independent-Sample Stochastic Extragradient for Monotone Variational Inequalities](https://mlsanity.com/paper/2608.06182/) (2026)
- [Provable Parameter-Free Fixed-Point Algorithms with Linear Convergence Rates](https://mlsanity.com/paper/2608.09022/) (2026)
- [Polyak-Type Extragradient Methods for Monotone Root-Finding Problems](https://mlsanity.com/paper/2609.26581/) (2026)

## Abstract (arXiv)

We derive an equivalent form of Halpern's fixed-point iteration scheme for solving a co-coercive equation (also called a root-finding problem), which can be viewed as a Nesterov's accelerated interpretation. We show that one method is equivalent to another via a simple transformation, leading to a straightforward convergence proof for Nesterov's accelerated scheme. Alternatively, we directly establish convergence rates of Nesterov's accelerated variant, and as a consequence, we obtain a new convergence rate of Halpern's fixed-point iteration. Next, we apply our results to different methods to solve monotone inclusions, where our convergence guarantees are applied. Since the gradient/forward scheme requires the co-coerciveness of the underlying operator, we derive new Nesterov's accelerated variants for both recent extra-anchored gradient and past-extra anchored gradient methods in the literature. These variants alleviate the co-coerciveness condition by only assuming the monotonicity and Lipschitz continuity of the underlying operator. Interestingly, our new Nesterov's accelerated interpretation of the past-extra anchored gradient method involves two past-iterate correction terms. This formulation is expected to guide us developing new Nesterov's accelerated methods for minimax problems and their continuous views without co-coericiveness. We test our theoretical results on two numerical examples, where the actual convergence rates match well the theoretical ones up to a constant factor.