Provable Parameter-Free Fixed-Point Algorithms with Linear Convergence Rates

2026-08-09

AI Summary

This work addresses the challenge of computing fixed points of contractive mappings without requiring prior knowledge of the contraction factor or manual hyperparameter tuning. We propose a fully adaptive Halpern-type algorithm that operates without line searches, bisection procedures, or any user-specified parameters, automatically exploiting the inherent contractiveness of the mapping to achieve explicit linear convergence even in the absence of a priori estimates of the contraction constant. Theoretical analysis establishes linear convergence rates both in terms of fixed-point residuals and distance to the solution, with an iteration complexity of 𝒪(ε⁻¹ln(ε⁻¹)) for solving cocoercive equations. By integrating Tikhonov regularization and Nesterov acceleration, we further extend the algorithm’s applicability. Numerical experiments confirm its superiority over existing adaptive methods, offering strong theoretical guarantees alongside low computational overhead.

Abstract

In this paper, we develop provable parameter-free and adaptive fixed-point algorithms for contractive mappings, with an emphasis on automatically exploiting hidden contractivity without requiring prior knowledge of the contraction factor. Our first method is a completely parameter-free variant of the Halpern fixed-point iteration. It requires no line search, bisection, or prior estimate of the contraction factor, while retaining essentially the same per-iteration computational cost as classical fixed-point schemes. We establish explicit linear convergence rates for both the fixed-point residual and the distance to the unique fixed point. The second algorithm is an adaptive Halpern method that requires only an upper bound on the contraction factor and reduces to an existing adaptive Halpern scheme in the nonexpansive case. This method also enjoys explicit linear convergence guarantees. We further extend these ideas in two directions. First, by combining the proposed fixed-point schemes with Tikhonov regularization, we obtain a parameter-free method for solving co-coercive equations and establish an iteration complexity of $\mathcal{O}({ε^{-1}\ln(ε^{-1})})$ for computing an $ε$-solution. Second, using the relation between Halpern iterations and Nesterov's accelerated fixed-point schemes, we derive parameter-free Nesterov's accelerated variants that inherit linear convergence in the contractive setting. Numerical experiments on several examples demonstrate that the proposed algorithms are competitive with, and often outperform, existing adaptive fixed-point methods. In particular, the methods successfully exploit contractive behavior when it is present while remaining effective on nonexpansive problems.

Problem

• fixed-point algorithms
• parameter-free
• contractive mappings
• linear convergence
• adaptive methods

Innovation

• parameter-free
• fixed-point algorithms
• linear convergence
• adaptive methods
• Halpern iteration