Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pushing the Complexity Boundaries of Fixed-Point Equations: Adaptation to Contraction and Controlled Expansion

Jelena Diakonikolas

TL;DR

This work studies ε-approximate fixed points of Lipschitz operators on general normed and geodesic spaces, focusing on mildly expansive regimes where γ>1. It shows that a fixed-step Halpern iteration can achieve provable convergence guarantees by carefully selecting or adapting the step size λ, and interprets the iteration as a resolvent/regularization scheme. Building on this, the authors introduce adaptive algorithms (GHAL and AdaGHAL) that unify contractive, nonexpansive, and mildly/gradually expansive settings, achieving near-optimal oracle complexity across regimes and extending to a new class of α-gradually expansive operators with expansion tied to fixed-point error. The results bridge fixed-point computation and hardness bounds, provide practical adaptive methods, and extend to infinite-dimensional Banach spaces and non-positively curved geodesic spaces, offering a broad, robust framework for fixed-point problems with Lipschitz operators.

Abstract

Fixed-point equations with Lipschitz operators have been studied for more than a century, and are central to problems in mathematical optimization, game theory, economics, and dynamical systems, among others. When the Lipschitz constant of the operator is larger than one (i.e., when the operator is expansive), it is well known that approximating fixed-point equations becomes computationally intractable even in basic finite-dimensional settings. In this work, we aim to push these complexity boundaries by introducing algorithms that can address problems with mildly expansive (i.e., with Lipschitz constant slightly larger than one) operators not excluded by existing lower bounds, attaining the best possible fixed-point error up to universal constants. We further introduce a class of \emph{gradually expansive operators} that allow for constant (up to $\approx 1.4$) expansion between points, for which we prove convergence to $ε$-approximate fixed points in order-$(1/ε)$ iterations for $ε> 0.$ Our algorithms automatically adapt to the Lipschitz constant of the operator and attain optimal oracle complexity bounds when the input operator is nonexpansive or contractive. Our results apply to general, possibly infinite-dimensional normed vector spaces and can be extended to non-positively curved geodesic metric spaces.

Pushing the Complexity Boundaries of Fixed-Point Equations: Adaptation to Contraction and Controlled Expansion

TL;DR

This work studies ε-approximate fixed points of Lipschitz operators on general normed and geodesic spaces, focusing on mildly expansive regimes where γ>1. It shows that a fixed-step Halpern iteration can achieve provable convergence guarantees by carefully selecting or adapting the step size λ, and interprets the iteration as a resolvent/regularization scheme. Building on this, the authors introduce adaptive algorithms (GHAL and AdaGHAL) that unify contractive, nonexpansive, and mildly/gradually expansive settings, achieving near-optimal oracle complexity across regimes and extending to a new class of α-gradually expansive operators with expansion tied to fixed-point error. The results bridge fixed-point computation and hardness bounds, provide practical adaptive methods, and extend to infinite-dimensional Banach spaces and non-positively curved geodesic spaces, offering a broad, robust framework for fixed-point problems with Lipschitz operators.

Abstract

Fixed-point equations with Lipschitz operators have been studied for more than a century, and are central to problems in mathematical optimization, game theory, economics, and dynamical systems, among others. When the Lipschitz constant of the operator is larger than one (i.e., when the operator is expansive), it is well known that approximating fixed-point equations becomes computationally intractable even in basic finite-dimensional settings. In this work, we aim to push these complexity boundaries by introducing algorithms that can address problems with mildly expansive (i.e., with Lipschitz constant slightly larger than one) operators not excluded by existing lower bounds, attaining the best possible fixed-point error up to universal constants. We further introduce a class of \emph{gradually expansive operators} that allow for constant (up to ) expansion between points, for which we prove convergence to -approximate fixed points in order- iterations for Our algorithms automatically adapt to the Lipschitz constant of the operator and attain optimal oracle complexity bounds when the input operator is nonexpansive or contractive. Our results apply to general, possibly infinite-dimensional normed vector spaces and can be extended to non-positively curved geodesic metric spaces.

Paper Structure

This paper contains 29 sections, 12 theorems, 70 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions and Paper Overview
  3. Fixed-step Halpern iteration as a surprising algorithmic tool
  4. Adaptive algorithms (GHAL and AdaGHAL)
  5. A new tractable class: gradually expansive operators
  6. Generality and robustness
  7. Further Notes on Related Work
  8. Convergence of the Fixed-Step Halpern Iteration
  9. Convergence for Nonexpansive Operators
  10. Convergence for Mildly Expansive Operators
  11. Finite-dimensional vector space
  12. Infinite-dimensional space.
  13. Extension to Geodesic Metric Spaces
  14. Connection to Resolvents and Regularization
  15. Gradual Halpern Algorithm and its Adaptive Version
  16. ...and 14 more sections

Key Result

Lemma 1

Given ${\bm{x}}_0 \in {\mathcal{E}}$ and iterates ${\bm{x}}_{k+1}$ defined by eq:main-iteration for a $\gamma$-Lipschitz operator ${\bm{T}}$ with $\gamma > 0,$ we have that for all $k \geq 1,$ If $\gamma \in (0, 1]$ and ${\bm{x}}_*$ is a fixed point of ${\bm{T}},$ then we further have As a consequence, for $\gamma \in (0, 1]$ and given any $\epsilon > 0,$ if $\lambda = \frac{\epsilon}{4D_* + \eps

Figures (4)

  • Figure 1: Comparison of fixed point algorithms for the operator defined by \ref{['eq:hard-rotation-op']} with $d = 500$.
  • Figure 2: Comparison of fixed point algorithms for an operator that is contractive near its fixed point and nonexpansive far from the fixed point.
  • Figure 3: Comparison of fixed point algorithms for an operator that is nonexpansive near its fixed point and contractive far from the fixed point.
  • Figure 4: Comparison of fixed point algorithms for an operator that is $\gamma$-Lipschitz, defined by ${\bm{T}} = {\bm{P}}_{\mathcal{B}} \circ {\bm{R}} \circ {\bm{S}},$ with operators ${\bm{P}}_{\mathcal{B}}$ being the projection onto the centered unit $\ell_2$ ball, ${\bm{R}}$ defined by \ref{['eq:hard-rotation-op']} with $\gamma = 1$, and ${\bm{S}}$ defined by \ref{['eq:piecewise-S']}. GHAL exhibits an overall more robust convergence to approximate fixed points of ${\bm{T}}$ compared to other algorithms.

Theorems & Definitions (25)

  • Lemma 1: Convergence of the Fixed-Stepsize Iteration
  • proof
  • Lemma 2: Convergence of \ref{['eq:main-iteration']} for a mildly expansive operator.
  • proof
  • Corollary 1
  • proof
  • Lemma 3: Convergence of the Fixed-Stepsize Iteration in a Busemann Space
  • proof
  • Proposition 1
  • proof
  • ...and 15 more