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

TL;DR

This paper develops algorithms to solve fixed-point equations with mildly expansive operators, pushing the computational complexity boundaries and achieving near-optimal convergence rates in diverse mathematical spaces.

Contribution

It introduces adaptive algorithms for mildly expansive operators, extending fixed-point computation methods beyond nonexpansive cases with optimal complexity bounds.

Findings

Algorithms attain the best possible fixed-point error up to universal constants.

Convergence to ε-approximate fixed points in order-1/ε iterations for gradually expansive operators.

Methods apply to infinite-dimensional spaces and non-positively curved metric spaces.

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 $\epsilon$-approximate fixed points in order-$(1/\epsilon)$ iterations for $\epsilon > 0.$ 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.