A converse to Banach's fixed point theorem and its CLS-completeness

Abstract

Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in non-convex problems. It is a common experience, however, that iterative maps fail to be globally contracting under the natural metric in their domain, making the applicability of Banach's theorem limited. We explore how generally we can apply Banach's fixed point theorem to establish the convergence of iterative methods when pairing it with carefully designed metrics.Our first result is a strong converse of Banach's theorem, showing that it is a universal analysis tool for establishing global convergence of iterative methods to unique fixed points, and for bounding their convergence rate. In other words, we show that, whenever an iterative map globally converges to a unique fixed point, there exists a metric under which the iterative map is contracting and which can be used to bound the number of iterations until convergence. We illustrate our approach in the widely used power method, providing a new way of bounding its convergence rate through contraction arguments.We next consider the computational complexity of Banach's fixed point theorem. Making the proof of our converse theorem constructive, we show that computing a fixed point whose existence is guaranteed by Banach's fixed point theorem is CLS-complete. We thus provide the first natural complete problem for the class CLS, which was defined in [DP11] to capture the complexity of problems such as P-matrix LCP, computing KKT-points, and finding mixed Nash equilibria in congestion and network coordination games. and, given ε > 0, we can find an integer N ∈ N 1 (depending maybe on ε) such that

The Complexity of Constrained Min-Max Optimization

“…Beyond equilibrium computation and its applications to Economics and Game Theory, the study of total search problems has found profound connections to many scientific fields, including continuous optimization [DP11,DTZ18], combinatorial optimization [SY91], query complexity [BCE + 95], topology [GH19], topological combinatorics and social choice theory [FG18,FG19,FRHSZ20b,FRHSZ20a], algebraic combinatorics [BIQ + 17, GKSZ19], and cryptography [Jeř16,BPR15,SZZ18]. For a more extensive overview of total search problems we refer the reader to the recent survey by Daskalakis [Das18].…”

Secure Fast Evaluation of Iterative Methods: With an Application to Secure PageRank

“…These are results which show that if an iterative method converges for some set X on a metric space with metric d, then there is a (potentially different) metric d for which the map is a contraction mapping for any Lipschitz constant q. In particular in [9] the following converse theorem is proved Theorem A.1 (Theorem 1 of [9]). Let (X, d) be a complete, proper metric space and F : X −→ X be continuous with respect to d such that F has a unique fixed point x * , and the iteration x i ← F (x i−1 ) converges to x * with respect to d, and there exists an open neighbourhood U of x * such that…”

“…The authors of [9] illustate this in terms of the power method for matrices. Indeed in [9][Proposition 1] it is shown that if we restrict to real matrices A (with eigenvalues |λ 1 | > |λ 2 | ≥ . .…”

The Hairy Ball problem is PPAD-complete

“…Its corresponding computational problem Contraction, is to find a fixed point of a given contraction map. Some versions of Contraction have been shown complete for CLS, a subclass of PPAD [11,12,16]. The search for Brouwer fixpoints (including discretised versions of Brouwer functions) is PPAD-complete for most variants of the problem [28,7], which is why we say the HBT is "Brouwer-like".…”