  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 [Daskalakis, Papadimitriou 2011] 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.
