On the probabilistic continuous complexity conjecture

TL;DR

This paper proves the probabilistic continuous complexity conjecture, showing that under uniform convexity, probabilistic and worst-case complexities align, but providing a counterexample in non-uniformly convex spaces.

Contribution

It establishes the conjecture's validity under uniform convexity and presents a counterexample in non-uniformly convex spaces, advancing understanding in continuous complexity theory.

Findings

Probabilistic complexity converges to worst-case complexity in uniformly convex spaces.

Counterexample in Wiener space shows convergence to half the worst-case complexity.

Proves the conjecture holds iff the space is uniformly convex.

Abstract

In this paper we prove the probabilistic continuous complexity conjecture. In continuous complexity theory, this states that the complexity of solving a continuous problem with probability approaching 1 converges (in this limit) to the complexity of solving the same problem in its worst case. We prove the conjecture holds if and only if space of problem elements is uniformly convex. The non-uniformly convex case has a striking counterexample in the problem of identifying a Brownian path in Wiener space, where it is shown that probabilistic complexity converges to only half of the worst case complexity in this limit.