  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.

Machine Learning