  The authors present empirical distributions for the halting time (measured by
the number of iterations to reach a given accuracy) of optimization algorithms
applied to two random systems: spin glasses and deep learning. Given an
algorithm, which we take to be both the optimization routine and the form of
the random landscape, the fluctuations of the halting time follow a
distribution that, after centering and scaling, remains unchanged even when the
distribution on the landscape is changed. We observe two qualitative classes: A
Gumbel-like distribution that appears in Google searches, human decision times,
the QR eigenvalue algorithm and spin glasses, and a Gaussian-like distribution
that appears in conjugate gradient method, deep network with MNIST input data
and deep network with random input data. This empirical evidence suggests
presence of a class of distributions for which the halting time is independent
of the underlying distribution under some conditions.
