  Optimization seeks extremal points in a function. When there are
superextensively many optima, optimization algorithms are liable to get stuck.
Under these conditions, generic algorithms tend to find marginal optima, which
have many nearly flat directions. In a companion paper, we introduce a
technique to count marginal optima in random landscapes. Here, we use the
statistics of marginal optima calculated using this technique to produce
generic bounds on optimization, based on the simple principle that algorithms
will overwhelmingly tend to get stuck only where marginal optima are found. We
demonstrate the idea on a simple non-Gaussian problem of maximizing the sum of
squared random functions on a compact space. Numeric experiments using both
gradient descent and generalized approximate message passing algorithms fall
inside the expected bounds.
