Algorithm-independent bounds on complex optimization through the statistics of marginal optima

TL;DR

This paper develops a method to derive bounds on optimization performance based on the statistics of marginal optima, explaining why algorithms tend to get stuck in these regions in complex landscapes.

Contribution

It introduces a technique to count marginal optima in random landscapes and uses this to establish generic bounds on optimization algorithms' behavior.

Findings

Numeric experiments confirm bounds with gradient descent.

Generalized approximate message passing results align with predictions.

Marginal optima dominate where algorithms tend to get stuck.

Abstract

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.