Empirical Tests of Optimization Assumptions in Deep Learning

There is a significant gap between our theoretical understanding of optimization algorithms used in deep learning and their practical performance. Theoretical development usually focuses on proving convergence guarantees under a variety of different assumptions, which are themselves often chosen based on a rough combination of intuitive match to practice and analytical convenience. The theory/practice gap may then arise because of the failure to prove a theorem under such assumptions, or because the assumptions do not reflect reality. In this paper, we carefully measure the degree to which these assumptions are capable of explaining modern optimization algorithms by developing new empirical metrics that closely track the key quantities that must be controlled in theoretical analysis. All of our tested assumptions (including typical modern assumptions based on bounds on the Hessian) fail to reliably capture optimization performance. This highlights a need for new empirical verification of analytical assumptions used in theoretical analysis.