A mathematical model for automatic differentiation in machine learning

TL;DR

This paper develops a mathematical framework for automatic differentiation tailored to modern machine learning, addressing issues like nonsmooth functions and artificial critical points.

Contribution

It introduces a simple class of functions and a nonsmooth calculus, linking program differentiation with nonsmooth function differentiation, and analyzes the impact on stochastic methods.

Findings

Addresses nonsmooth functions in differentiation

Shows how to avoid artificial critical points

Provides a new mathematical model for automatic differentiation

Abstract

Automatic differentiation, as implemented today, does not have a simple mathematical model adapted to the needs of modern machine learning. In this work we articulate the relationships between differentiation of programs as implemented in practice and differentiation of nonsmooth functions. To this end we provide a simple class of functions, a nonsmooth calculus, and show how they apply to stochastic approximation methods. We also evidence the issue of artificial critical points created by algorithmic differentiation and show how usual methods avoid these points with probability one.