Automatic differentiation, as implemented today, does not have a simple mathematical model adapted to the needs of modern machine learning.

We thank the referees for their comments. Following the suggestions of R1 and R2, we will add a conclusion to the paper and modify the broader impact statement. This will clarify our contribution, its practical impact, but also the results described in Section 5. We will, of course, Our "trap-avoidance" result suggests that, at high precision, artificial critical points are not met in practice. On the other hand, at low precision, "genericity results" may partly collapse. Y et the theory says uniqueness is generic.

Automatic differentiation, as implemented today, does not have a simple mathematical model adapted to the needs of modern machine learning.

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.