Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning
This provides a foundational framework for analyzing and improving nonsmooth optimization algorithms, such as those used in deep learning, benefiting researchers and practitioners in machine learning and numerical analysis.
The paper tackles the challenge of nonsmooth optimization in AI and numerical analysis by introducing conservative fields and path differentiable functions, enabling a flexible calculus and variational formulas for nonsmooth automatic differentiation, and applies this to prove convergence of stochastic gradient methods in practice.
Modern problems in AI or in numerical analysis require nonsmooth approaches with a flexible calculus. We introduce generalized derivatives called conservative fields for which we develop a calculus and provide representation formulas. Functions having a conservative field are called path differentiable: convex, concave, Clarke regular and any semialgebraic Lipschitz continuous functions are path differentiable. Using Whitney stratification techniques for semialgebraic and definable sets, our model provides variational formulas for nonsmooth automatic differentiation oracles, as for instance the famous backpropagation algorithm in deep learning. Our differential model is applied to establish the convergence in values of nonsmooth stochastic gradient methods as they are implemented in practice.