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Modern problems in AI or in numerical analysis require nonsmooth approacheswith a flexible calculus. We introduce generalized derivatives calledconservative fields for which we develop a calculus and provide representationformulas. Functions having a conservative field are called path differentiable:convex, concave, Clarke regular and any semialgebraic Lipschitz continuousfunctions are path differentiable. Using Whitney stratification techniques forsemialgebraic and definable sets, our model provides variational formulas fornonsmooth automatic differentiation oracles, as for instance the famousbackpropagation algorithm in deep learning. Our differential model is appliedto establish the convergence in values of nonsmooth stochastic gradient methodsas they are implemented in practice.

Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning