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Motion planning, rough controls, and deep neural networks

le 14 mai 2025

11h - Groupe de travail "Applications des Mathématiques"

ENS Rennes Salle 9

Séminaire de Florin Suciu (Université Paris Dauphine), au groupe de travail "Applications des mathématiques"

Groupe de travail

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The motion planning control problem consists in finding a path joining two given points, with (state-dependent) constraints on the admissible directions of movements. When these form a strict subspace of ambient space, it is typically difficult to obtain convergence guarantees for gradient methods, due to the possible degeneracies (singular controls) of the endpoint map. We consider the impact of rough (possibly probabilistic) controls as initial conditions in this context. We show that one advantage of these initialisations is that the saddle points are moved to infinity, while minima remain at a finite distance from the starting point. In the step 2-nilpotent case, we further manage to prove that the gradient flow converges to a solution, if the initial condition is the path of a Brownian motion (or rougher). The main motivation for our study comes from the training of deep Residual Neural Networks, in the regime when the number of trainable parameters per layer is smaller than the dimension of the data vector. In this context, we further obtain a local convergence result to zero loss for i.i.d. layer initializations and sufficiently deep networks. The analysis is based on combining the rough path toolkit with ideas from Malliavin calculus and Łojasiewicz inequalities. Based on joint works with Paul Gassiat and Zhenjie Ren.
Thématique(s)
Recherche - Valorisation
Contact
Adrien LAURENT

Mise à jour le 13 mai 2025