Abstract
Consider a control problem consisting in finding a trajectory connecting an initial point x to a target point y, with the system moving only in certain admissible directions. It is assumed that the corresponding vector fields satisfy the Hörmander condition, so that by a classical theorem (Chow-Rashevskii), there exist trajectories that satisfy this constraint. A natural way of trying to solve this problem is via a gradient flow on the control space.
However, the corresponding dynamics may have saddle points, and to obtain a convergence result we must therefore make (e.g. probabilistic) assumptions about the initial condition. In this work, we consider the case where this initialization is irregular, which we formulate using Lyons' theory of rough trajectories. In simple cases, we prove that the gradient stream converges to a solution, if the initial condition is a trajectory of Brownian motion (or of a process of weaker regularity). The proof combines ideas from Malliavin calculus with Łojasiewicz inequalities. A possible motivation for our work comes from training deep residual neural networks, in a regime where the number of parameters per layer is fixed, and the dimension of the data vector is high.
This is a collaborative project with Florin Suciu (Paris-Dauphine).
