Adrian Lewis

Adrian Lewis, Russell Luke, Jerome Malick

The idea of a finite collection of closed sets having "strongly regular intersection" at a given point is crucial in variational analysis. We show that this central theoretical tool also has striking algorithmic consequences. Specifically, we consider the case of two sets, one of which we assume to be suitably "regular" (special cases being convex sets, smooth manifolds, or feasible regions satisfying the Mangasarian-Fromovitz constraint qualification). We then prove that von Neumann's method of "alternating projections" converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity. As a consequence, in the case of several arbitrary closed sets having strongly regular intersection at some point, the method of "averaged projections" converges locally at a linear rate to a point in the intersection. Inexact versions of both algorithms also converge linearly.

Adrian Lewis, Tonghua Tian

The classical Clarke subdifferential alone is inadequate for understanding automatic differentiation in nonsmooth contexts. Instead, we can sometimes rely on enlarged generalized gradients called "conservative fields", defined through the natural path-wise chain rule: one application is the convergence analysis of gradient-based deep learning algorithms. In the semi-algebraic case, we show that all conservative fields are in fact just Clarke subdifferentials plus normals of manifolds in underlying Whitney stratifications.