A Bregman firmly nonexpansive proximal operator for baryconvex optimization
Mastane Achab
TL;DR
The paper introduces a baryconvex generalization of the proximal operator by minimizing a minimax objective over a family of convex losses, using a hybrid Euclidean–KL Bregman geometry. It proves Bregman firm nonexpansiveness of the generalized prox, links fixed points to critical points of a nonconvex function tied to an exponential-family barycentric representation, and derives continuous barygradient flows (min-max and min-min) with explicit expressions for their entropy dynamics and monotonicity properties. These results extend classical proximal methods to a richer, information-geometric setting and provide new tools for convergence analysis and continuous-time optimization in baryconvex landscapes. The practical impact lies in enabling stable fixed-point algorithms and flow-based interpretations for learning and variational problems with learnable probability weights over convex objectives.
Abstract
We present a generalization of the proximal operator defined through a convex combination of convex objectives, where the coefficients are updated in a minimax fashion. We prove that this new operator is Bregman firmly nonexpansive with respect to a Bregman divergence that combines Euclidean and information geometries; and that its fixed points are given by the critical points of a certain nonconvex function. Finally, we derive the associated continuous flows.