Dynamical mean field theory for models of confluent tissues and beyond

TL;DR

This paper develops a dynamical mean field theory for a model of confluent tissues, analyzing various dynamics including gradient descent, and reveals that gradient descent can find zero energy states despite replica symmetry breaking.

Contribution

The paper derives DMFT equations for the tissue model and demonstrates that gradient descent bypasses the RSB transition, providing insights into high-dimensional optimization.

Findings

Gradient descent finds zero energy states despite RSB.

DMFT equations accurately describe tissue dynamics.

Gradient descent is insensitive to the RSB transition.

Abstract

We consider a recently proposed model to understand the rigidity transition in confluent tissues and we derive the dynamical mean field theory (DMFT) equations that describes several types of dynamics of the model in the thermodynamic limit: gradient descent, thermal Langevin noise and active drive. In particular we focus on gradient descent dynamics and we integrate numerically the corresponding DMFT equations. In this case we show that gradient descent is blind to the zero temperature replica symmetry breaking (RSB) transition point. This means that, even if the Gibbs measure in the zero temperature limit displays RSB, this algorithm is able to find its way to a zero energy configuration. We include a discussion on possible extensions of the DMFT derivation to study problems rooted in high-dimensional regression and optimization via the square loss function.