Control for Deep and Federated Learning

Machine Learning (ML) is forging a new era in Applied Mathematics (AM), leading to innovative and powerful methods. But the need for theoretical guarantees generates challenging, fundamental, deep mathematical questions. This great challenge can be addressed from the perspective of other, more mature areas of AM.

CoDeFeL seeks to do so from the rich interface between Control Theory (CT) and ML, contributing to the analytical foundations of ML methods, significantly enlarging, and updating the range of applications of CT.As our recent results show, classification, regression, and prediction problems in Supervised Learning (SL) and the Universal Approximation Theorem can be successfully recast as the simultaneous or ensemble controllability property of Residual Neural Networks (ResNets).

Following this path, we will develop ResNets of minimal complexity and cost, addressing the deep, intricate issue of linking the structure of the data set to be classified with the dynamics of the networks trained. Taking the turnpike principle as our inspiration, we will build new simplified ResNet architectures.

This, however, raises major challenges for the genuinely nonlinear dynamics that ResNets represent.Adjoint methods will also be developed and applied, to understand the sensitivity of ResNets, and proposing techniques for Adversarial Training and computing Saliency Maps, applicable in Unsupervised Learning.The project is strongly inspired on the challenges arising in relevant applications in digital medicine and internet recommendation systems, among other areas.

Accordingly, we will also develop a body of rich, hybrid, cutting-edge methods for data-aware modelling, combining ResNet surrogate models and those inspired on Mechanics, with the aid of Model Predictive Control strategies. New Federated Learning methodologies with privacy preservation guarantees will also be developed.

The computational counterparts will be brought together in a new CoDeFeL GitHub repository.