CoDeFeL. Control for Deep and Federated Learning

ERC Advanced Grant 2022. Control for Deep and Federated Learning (CoDeFeL)
Principal Investigator (PI): Prof. Enrique Zuazua
Host Institution: Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU, Germany) and DeustoCCM (Deusto University, Spain)
Duration: 5 years (2024 – 2029)

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.

Principal Investigator

Enrique Zuazua

Working Packages

WP1 – Asymptotics and Turnpike for Deep Neural Networks
Turnpike theory, originally an economic concept, describes how optimal solutions follow a stable, efficient trajectory (the “turnpike”) for most of the process before diverging toward specific final configurations. In machine learning, this principle applies to optimization dynamics, particularly in training deep neural networks, offering insights with both theoretical and practical significance. By leveraging turnpike principles, we can design more efficient training strategies, enabling machine learning models to achieve optimal performance faster and with fewer computational resources.

WP2 – Complexity of ResNet Dynamics
Residual Neural Networks (ResNets) incorporate shortcut connections that facilitate gradient flow, leading to faster convergence and greater robustness to noise compared to standard deep networks. These residual connections make ResNets behave like time-discrete ordinary differential equations, introducing unique dynamical properties that must be thoroughly understood to unlock their full potential. Building on the foundational pillars of simultaneous control theory and universal approximation principles, we aim to design ResNet architectures with improved explainability and performance.

WP3 – Federated Learning
Federated Learning (FL) is a decentralized machine learning approach that allows multiple devices to collaboratively train a model without exchanging raw data. This method is especially valuable in privacy-sensitive domains such as healthcare and finance, where strict regulations limit data sharing. As a relatively new paradigm, FL still presents a significant gap between practical algorithmic performance and theoretical understanding. By integrating a control theory perspective, we aim to develop novel FL methodologies with strong mathematical rigor and solid statistical foundations.

WP4 – Modelling through Control and Machine Learning
Control theory focuses on regulating dynamic systems to achieve desired outcomes, while machine learning excels at identifying patterns and making predictions from data. By integrating these two fields, we aim to develop efficient, robust, and adaptive systems that not only learn from data but also dynamically adjust their behavior based on real-time feedback. This synergy ensures optimal performance in dynamic and uncertain environments.
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The CoDeFeL project (ERC-2022-ADG) has received funding from the European Union’s Horizon ERC Grants programme under grant agreement No. 101096251.
The views and opinions expressed are solely those of the author(s) and do not necessarily reflect those of the European Research Council Executive Agency (ERCEA), the European Union or the granting authority who cannot be held responsible for them

|| Check the news at FAU on our Math to go! section: FAU mathematician receives ERC Advanced Grant