Progress and future directions in machine learning through control theory - Paper

Aug 3, 2026

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This paper outlines recent advances at the intersection of machine learning and control theory. It uses control-theoretic tools to clarify why common ML methods work, improving explainability and practical performance. The study analyzes residual neural networks by reframing memorization, representation, classification, and approximation as control problems, deriving nonlinear constructive training methods and complexity insights. It further investigates neural ODEs, optimal MLP architectures, nonconvex mean-field optimization for robustness and generalization, and extends the ideas to attention dynamics and federated learning.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages8

Reading Time20 min

AccessFree to download

Abstract

Introduction

Control-based supervised learning via neural networks

Residual neural networks

Neural ODEs

Multilayer perceptrons

Mean-field optimization and generalization

Attention dynamics in large language models

Federated learning

French-German-Spanish Conference on Optimization Gijón, June 18-21, 2024 (pp. 116-123) Progress and future directions in machine learning through control theory Enrique Zuazua1,2,3 1. enrique . zuazua@fau . de Chair for Dynamics, Control, Machine Learning, and Numerics, Alexander von Humboldt-Professorship, Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany 2. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain 3. Chair of Computational Mathematics, Fundación Deusto. Av. de las Universidades, 24, 48007 Bilbao, Basque Country, Spain Abstract This paper presents our recent advancements atthe intersection of machine learning and control theory. We focus specifically on utilizing control theoretical tools to elucidate the underlying mechanisms driving the success of machine learning algorithms. By enhancing the explainability of these algorithms, we aim to contribute to their ongoing improvement and more effective application. Our research explores several critical areas: Firstly, we investigate the memorization, representation, classification, and approximation properties of residual neural networks (ResNets) . By framing these tasks as simultaneous or ensemble control problems, we have developed nonlinear and constructive algorithms for training. Our work provides insights into the parameter complexity and computational requirements of ResNets. Similarly, we delve into the properties of neural ODEs (NODEs) . We demonstrate that autonomous NODEs of sufficient width can ensure approximate memorization properties. Furthermore, we prove that by allowing biases tobe time-dependent, NODEs can track dynamic data. This showcases their potential for synthetic model generation and helps elucidate the success of methodologies such as Reservoir Computing. Next, we analyze the optimal architectures of multilayer perceptrons (MLPs) . Our findings offer guidelines for designing MLPs with minimal complexity, ensuring efficiency and effectiveness for supervised learning tasks. The generalization and prediction capacity of trained networks plays a crucial role. To address these properties, we present two nonconvex optimization problems related to shallow neural networks, capturing the ”sparsity” of parameters and robustness of representation. We introduce a ”mean-field” model, proving, via representer theorems, the absence of a relaxation gap. This aids in designing an optimal tolerance strategy for robustness and, through convexification, efficient algorithms for training. In the context of large language models (LLMs), we explore the integration of residual networks with self-attention layers for context capture. We treat ”attention” as a dynamical system acting on a collection of points and characterize their asymptotic dynamics, identifying convergence towards special points called leaders. These theoretical insights have led to the development of an interpretable model for sentiment analysis of movie reviews, among other possible applications. Lastly, we address federated learning, which enables multiple clients to collaboratively train models without sharing private data, thus addressing data collection and privacy challenges. We examine training efficiency, incentive mechanisms, and privacy concerns within this framework, proposing solutions to enhance the effectiveness and security of federated learning methods. Our work underscores the potential of applying control theory principles to improve machine learning models, resulting in more interpretable and efficient algorithms. This interdisciplinary approach opens up a fertile ground for future research, raising profound mathematical questions and application-oriented challenges and opportunities. 1. Introduction The impact of machine learning (ML) and artificial intelligence (AI) in science is leading to rich and innovative lines of research in applied mathematics. There i

FAQs

How does the paper use control theory to improve machine learning explainability?

It employs control-theoretic tools to identify the mechanisms behind ML algorithm success, aiming to enhance explainability and guide model improvement and application.

What are the main results regarding residual neural networks (ResNets)?

The paper characterizes memorization, representation, classification, and approximation properties by framing them as simultaneous or ensemble control problems, yielding nonlinear constructive training algorithms and insights into parameter complexity.

How does the paper extend these ideas to federated learning?

It examines training efficiency, incentive mechanisms, and privacy concerns in federated learning and proposes solutions to strengthen both effectiveness and security without sharing private data.

Aug 3, 2026

Upload by Riley