A priori guarantees of finite-time convergence for Deep Neural Networks

TL;DR

This paper introduces a control-theoretic approach to analyze deep neural networks, providing explicit a priori bounds on their finite-time convergence and robustness to input perturbations.

Contribution

It offers the first a priori finite-time convergence guarantees for deep neural networks using Lyapunov analysis within a control framework.

Findings

Derived an analytical formula for finite-time upper bounds on convergence

Proved robustness of the loss function against input perturbations

Established a control-theoretic framework for deep learning analysis

Abstract

In this paper, we perform Lyapunov based analysis of the loss function to derive an a priori upper bound on the settling time of deep neural networks. While previous studies have attempted to understand deep learning using control theory framework, there is limited work on a priori finite time convergence analysis. Drawing from the advances in analysis of finite-time control of non-linear systems, we provide a priori guarantees of finite-time convergence in a deterministic control theoretic setting. We formulate the supervised learning framework as a control problem where weights of the network are control inputs and learning translates into a tracking problem. An analytical formula for finite-time upper bound on settling time is computed a priori under the assumptions of boundedness of input. Finally, we prove the robustness and sensitivity of the loss function against input perturbations.