Machine Learning from a Continuous Viewpoint

TL;DR

This paper introduces a continuous mathematical framework for machine learning, connecting traditional models to differential equations, and proposes new models and algorithms inspired by this perspective.

Contribution

It formulates machine learning as a calculus of variations problem, unifies existing models, and introduces novel models and algorithms based on continuous formulations.

Findings

Conventional models are recoverable as discretizations of continuous formulations.

New models like flow-based random features are proposed.

Algorithms such as the smoothed particle and spectral methods are introduced.

Abstract

We present a continuous formulation of machine learning, as a problem in the calculus of variations and differential-integral equations, in the spirit of classical numerical analysis. We demonstrate that conventional machine learning models and algorithms, such as the random feature model, the two-layer neural network model and the residual neural network model, can all be recovered (in a scaled form) as particular discretizations of different continuous formulations. We also present examples of new models, such as the flow-based random feature model, and new algorithms, such as the smoothed particle method and spectral method, that arise naturally from this continuous formulation. We discuss how the issues of generalization error and implicit regularization can be studied under this framework.