Least Action Principles and Well-Posed Learning Problems

TL;DR

This paper explores the application of the Least Cognitive Action principle to machine learning, establishing well-posedness and existence of minima, leading to fourth-order differential equations that model the learning process.

Contribution

It introduces a novel formulation of learning based on the Least Cognitive Action principle and proves the existence of minima, extending the theoretical understanding of learning dynamics.

Findings

Existence of minima for a special form of cognitive action.

Derivation of fourth-order differential equations for learning.

Discussion of dissipative behavior in learning equations.

Abstract

Machine Learning algorithms are typically regarded as appropriate optimization schemes for minimizing risk functions that are constructed on the training set, which conveys statistical flavor to the corresponding learning problem. When the focus is shifted on perception, which is inherently interwound with time, recent alternative formulations of learning have been proposed that rely on the principle of Least Cognitive Action, which very much reminds us of the Least Action Principle in mechanics. In this paper, we discuss different forms of the cognitive action and show the well-posedness of learning. In particular, unlike the special case of the action in mechanics, where the stationarity is typically gained on saddle points, we prove the existence of the minimum of a special form of cognitive action, which yields forth-order differential equations of learning. We also briefly discuss the dissipative behavior of these equations that turns out to characterize the process of learning.