Least Action Principles and Well-Posed Learning Problems
This work provides a theoretical foundation for time-aware learning models, though it appears incremental in extending mechanics principles to cognitive domains.
The paper tackles the problem of formulating learning as a time-dependent process by introducing a Least Cognitive Action principle, proving the existence of a minimum for a special form of this action and deriving fourth-order differential equations for learning.
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.