Information evolution of optimal learning

It is widely accepted that learning is closely related to theories of optimisation and information. Indeed, there is no need to learn if there is nothing to optimise; if one possesses full information, then there is simply nothing new to learn. The paper considers learning as an optimisation problem with dynamical information constraints. Unlike the standard approach in the optimal control theory, where the solutions are given by the Hamilton–Jacobi–Bellman equation for Markov time evolution, the optimal solution is presented as the system of canonical Euler equations defining the optimal information–utility trajectory in the conjugate space. The optimal trajectory is parameterised by theinformation–utility constraints, which are illustrated on examples for finite and infinite–dimensional cases.