Bounds of optimal learning

Abstract

Learning is considered as a dynamic process described by a trajectory on a statistical manifold, and a topology is introduced defining trajectories continuous in information. The analysis generalises the application of Orlicz spaces in nonparametric information geometry to topological function spaces with asymmetric gauge functions (e.g. quasi-metric spaces defined in terms of KL divergence). Optimality conditions are formulated for dynamical constraints, and two main results are outlined: 1) Parametrisation of optimal learning trajectories from empirical constraints using generalised characteristic potentials; 2) A gradient theorem for the potentials defining optimal utility and information bounds of a learning system. These results not only generalise some known relations of statistical mechanics and variational methods in information theory, but also can be used for optimisation of the explorationexploitation balance in online learning systems. © 2009 IEEE.