Combinatorial Optimization with Information Geometry: The Newton Method

Luigi Malagò,Giovanni Pistone

doi:10.3390/e16084260

Abstract

We discuss the use of the Newton method in the computation of max(p → Εp [f]), where p belongs to a statistical exponential family on a finite state space. In a number of papers, the authors have applied first order search methods based on information geometry. Second order methods have been widely used in optimization on manifolds, e.g., matrix manifolds, but appear to be new in statistical manifolds. These methods require the computation of the Riemannian Hessian in a statistical manifold. We use a non-parametric formulation of information geometry in view of further applications in the continuous state space cases, where the construction of a proper Riemannian structure is still an open problem.

Highlights

In this paper, statistical exponential families [1] are thought of as differentiable manifolds along the approach called information geometry [2] or the exponential statistical manifold [3]
In this formalism, a vector field is a mapping from densities to centered random variables, that is, it is what in statistics is called a pivot of the statistical model
We introduced second-order calculus over a statistical manifold, following the approach described in [4], which has been adapted to the special case of exponential statistical models [2,3]

Summary

Introduction

Statistical exponential families [1] are thought of as differentiable manifolds along the approach called information geometry [2] or the exponential statistical manifold [3]. Each point of the manifold has a different tangent space, and this tangent space can be used as a non-parametric model space of the manifold In this formalism, a vector field is a mapping from densities to centered random variables, that is, it is what in statistics is called a pivot of the statistical model. To avoid confusion with the product of random variables, we do not use the standard notation for the action of a vector field on a real function This approach is possibly unusual in differential geometry, but it is fully natural from the statistical point of view, where the Fisher score has a central place. This approach scales nicely from the finite state space to the general state space; see the discussion in [9] and the review in [3]

Objectives

Methods

Conclusion