Abstract
We consider a system of nonlinear ordinary differential equations for the solution of linear programming (LP) problems that was first proposed in the mathematical biology literature as a model for the foraging behavior of acellular slime mold Physarum polycephalum, and more recently considered as a method to solve LP instances. We study the convergence time of the continuous Physarum dynamics in the context of the linear programming problem, and derive a new time bound to approximate optimality that depends on the relative entropy between projected versions of the optimal point and of the initial point. The bound scales logarithmically with the LP cost coefficients and linearly with the inverse of the relative accuracy, establishing the efficiency of the dynamics for arbitrary LP instances with positive costs.
Highlights
The theoretical analysis of natural systems has historically been the domain of mathematical biology, dynamical systems theory, and physics, but certain natural processes are capable of exhibiting remarkable information processing abilities which are often best understood from an optimization perspective
The contribution of this paper is to study the convergence time of the continuous Physarum dynamics in the context of the linear programming problem, and to derive a new time bound to approximate optimality that does not depend on the maximum subdeterminant of the constraint matrix, and depends only logarithmically on the LP costs, establishing efficiency for any LP instance with positive costs
The appearance of the relative entropy term in our potential function is not an accident: it can be shown that when the feasible LP region is the unit simplex, independently of whether the dynamics is initialized with a feasible point or not, its trajectories coincide with those of the continuous Mirror Descent method of Nemirovski and Yudin [16] in a metric with geometry dictated by the negative entropy function – known as the information geometry metric [1]
Summary
The theoretical analysis of natural systems has historically been the domain of mathematical biology, dynamical systems theory, and physics, but certain natural processes are capable of exhibiting remarkable information processing abilities which are often best understood from an optimization perspective. Several convex optimization methods can be interpreted as the discretization of an ordinary differential equation system, the solutions of which are guaranteed to converge to the set of minimizers; a well-known example is the interior point method [3, 13] There is another compelling reason to study the convergence properties of the Physarum dynamics. It has been showed that, at least when started from a feasible point, this dynamics can be interpreted as a natural gradient descent algorithm in a space endowed with a nonEuclidean metric obtained from an entropy-like function [19] This is the case for certain incarnations of well-known meta-algorithms, such as Mirror Descent [5, 16], which are at the basis of very effective approximation algorithms for machine learning and convex optimization problems [2]. A full characterization of the meta-algorithm behind the dynamics remains open; we believe it deserves to be investigated, due to its potential to suggest a novel iterative approach to linear optimization problems
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