Core Concepts
The Fisher-Rao gradient flow of a linear program converges linearly with an exponential rate that depends on the geometry of the linear program. This yields improved bounds on the error induced by entropic regularization of linear programs.
Abstract
The paper studies the convergence properties of Fisher-Rao gradient flows of general linear programs. The key insights are:
Fisher-Rao gradient flows of linear programs converge linearly in KL-divergence and function value with an exponential rate that depends on the geometry of the linear program. This improves upon existing results.
In the case of non-unique optimizers, the Fisher-Rao gradient flow converges to the information projection of the initial condition to the set of optimizers, characterizing its implicit bias.
The linear convergence results for Fisher-Rao gradient flows of linear programs yield improved bounds on the regularization error in entropy-regularized linear programming.
The general convergence results for Fisher-Rao gradient flows are then applied to study natural gradient methods in multi-player games and Markov decision processes, providing sublinear and linear convergence guarantees.