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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.

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Deeper Inquiries

The improved convergence rates for entropy-regularized linear programming have various potential applications beyond reinforcement learning. One such application is in the field of computational optimal transport, where entropy regularization is commonly used to solve transportation and assignment problems efficiently. By improving the convergence rates of these optimization methods, tasks such as image registration, shape matching, and data association can be performed more effectively. Additionally, in machine learning and data science, entropy-regularized linear programming can be applied to clustering, classification, and regression problems, where the faster convergence can lead to quicker model training and improved performance. Furthermore, in operations research and logistics, the enhanced convergence rates can optimize resource allocation, supply chain management, and network flow problems more efficiently.

The convergence rates of Fisher-Rao gradient flows for linear programs offer a different perspective compared to traditional optimization methods like the simplex method or interior point methods. While the simplex method and interior point methods are widely used for solving linear programs, they may face challenges with large-scale problems or in cases where the objective function is non-smooth or non-convex. In contrast, Fisher-Rao gradient flows leverage the geometry of the problem space through the Fisher-Rao metric, allowing for efficient optimization even in high-dimensional or complex scenarios. The linear convergence rates with respect to KL-divergence and function value provide insights into the optimization process that may not be apparent with traditional methods. Additionally, the ability to estimate regularization errors and characterize implicit biases adds a new dimension to the analysis of optimization algorithms for linear programs.

The techniques used to analyze Fisher-Rao gradient flows can be extended to study the convergence of other types of gradient flows, such as those induced by Bregman divergences or more general Hessian geometries. By adapting the framework developed for Fisher-Rao gradient flows, researchers can investigate the convergence properties, optimization behavior, and regularization effects of these alternative gradient-based methods. The analysis may involve establishing convergence rates, characterizing implicit biases, and estimating regularization errors, similar to the approach taken for Fisher-Rao gradient flows. This extension can provide valuable insights into the optimization landscape of different types of gradient flows and contribute to the development of efficient optimization algorithms for a wide range of problems.

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