# Fast and Interpretable Dynamics for Fisher Markets via Block-Coordinate Updates

## TL;DR

This paper introduces scalable block-coordinate first-order methods for Fisher market equilibrium computation, offering interpretable dynamics and fast convergence suitable for large-scale problems.

## Contribution

It develops novel block-coordinate first-order algorithms with interpretations as market dynamics, improving scalability and convergence analysis for Fisher market equilibria.

## Key findings

- Methods have provably fast convergence rates.
- Algorithms are interpretable as market dynamics.
- Applicable to large-scale Fisher market problems.

## Abstract

We consider the problem of large-scale Fisher market equilibrium computation through scalable first-order optimization methods. It is well-known that market equilibria can be captured using structured convex programs such as the Eisenberg-Gale and Shmyrev convex programs. Highly performant deterministic full-gradient first-order methods have been developed for these programs. In this paper, we develop new block-coordinate first-order methods for computing Fisher market equilibria, and show that these methods have interpretations as t\^atonnement-style or proportional response-style dynamics where either buyers or items show up one at a time. We reformulate these convex programs and solve them using proximal block coordinate descent methods, a class of methods that update only a small number of coordinates of the decision variable in each iteration. Leveraging recent advances in the convergence analysis of these methods and structures of the equilibrium-capturing convex programs, we establish fast convergence rates of these methods.