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.

翻译：我们考虑通过可扩展一阶优化方法求解大规模Fisher市场均衡计算的问题。众所周知，市场均衡可通过结构化凸规划（如Eisenberg-Gale和Shmyrev凸规划）来刻画。针对这些规划，已有高性能确定性全梯度一阶方法。本文提出用于计算Fisher市场均衡的新型块坐标一阶方法，并证明这些方法可解释为逐步出现买家或商品的市场叫价式或比例响应式动力学过程。我们重新表述这些凸规划，采用近端块坐标下降法（一类每次迭代仅更新决策变量少量坐标的方法）进行求解。利用此类方法收敛性分析的最新进展以及均衡刻画凸规划的结构特性，我们建立了这些方法的快速收敛速率。