  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.
