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

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 tâtonnement-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.

💡 Research Summary

This paper addresses the computational challenge of finding market equilibria in large‑scale Fisher markets by introducing stochastic block‑coordinate first‑order methods. Traditional approaches solve the equilibrium‑capturing convex programs—namely the Eisenberg‑Gale (EG) and Shmyrev formulations—using full‑gradient techniques such as projected gradient, Frank‑Wolfe, or mirror descent. While these methods enjoy strong theoretical guarantees, each iteration requires updating all n·m allocation variables, which becomes prohibitive when the number of buyers (n) and items (m) reaches millions.

The authors propose two algorithms that update only a small subset of variables per iteration:

1. BCDEG (Proximal Block‑Coordinate Descent for the EG program).
   The EG program is rewritten as a smooth part f(x)=−∑i B_i log⟨v_i,x_i⟩ plus a separable nonsmooth part ψ(x) that encodes the item‑capacity constraints. Because ψ decomposes across items, the decision vector can be viewed as a collection of m blocks, each block x·j containing the allocations of item j to all buyers. At each iteration a random item j_k is selected, its partial gradient ∇{·j_k}f(x) is computed, and the block is updated by a Euclidean projection onto the n‑dimensional simplex:

   x_{·j_k}^{new}=Proj_{Δ_n}(x_{·j_k}^{old}−η_{j_k}∇_{·j_k}f(x)).

   The stepsize η_{j_k}=1/L_{j_k} is chosen based on a block‑wise Lipschitz constant L_{j}=max_i B_i v_{ij}^2 / u_i^2, where u_i is the current utility of buyer i. A simple line‑search scheme can adapt η_{j_k} on the fly, further accelerating convergence. The authors prove a linear (geometric) convergence rate for the expected objective value:

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