Distributionally robust optimization protects against worst-case distributions — not just worst-case data points, but worst-case probability measures within a neighborhood of the empirical distribution. The Wasserstein ball defines this neighborhood. Computing the worst-case expectation over this ball is an infinite-dimensional optimization problem: you're optimizing over all probability measures within transport distance of the data.
Chen, Fattahi, and Shafiee (arXiv:2602.20403) show that for piecewise concave loss functions, this infinite-dimensional problem reduces to a classical budget allocation problem. The worst-case distribution concentrates its mass on finitely many support points, and distributing probability among them is equivalent to allocating a fixed budget across discrete items — a problem solved decades ago.
The connection enables significant speedups over commercial solvers like Gurobi. The budget allocation problem has combinatorial structure that specialized algorithms exploit. Gurobi sees a generic optimization problem; the reduction reveals the specific structure.
The online version — making sequential decisions while protecting against adversarial distributions — formulates as a saddle-point game between a decision-maker and an adversary. The framework converges to a robust Nash equilibrium that matches the offline Wasserstein DRO solution. The game converges because both players are solving budget allocation problems at each step.
The general observation: an intractable infinite-dimensional problem can hide a tractable finite-dimensional problem if you recognize the structure. The worst-case distribution over a Wasserstein ball is not arbitrary — it has the form of a budget allocation because the transport constraint limits how mass can be moved. The difficulty was in the formulation, not the problem.