Multinomial Subset Routing with Sum-Max Rewards and Operational Constraints

Abstract

We study online routing to subsets of experts under aggregate bandit feedback and long-run operational constraints. Expert contributions can be complementary: for each task dimension, the best selected expert determines the contribution, and the total reward aggregates these dimension-wise maxima. At the same time, capacity, budget, and fairness requirements impose lower and upper bounds on long-run expert activation frequencies. A fixed deterministic subset cannot generally satisfy such heterogeneous requirements, motivating a stochastic routing policy. We formalize this problem as Multinomial Subset Routing (MSR). The learner maintains a distribution \(q\) over \(K\) experts, samples an expert independently \(M\) times from \(q\), and routes each task to the distinct sampled experts. We propose OMD-Approachability, which combines online mirror descent with Blackwell's approachability to optimize MSR under two-sided operational constraints. We establish \(O(T^{-1/2})\) average reward regret and expected constraint violation. We also quantify the approximation gap induced by a linear surrogate, and evaluate the approach on a real-world crowdsourcing dataset.