Multiclass classification often relies on aggregating a dictionary of score functions to mimic the best possible combination. Abramovich and Davidovitch-Servadio extend classical aggregation theory from binary to multiclass settings, asking how the number of classes affects the achievable risk. They focus on model selection and convex aggregation under both 0/1 and multinomial logistic losses.
Under the 0/1 loss, the optimal aggregation rates are surprisingly independent of K, and empirical risk minimization is rate-optimal for model selection and nearly optimal for convex aggregation, up to a sqrt(log K) factor. The multinomial logistic loss is more delicate: although the optimal rates still do not depend on K, restricting aggregation to proper aggregates in the convex hull of the score functions degrades the achievable rates in some regimes as K grows.
To avoid this limitation, the authors propose improper probability-mixture aggregates that average the induced class probabilities and map the result back to score-function space. These aggregates recover optimal or nearly optimal rates independent of K. The paper also introduces a separate framework where both the aggregation class and the oracle are defined directly in probability space, offering an alternative route for multiclass aggregation.