Fast rates for GLMs via strong convexity on the simplex

Establish that exploiting strong convexity of the generalized linear model loss over the probability simplex Δ_d yields fast excess prediction risk rates of order O(log d / n) for procedures based on exponentiated gradient descent or KL-divergence regularization in the model aggregation setting, thereby improving the slow-rate bounds currently derived under general convexity.

Background

In the model aggregation setting over the simplex, the paper derives slow-rate bounds of order √((log d)/n) for exponentiated gradient descent and KL-regularized estimators without requiring exponential concavity or i.i.d. assumptions. Prior works show that fast rates O((log d)/n) can be achieved under stronger assumptions (e.g., exponential concavity or strong convexity).

The authors explicitly conjecture that leveraging strong convexity of the GLM loss on Δ_d may recover such fast rates for their framework, and they note that developing this refinement is left for future work.