Effective-dimension reduction under restrictive geometry

Characterize whether more restrictive domain or loss geometry can reduce the effective dimension in dynamic regret for exp-concave online learning.

Background

The paper establishes dimension-optimal dynamic-regret rates for general exp-concave online learning, including a sharp d^{1/3} dependence in the curved regime. Its lower bound shows that this dependence is unavoidable for the broad class of compact convex domains and exp-concave losses considered in the main results.

The conclusion identifies a remaining issue: additional structure in the decision domain or loss functions might reduce the effective dimension governing tracking difficulty. Determining which geometric or curvature restrictions enable such a reduction, and quantifying the resulting regret dependence, is left unresolved.