Weakening mirror-descent regularity assumptions

Determine whether the relative-curvature, Lipschitz, and ambient-regularity assumptions used in the positive result for standard constrained mirror descent can be weakened while retaining control of the boundary residual and sublinear regret.

Background

The paper establishes sublinear-regret guarantees for standard constrained mirror descent against mirror-exact deviations under relative-curvature, Lipschitz, and ambient-regularity conditions. These assumptions are used to control the normal-cone or boundary residual that does not arise in the Legendre-interior analysis.

The authors note that the circulation-based lower bound requires only continuity of the deviation field. The unresolved problem is therefore to narrow the gap between the sufficient hypotheses for the positive result and the weaker regularity needed for the obstruction result.