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.

References

First, can the relative-curvature, Lipschitz, and ambient regularity assumptions in the positive mirror-descent result be weakened while retaining control of the boundary residual and hence sublinear regret?

— Exact-Form Regret for Gradient Descent, Mirror Descent and Follow-the-Regularized-Leader  (2609.09466 - Soleymani et al., 8 Sep 2026) in Section Conclusion

Solver step sizes are per-game: four of the five 1-D games and polytope4 require a smaller mirror step for the dynamics to express the regularizer's selection at all, and although the step was selected by a baseline-only rule and applied uniformly, sensitivity of selection to the step size on stiff families is itself a finding (the boundary undershoot of Sec.~\ref{sec:control}) and remains only partially characterized;

— Steering Equilibrium Selection in Regularized Self-Play via the Reference Policy  (2609.19820 - Leal, 17 Sep 2026) in Limitations section