Establish the true regret dependence of the contextual-recommendation cutting-plane algorithm

Determine whether the regret of the contextual-recommendation cutting-plane algorithm of Gollapudi et al. can have polynomial dependence on the dimension, rather than the currently established exponential dependence.

Background

The paper discusses a prior reduction from contextual recommendation to a cutting-plane algorithm that yields a regret bound of exp(O(d log d)) without assuming a margin or gap condition.

The cited prior work is described as leaving open the actual regret of that algorithm, specifically whether a polynomial dependence on the dimension is attainable. This is an explicitly unresolved question attributed to the prior literature, and is therefore included despite not being a question about the new SGS algorithms.

References

Its assumptions are weaker than ours in that it requires neither a margin nor a gap condition, but it is exponential in the dimension, and the authors themselves leave the true regret of that algorithm---in particular whether a polynomial dependence on the dimension is attainable---as an open question.

— Online Inverse Integer Linear Optimization via Small-Gradient Skipping: Constant Regret and Finite Mistakes  (2609.09809 - Kitaoka, 9 Sep 2026) in Appendix, Section 'Related work in detail', paragraph 'Finite regret in online inverse optimization'