Close the dimension and coordinate-range gap in regret bounds

Determine whether the factor- dimension gap and the dependence on the coordinatewise action ranges can be improved in either the upper bound or the known lower bound for cumulative decision regret in online inverse linear optimization with integer-linear-program forward problems.

Background

For integer-linear-program forward problems, the paper obtains a cumulative decision-regret upper bound of O(d^2 ||M||_2 log(2||M||_2)) for ONS and growing-grid SGS-MetaGrad under a uniform margin. The cited lower bound is a9(d), leaving both a factor-d gap in the dimension and a discrepancy involving the coordinatewise range parameter ||M||_2.

The authors explicitly leave unresolved whether the upper bound should be sharpened, the lower bound strengthened, or both. This problem concerns identifying the correct dependence on dimension and action-range parameters.