Necessity of the curvature-induced regret dependence in GLBM

Determine whether the curvature-induced dependence in the regret upper bound for generalized linear bandits with memory is necessary, by establishing a matching lower bound or sharpening the upper bound.

Background

For generalized linear bandits with memory, the paper proves an upper bound containing a curvature-induced term of order $\widetilde O(d^2\kappa v\rho^2\Delta_\mu)$, where $\rho$ measures relative variation in the link function’s slope and $\kappa$ measures its worst-case inverse slope. The accompanying lower bound matches the memory and statistical-estimation terms but does not include a corresponding curvature-dependent term.

Consequently, it is unresolved whether the curvature dependence in the upper bound reflects an inherent identification cost or is an artifact of the analysis. Resolving this gap requires either sharpening the upper bound to reduce or eliminate the curvature term or proving a matching lower bound demonstrating that such dependence is unavoidable.