Dimension dependence of the transition-diameter regret term

Determine whether the additional factor of the reward dimension d in the rare-switching regret term \(\widetilde{\mathcal O}(\tau d)\) is intrinsic, or whether it can be removed through a sharper algorithm or analysis for linear bandits under exact sliding-window constraints.

Background

For convex, cyclically invariant sliding-window constraints, the paper introduces the transition diameter τ\tau, which measures the maximum number of feasible updates needed to move between stationary feasible histories. The feasible rare-switching OFUL algorithm achieves regret O~(dT+τd+w)\widetilde{\mathcal O}(d\sqrt{T}+\tau d+w).

The paper proves only an Ω(τ)\Omega(\tau) lower bound for the transition cost, even in a one-dimensional instance and even when the learner receives the true reward parameter immediately after its first action. Consequently, the upper and lower bounds differ by a factor of dd in the transition term, leaving unresolved whether this dimension dependence reflects a fundamental statistical cost or an artifact of the analysis or algorithm.

References

A further open question concerns the dependence on the transition diameter. Our rare-switching regret bound contains a transition term of order \widetilde{\mathcal O}(\tau d), whereas Proposition~\ref{prop:tau-lower-bound} establishes only an \Omega(\tau) lower bound. It remains open whether the additional factor of d is intrinsic or can be removed through a sharper algorithm or analysis.

— Linear Bandits under Exact Sliding-Window Constraints  (2610.08745 - Hosseini et al., 6 Oct 2026) in Appendix, Section 1, “Limitations and Extensions” (also reiterated in Remark \ref{rem:statistical-transition-lower-bound})