Remove the LinUCB regret gap under partial offline coverage

Develop a refined analysis of LinUCB for offline-to-online linear bandits that eliminates the multiplicative \(\sqrt{d/k}\) gap in its regret guarantee in the \(k\)-uncovered offline-coverage regime.

Background

The paper analyzes LinUCB initialized with offline data when only kk of the dd feature directions have weak offline coverage. Its standard confidence-set analysis yields a regret upper bound of order O~(kdT)\widetilde O(\sqrt{kdT}), whereas the paper’s Gaussian-width phased-elimination method achieves the sharper O~(kT)\widetilde O(k\sqrt T) rate in the same regime.

The unresolved issue is whether LinUCB itself, through a more refined confidence or regret analysis, can attain the dimension dependence suggested by the lower bound rather than retaining the extra factor d/k\sqrt{d/k}.

References

A more refined analysis to fix the issue is left for future work.

— Gaussian-Width Bounds for Offline-to-Online Linear Bandits  (2610.12990 - Chandak et al., 9 Oct 2026) in Appendix, Section “Missing Results and Details for Section 3,” subsection “LinUCB Analysis for Offline-to-Online Learning”

Finally, designing optimal offline-to-online algorithms for adaptive offline data and the full reinforcement learning setting remains open.

— Gaussian-Width Bounds for Offline-to-Online Linear Bandits  (2610.12990 - Chandak et al., 9 Oct 2026) in Section 6, “Conclusion”