Establish sharp lower bounds for rational-quadratic and one-dimensional Dirichlet kernel bandits

Establish sharp minimax-regret lower bounds for kernel bandits using the rational-quadratic kernel and the one-dimensional Dirichlet kernel, thereby determining whether the currently known bounds are optimal.

Background

The paper develops a general minimax-regret lower bound of order Ω(TγT/log⁡T)\Omega(\sqrt{T\gamma_T/\log T}) for continuous non-constant kernels on compact domains. It compares this result with kernel-specific bounds for several families, including the rational-quadratic and one-dimensional Dirichlet kernels.

For the rational-quadratic and one-dimensional Dirichlet kernels, the paper provides lower bounds with a logarithmic gap relative to corresponding upper bounds. The authors explicitly state that sharp lower bounds had not been established for these kernels, leaving their precise minimax-regret scaling unresolved.

References

To our knowledge, sharp lower bounds have not been established for the rational quadratic kernel and one-dimensional Dirichlet kernel (among many others).

— A General $\widetildeΩ(\sqrt{T γ_T})$ Lower Bound for Kernel Bandits  (2610.11082 - Ma et al., 8 Oct 2026) in Section 1, Related Work, paragraph “Lower Bounds for Kernel Bandits”

Failing \cref{eq:growth_condition} does not eliminate the possibility of \Omega(\sqrt{T\gamma_T}) regret for these kernels, but proving (or disproving) it would require a different argument.

— A General $\widetildeΩ(\sqrt{T γ_T})$ Lower Bound for Kernel Bandits  (2610.11082 - Ma et al., 8 Oct 2026) in Section 3, subsection “When can the Log Factor be Removed?”, paragraph “A Sufficient Condition for \(\Omega(\sqrt{T\gamma_T})\) Regret”