Determine whether the transfer lower bound is tight in full generality

Determine whether the asymptotic lower bound for transfer learning with process-generated samples is tight in full generality, and establish whether closing the gap between that lower bound and the KL-UCB-Transfer upper bound requires a fundamentally different proof technique.

Background

The paper derives an instance-dependent lower bound for stochastic bandits that use prior observations generated by an unknown, potentially adaptive source policy. The lower bound depends on the source sample size through its expectation, whereas the KL-UCB-Transfer upper bound depends on the realized source information. The authors exhibit instances where this expectation-based lower bound is arbitrarily loose, although they show that the gap disappears under a mild concentration condition on the source sample count, except at a boundary case.

The authors explicitly conjecture that closing the gap in general requires a fundamentally different change-of-measure argument. Resolving this issue would characterize the optimal regret rate when source sample counts can be highly variable or data-dependent.