Identify the selected D-optimal design in the nonunique case

Determine which D-optimal design the empirical sampling distribution generated by LinUCB converges to when the D-optimal design is not unique, thereby identifying the exact leading-order constant of the pseudo-regret in the nonunique case.

Background

The paper proves that, in the small-gap regime, LinUCB’s empirical sampling distribution converges to the set of D-optimal designs. When that set contains multiple designs, this set-valued convergence determines only an envelope of possible limits for the normalized pseudo-regret, because different D-optimal designs may induce different average scaled reward gaps.

The unresolved issue is to characterize the finer adaptive dynamics that select a particular D-optimal design. The authors indicate that resolving it may require a fluid- or diffusion-level analysis of LinUCB’s sampling, regret, or reward processes, which would in turn determine the exact asymptotic regret constant.

References

Pinning down the exact constant would require identifying which D-optimal design is selected by the adaptive dynamics, a finer question that may require a fluid- or even diffusion-level characterization of the sampling dynamics~\citep{KalvitZeevi2021,Lattimore2026DiffusionPolicyGradient}. We leave this characterization as an open problem.

— Sampling Allocation of LinUCB: Optimal Design Limits in the Small-Gap Regime  (2610.06213 - Liu et al., 5 Oct 2026) in Section 3.2, immediately after Proposition 3.2 (Pseudo-regret in the small-gap regime)

Several directions remain open. First, in settings where the D-optimal design is not unique, it remains important to determine which D-optimal design the empirical sampling distribution converges to in the limit. Addressing this question may require developing fluid or diffusion approximations for the regret or reward processes of LinUCB in the small-gap regime, and could yield a sharper characterization of the leading-order constant for the pseudo-regret in the nonunique case. Second, a natural next step is to extend the analysis beyond the small-gap regime by developing a unified mean-field description of the full phase transition across the small-gap, critical-gap, and fixed-gap regimes, in the spirit of the two-arm bandit analysis of \citet{KalvitZeevi2021}. Such a theory would provide a fuller answer to the challenging problem of sampling allocation in correlated linear bandits, and would clarify how this allocation encodes information beyond regret, such as critical thresholds for valid statistical inference on the reward parameter.

— Sampling Allocation of LinUCB: Optimal Design Limits in the Small-Gap Regime  (2610.06213 - Liu et al., 5 Oct 2026) in Section 5, Conclusion