Remove the logarithmic gap in the positive-variance minimax bound

Remove the remaining logarithmic factor between the upper and lower bounds for the expected minimax dynamic regret when the variance budget Q is positive in the variance-aware non-stationary multi-armed bandit model with supplied variance and variation budgets.

Background

Theorem \ref{thm:jointminimax} characterizes the expected minimax regret for supplied variance budget Q and variation budget B up to a multiplicative factor of log(AT) when Q>0. The deterministic boundary Q=0 is characterized without logarithmic loss, but the positive-variance result does not achieve a constant-factor characterization. The unresolved issue is whether the logarithmic factor is an artifact of the proof techniques, including calibration, concentration, and restarting, or is genuinely necessary.

References

For $Q>0$, this is a characterization up to one logarithmic factor, not a constant-factor characterization. Whether that logarithm can be removed is not resolved here.

— Variance-Aware Non-Stationary Bandits:All-Budget Regret Bounds and Adaptation Costs  (2610.26116 - Zhang, 19 Oct 2026) in Section 5, immediately following Theorem 2 (Theorem \ref{thm:jointminimax})