Develop a sharp unknown-budget low-noise adaptation guarantee

Establish a sharp all-budget variance-sensitive dynamic-regret guarantee for a single policy that does not receive the variance budget Q or the mean-variation budget B, thereby adapting to unknown budgets in the general non-stationary multi-armed bandit model.

Background

The paper’s main minimax characterization assumes that the learner is supplied with both the variance and variation budgets. Its adaptation results show an obstruction for policies that must operate without a supplied variation budget, but the paper does not provide a general policy that adapts simultaneously to unknown variance and variation budgets while retaining the sharp low-noise rates. The authors explicitly identify this general unknown-budget scheduling problem as unresolved.

References

The general unknown-budget low-noise schedule remains open; it is not a consequence of either characterization.

— Variance-Aware Non-Stationary Bandits:All-Budget Regret Bounds and Adaptation Costs  (2610.26116 - Zhang, 19 Oct 2026) in Section 1, paragraph “Scope of the claims”

Removing the logarithmic gap in the noisy minimax bound, and obtaining matching adaptation frontiers for unrestricted noisy environments, remain open.

— Variance-Aware Non-Stationary Bandits:All-Budget Regret Bounds and Adaptation Costs  (2610.26116 - Zhang, 19 Oct 2026) in Conclusion