Variance-Aware Non-Stationary Bandits:All-Budget Regret Bounds and Adaptation Costs

Xiaojin Zhang

Low reward variance reduces estimation cost, but does not eliminate monitoring of arms that may improve while unobserved. We refine the variance--variation tradeoff and characterize how both costs depend on supplied variance and variation budgets in bounded, non-stationary multi-armed bandits. Let $A$ be the number of arms, $T$ the horizon, $B$ a bound on consecutive mean variation excluding the initial vector, and $Q$ a bound on the sum of the largest per-round reward variances. For every admissible pair $Q,B$, the minimax expected dynamic regret is, within a single logarithmic factor, \[ \min\{T,\;A+\sqrt{AQ}+(AQBT)^{1/3} +\sqrt{AT}\min\{B,\sqrt B\}\}. \] The deterministic boundary $Q=0$ is characterized without logarithms. For small positive budgets, a safe calibration phase identifies a common interval containing all future means of surviving arms. Clipping observations to a wider interval reduces the cost of restarting a prior hybrid static-regret algorithm; an original-mean analytical reference controls its residuals without charging the variation of clipped means. Balancing clipping bias and monitoring closes the small-budget noisy regime. Fixed oblivious constructions…