The Price of Current-Effect Inference in Nonstationary Multi-Armed Bandits

Sharan Sahu, Martin T. Wells

Adaptive experiments use bandit algorithms to favor interventions with higher expected rewards. However, decision-makers also require reliable inference on current treatment effects: differences between intervention and reference-arm mean rewards at the present time, rather than historical averages. These objectives conflict: reward maximization favors better interventions, whereas current-effect inference requires continued exploration of potentially suboptimal interventions as outcomes change over time. We characterize this tradeoff for $K$-arm bandits with Hölder-smooth mean rewards of exponent $β\in(0,1]$ and constant $L$. Requiring high-probability simultaneous coverage of all current effects by confidence intervals of width at most $w$, we establish matching minimax bounds, up to logarithmic factors, on expected dynamic regret. Over horizon $T$, our algorithm TRACE (Tracking Rewards and Current Effects) achieves regret of order $\widetilde{O}_β\left(\sqrt{KT}+K^{β/(2β+1)}L^{1/(2β+1)}T^{(β+1)/(2β+1)}+Kw^{-2}(1+L^{1/β}w^{-1/β})\right)$, where the notation hides logarithms and constants depending only on $β$. This regret--precision tradeoff separates reward-learning regret…