On the Peril of (Even a Little) Nonstationarity in Satisficing Regret Minimization
This work addresses a fundamental limitation in bandit algorithms for decision-makers facing nonstationary environments, revealing that incremental changes can drastically worsen regret guarantees.
The paper tackles the problem of satisficing regret minimization in nonstationary K-armed bandits, showing that even minimal nonstationarity (L≥2 segments) leads to optimal regret scaling as Θ(L log T), in contrast to the stationary case (L=1) where constant regret is achievable.
Motivated by the principle of satisficing in decision-making, we study satisficing regret guarantees for nonstationary $K$-armed bandits. We show that in the general realizable, piecewise-stationary setting with $L$ stationary segments, the optimal regret is $Î(L\log T)$ as long as $L\geq 2$. This stands in sharp contrast to the case of $L=1$ (i.e., the stationary setting), where a $T$-independent $Î(1)$ satisficing regret is achievable under realizability. In other words, the optimal regret has to scale with $T$ even if just a little nonstationarity presents. A key ingredient in our analysis is a novel Fano-based framework tailored to nonstationary bandits via a \emph{post-interaction reference} construction. This framework strictly extends the classical Fano method for passive estimation as well as recent interactive Fano techniques for stationary bandits. As a complement, we also discuss a special regime in which constant satisficing regret is again possible.