On the Peril of (Even a Little) Nonstationarity in Satisficing Regret Minimization

TL;DR

This paper investigates the impact of even minimal nonstationarity on satisficing regret in multi-armed bandits, revealing that regret necessarily scales with time in nonstationary environments.

Contribution

It introduces a novel Fano-based analytical framework for nonstationary bandits and characterizes the regret scaling with the number of stationary segments.

Findings

Optimal regret scales as Θ(L log T) with L stationary segments.

In stationary case (L=1), constant satisficing regret is achievable.

Even slight nonstationarity causes regret to grow with T.

Abstract

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 $\Theta(L\log T)$ 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 $\Theta(1)$ 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.