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

Motivated by the concept of satisficing in decision-making, we consider the problem of satisficing regret minimization in bandit optimization. In this setting, the learner aims at selecting satisficing arms (arms with mean reward exceeding a certain threshold value) as frequently as possible. The performance is measured by satisficing regret, which is the cumulative deficit of the chosen arm's mean reward compared to the threshold. We propose SELECT, a general algorithmic template for Satisficing REgret Minimization via SampLing and LowEr Confidence bound Testing, that attains constant expected satisficing regret for a wide variety of bandit optimization problems in the realizable case (i.e., a satisficing arm exists). As a complement, SELECT also enjoys the same (standard) regret guarantee as the oracle in the non-realizable case. To further ensure stability of the algorithm, we introduce SELECT-LITE that achieves a light-tailed satisficing regret distribution plus a constant expected satisficing regret in the realizable case and a sub-linear expected (standard) regret in the non-realizable case. Notably, SELECT-LITE can operate on learning oracles with heavy-tailed (standard) regret distribution. More importantly, our results reveal the surprising compatibility between constant expected satisficing regret and light-tailed satisficing regret distribution, which is in sharp contrast to the case of (standard) regret. Finally, we conduct numerical experiments to validate the performance of SELECT and SELECT-LITE on both synthetic datasets and a real-world dynamic pricing case study.

Many online decision-making problems involve both round-specific feasible actions and drifting reward models: eligible ad impressions, feasible prices, and available treatments can change over time, while user preferences, demand curves, and patient responses may evolve. Motivated by these applications, we study non-stationary linear bandits with round-specific feasible decision sets. Existing methods that obtain the optimal $\widetilde O(T^{2/3}P_T^{1/3})$ dependence, where $P_T$ is the path length of the reward-parameter sequence, impose an orthogonal-structure assumption on round-specific decision sets, which can be restrictive in contextual applications. We address this gap through a unified misspecification-reduction viewpoint: after partitioning the horizon into blocks, we relate each block's dynamic regret to regret against a fixed-parameter linear bandit benchmark, with the within-block parameter drift entering as bounded misspecification. Restarting algorithms with misspecification-dependent regret guarantees then yields the optimal $T^{2/3}P_T^{1/3}$ dynamic-regret dependence for both linear bandits with general compact decision sets and $K$-armed contextual linear bandits.

Multi-armed bandit algorithms are evaluated by regret, yet comparable regret can coexist with different allocations across independent runs. We study the trade-off between worst-case regret $\mathcal{R}_{K,T}$ and instability $\mathcal S_{K,T}$, defined as the largest standard deviation of a terminal pull count, for $K$ arms and $T$ rounds. We prove the finite-time lower bound $\mathcal R_{K,T}\mathcal S_{K,T}\ge C T^{3/2}$, where $C$ is independent of $K$ and $T$, under a finite-time regret condition and without the regularity assumptions imposed in the prior asymptotic analysis. We also introduce Stabilized Lower-Envelope UCB (\textup{SLE-UCB}), a new tunable algorithm combining a running lower-envelope index with a decreasing pull-count stabilizer. \textup{SLE-UCB} satisfies $\mathcal R_{K,T}\mathcal S_{K,T}=O(T^{3/2}\log K)$, with an implicit constant independent of $K$ and $T$, matching the lower bound exactly in $T$ and within a logarithmic factor in $K$. To prove the instability bound, we develop a new offline top-prefix representation that removes path dependence from online decisions. Together with single-reward perturbations and the Efron--Stein inequality, this representation controls pull-count variance. Thus, regret and instability depend reciprocally on $K$, while their product has no polynomial dependence on $K$. These results resolve the open question raised in the literature concerning the sharp arm-dependent regret--instability frontier.