Bandits for Efficient Experimentation: Adapting to Control Group, Preferences, and Context Drifts

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.

We study the contextual multi-armed bandit problem with a finite context space (a.k.a. subpopulations), where the learner recommends a best action for each context and is evaluated by context-weighted simple regret. Our guarantees are worst-case over the reward distributions, while remaining instance-dependent with respect to the context distribution vector $p$. Akin to experimental design problems where the population of interest is fixed but the sampled subpopulation can be controlled, we allow the learner to actively choose which context to sample from. For a known $p$, we characterize tight regret rates: passive sampling where contexts are randomly revealed achieves regret of order $\sqrt{n/T \, \lVert p \rVert_{1/2}}$, whereas active sampling with allocation $q_j \propto p_j^{2/3}$ achieves the tight rate $\sqrt{n/T} \, \lVert p \rVert_{2/3}$. The resulting improvement can be as large as $Θ(k^{1/4})$, where $k$ is the number of contexts. We further extend the analysis to budgeted active sampling, characterize the corresponding tight rate, and identify when a limited active budget suffices to recover the fully active rate. When $p$ is unknown, we propose the Explore-Explore-Then-Commit (EETC) algorithm, which optimally balances estimating the context distribution and the time to switch to active allocation, such that for large horizons, it matches the known-$p$ active rate up to constants. Experiments on synthetic and real-world data support our theoretical findings.

A key goal in stochastic contextual linear bandits is to efficiently learn a near-optimal policy. Prior algorithms for this problem learn a policy by strategically sampling actions but naively (passively) sampling contexts from the underlying context distribution. However, in many practical scenarios -- including online content recommendation, survey research, and clinical trials -- practitioners can actively sample or recruit contexts based on prior knowledge of the context distribution. Despite this potential for active learning, the role of strategic context sampling in stochastic contextual linear bandits is underexplored. We propose an algorithm that learns a near-optimal policy by strategically sampling rewards of context-action pairs. We prove instance-dependent theoretical guarantees demonstrating that our active context sampling strategy can improve over the minimax rate by up to a factor of $\sqrt{d}$, where $d$ is the linear dimension. We show empirically that our algorithm reduces the number of samples needed to learn a near-optimal policy, in tasks such as warfarin dose prediction and joke recommendation.