Active Context Selection Improves Simple Regret in Contextual Bandits

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.

Contextual queueing bandits provide a framework for learning to schedule heterogeneous jobs under unknown context-dependent service rates. Under stochastic contexts, existing algorithms achieve $\widetilde{\mathcal{O}}(T^{-1/4})$ queue length regret, defined as the expected difference between the learner's and oracle's queue lengths at horizon $T$. In this paper, we improve this rate to $\widetilde{\mathcal{O}}(T^{-1/2})$. The key observation is that random exploration is needed only up to a carefully chosen cutoff round, rather than throughout the entire horizon. We propose CQB-$η$-2, a three-phase algorithm: (i) pure random exploration to construct an initial estimator, (ii) $η$-random exploration combined with a UCB rule to continue learning while maintaining negative drift, and (iii) pure UCB after the exploration cutoff. Our proof decomposes the queue length regret at the cutoff round. Before the cutoff, negative drift suppresses queue length differences caused by suboptimal choices. After the cutoff, the first two phases provide sufficient random exploration samples, ensuring that UCB decisions incur small departure-rate gaps. Combining these two bounds yields queue length regret of order $\widetilde{\mathcal{O}}(T^{-1/2})$. We further prove a minimax lower bound of order $Ω(T^{-1/2})$. The proof constructs two hard instances that are statistically indistinguishable up to the final service decision, and uses a queue-specific coupling argument to convert the resulting testing error into queue length regret. Together, our upper and lower bounds characterize the minimax dependence on the horizon $T$ up to logarithmic factors.

We propose an algorithmic framework, Offline Estimation to Decisions (OE2D), that efficiently reduces contextual bandit learning with general reward function approximation to offline regression. The framework allows near-optimal regret for contextual bandits with large action spaces with $O(\log T)$ calls to an offline regression oracle over $T$ rounds, and makes $O(\log\log T)$ calls when $T$ is known. The design of OE2D algorithm generalizes Falcon~\citep{simchi2022bypassing} and its linear reward version~\citep[][Section 4]{xu2020upper} in that it finds an action distribution that we term ``exploitative F-design'' that simultaneously guarantees low regret and good coverage, striking a balance between exploration and exploitation. Central to our regret analysis is a new complexity measure, the Decision-Offline Estimation Coefficient (DOEC), which we show is small in many settings, including bounded Eluder dimension per-context and the smoothed regret setting. We also establish a relationship between DOEC and Decision Estimation Coefficient (DEC)~\citep{foster2021statistical}, bridging the design principles of offline- and online-oracle efficient contextual bandit algorithms for the first time.