Optimal Online Regret for Context-Lumpable 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.

We study stochastic linear contextual bandits with arbitrary action menus that may depend on the fixed parameter and the interaction history. We establish matching upper and lower bounds, up to logarithmic factors. Let $d$ be the dimension, $K$ be the menu size, and $T$ the time horizon. For $2\le K\le d$, we prove an upper bound $\widetilde O(K^{1/4}\sqrt{dT})$. When $T\ge d^2$, we further prove a lower bound $Ω(K^{1/4}\sqrt{dT})$. Thus, for $T\ge d^2$ and $2\le K\le d$, the upper and lower bounds match up to logarithmic factors, and the polynomial dependence on $K$ is optimal. Compared with the previous $\widetilde O(\sqrt{dKT})$ bound, our upper bound improves the dependence on $K$ by a factor of $K^{1/4}$. For $K\ge d$, we prove an upper bound $\widetilde O_{d,T}\left(\sqrt{dT}\min\{\sqrt d,(d\log K)^{1/4}\}\right)$ and a lower bound $Ω\left(\sqrt{dT}\min\left\{\sqrt d,\left(\frac{d\log K}{\log(2d)}\right)^{1/4}\right\}\right)$. Here, $\widetilde O_{d,T}$ omits logarithmic factors only in $d$ and $T$. In particular, for polynomially large $K\ge d$, the upper and lower bounds both scale as $d^{3/4}\sqrt T$ up to logarithmic factors, improving the standard $\widetilde O(d\sqrt T)$ rate by a factor of $d^{1/4}$. As $K$ grows further, the regret smoothly recovers the $d\sqrt T$ scale once $\log K$ reaches order $d$.

We study linear contextual bandits with paid observations, where at each round the learner observes a context, selects an action, and may pay a fixed cost to observe feedback from a subset of arms. We propose two Follow-the-Regularized-Leader algorithms with Best-of-Both-Worlds guarantees. The first, Agg-SPB, extends the SPB-matching framework of Tsuchiya and Ito (2024) by aggregating context-dependent stability terms, achieving the characteristic $T^{2/3}$ adversarial regret rate and logarithmic dependence on $T$ in stochastic environments. The second, CE-SPB, combines arm-dependent observation probabilities with an entropy-adaptive learning rate inspired by Kuroki et al. (2024). It achieves an entropy-adaptive $\widetilde{O}(T^{2/3})$ adversarial guarantee and polylogarithmic stochastic regret, while avoiding the minimum-context-mass dependence arising in the stochastic analysis of Agg-SPB. Both algorithms further extend to corrupted stochastic environments with explicit corruption-dependent guarantees. These results establish that logarithmic stochastic regret is compatible with the $T^{2/3}$ adversarial regime for linear contextual bandits with paid observations, while highlighting a tradeoff between sharper horizon dependence in stochastic settings and path-dependent matching without explicit minimum-context-mass dependence.