Stochastic Linear Contextual Bandits with Bounded Noise: A Set-Membership Approach

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.

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 under rare parameter updates: the learner may incorporate reward feedback into its parameter estimate only at a small number of update times, while still observing contexts online and selecting actions sequentially. This viewpoint clarifies a practical distinction that is often blurred in the literature: many "strictly batched" methods additionally restrict within-interval context adaptivity, meaning that the action rule inside an interval cannot depend on the sequence of realized contexts/actions in that interval (beyond the current round's context). For linear contextual bandits, we propose two practical algorithms with only $O(\log\log T)$ parameter updates. Our first algorithm BLCE-G attains minimax-optimal regret (up to polylogarithmic factors in $T$) simultaneously in both the small-$K$ and large-$K$ regimes under a static schedule. Our second algorithm BLCE removes the near G-optimal design step -- a dominant computational bottleneck in prior strictly batched static-grid methods -- yet preserves minimax-optimal regret and achieves the lowest known runtime complexity among optimal algorithms. We further extend these rare-update and computational principles to generalized linear contextual bandits. Overall, our results yield statistically optimal algorithms under $O(\log\log T)$ parameter updates that are also computationally efficient in practice.