A Direct Approach for Handling Contextual Bandits with Latent State Dynamics

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.

Recent work shows that when contexts are drawn i.i.d., linear contextual bandits can be reduced to single-context linear bandits. This ``contexts are cheap'' perspective is highly advantageous, as it allows for sharper finite-time analyses and leverages mature techniques from the linear bandit literature, such as those for misspecification and adversarial corruption. However, this reduction crucially relies on the independence of contexts and does not extend to settings with temporally correlated (e.g., Markovian) contexts, which arise frequently in practice. Motivated by applications with temporally correlated availability, we extend this perspective to linear bandits with Markovian context processes, where the action set evolves via an exogenous Markov chain. Our main contribution is a reduction that applies under uniform geometric ergodicity. We construct a stationary surrogate action set to solve the problem using a standard linear bandit oracle, employing a delayed-update scheme to control the bias induced by the nonstationary conditional context distributions. We further provide a phased algorithm for unknown stationary distributions that learns the surrogate mapping online. In both settings, we obtain a high-probability worst-case regret bound matching that of the underlying linear bandit oracle in sufficiently fast mixing regimes. We then validate our results on a real-world instance, where we show practical gains over a LinUCB baseline.

We study contextual bandit problems with correlated arms and access to surrogate reward signals produced by a machine learning model, motivated by applications such as large language model (LLM) routing. Unlike classical contextual bandits that rely solely on bandit feedback and assume conditional independence across arms, our setting allows context-dependent inter-arm correlations and auxiliary reward information that may be noisy or misspecified. We propose algorithms that leverage such surrogate rewards through two complementary designs. A coupled reward-mixing approach pools true and surrogate rewards to accelerate learning when surrogate signals are reliable, while a decoupled prediction-mixing approach maintains separate estimators for bandit feedback and surrogate rewards and adaptively combines their predictions. This decoupling yields robustness to surrogate misspecification, recovering regret guarantees comparable to reward-only bandit methods in the worst case, while achieving improved regret when surrogate predictions are sufficiently informative. We provide theoretical regret analyses for both approaches and evaluate them on LLM routing benchmarks under varying accuracy versus cost trade-offs. The results demonstrate improved sample efficiency and consistently better accuracy-cost trade-offs compared to standard contextual bandit baselines and strong static routing methods.