Contextual bandits are a standard framework for sequential decision-making under uncertainty, with applications in clinical trials, dosage selection, recommendation systems, and autonomous systems. Safety is central in many of these applications, since a single unsafe decision in settings such as dosage selection or autonomous driving can have catastrophic consequences. A common way to model safety in bandit problems is to associate each action with both a reward signal and a cost signal, and to optimize reward subject to constraints on cost. Most existing safety-constrained bandit models enforce safety by requiring the expected cost of each action to remain below a prescribed threshold. However, this may be insufficient in heteroscedastic settings, where the chosen action affects not only the expected reward and cost, but also the variability of the observed outcomes. We study contextual bandits with one-dimensional continuous actions and stage-wise high-probability constraints on the realized cost. We propose High-Probability Constrained UCB, an optimistic-pessimistic algorithm that explores for reward while conservatively estimating the safe action set. For linear reward and cost models, we prove a tight $\tilde{\mathcal{O}}(d\sqrt{T})$ regret bound, and we extend the analysis to general function classes using the eluder dimension. Experiments show that enforcing realized-cost safety substantially reduces violations compared with expected-cost constrained baselines.
We introduce Gated Decoupled Compositional Bandits (GDCB), a family of contextual bandit algorithms with three structural innovations that jointly fall outside the taxonomy of LinUCB, LinTS, HierTS, factored bandits, neural contextual bandits, and RLHF. In a GDCB system: (i) the action delivered to the environment is the composition of a nominal arm, drawn by a discrete or hierarchical bandit, with a context-dependent scaler; (ii) the scaler parameter is learned in a separate supervised loop, not jointly with arm selection; and (iii) every action passes through a pre-execution gate that may modify or veto the composed action before it reaches the environment. We formalise this class of algorithms, prove four structural theorems characterising its statistical behaviour, and show that six industrially significant systems -- short-term rental dynamic pricing, clinical drug dosing, credit origination, grid demand response, content moderation, and LLM tool-use agents -- are all instances of GDCB, differing only in the composition operator, scaler family, and gate. The central result is the Decoupling Variance Reduction theorem: a well-calibrated scaler removes context-induced variance from the arm-to-reward mapping, turning a non-stationary bandit problem into an approximately stationary one. The Gate-Induced Equivalence theorem shows that under a stationary gate, historical data collected under any prior policy is a valid warm-up initialiser without importance-sampling correction, generalising the companion P-HITL result (arXiv:2606.02595) from human approval to arbitrary gates. In regulated, high-stakes domains, constraints usually treated as deployment frictions -- approval gates, compliance rules, safety shields -- are the mechanism that makes fast deployment possible, not an obstacle to it. The companion paper validates instance 1 (STR dynamic pricing) on real production data.
Online decision making often requires navigating a landscape shaped by both dynamic contexts and strategic interactions. In competitive pricing, for example, hotels must account for both dynamic contextual factors and rivals' strategic responses. Existing approaches address only part of this challenge: contextual bandits optimize single-agent decisions using observable features but ignore multi-player interactions, while online matrix games capture strategic behavior through Nash equilibrium but assume fixed payoffs, ignoring contextual information. How should agents act then when strategic payoffs evolve with contextual signals? We introduce \emph{online contextual matrix games} to integrate contextual information into multi-player online games. We further propose \emph{OnGameLearn}, an online learning algorithm that efficiently balances exploration and exploitation across both player actions and contexts. This approach comes with statistical guarantees: tail bounds for the estimated payoff matrix, the convergence of the estimated Nash equilibrium, the asymptotic normality of the parameter estimators, and the sublinear regret bound. We also develop the notion of \emph{policy value} in matrix games and develop a doubly robust, $\sqrt{T}$-consistent estimator for it. Across simulated studies and a real-world hotel pricing application, we find that OnGameLearn effectively navigates the intertwined challenges of strategic and contextual decision-making.
We study new algorithms for Contextual Bandits with Knapsack. In these problems, there are finitely many types of customers, products, and resources. Each product is made from a fixed combination of resources, and resources have finite capacity. A decision maker must assign each arriving customer one out of a set of multiple possible products. Every assignment of a customer to a product will generate a random reward, which equals an unknown linear function of customer and product features, plus a noise term. The objective is to jointly learn the mean reward function, and to make online assignments to minimize the expected revenue loss relative to an optimal policy that knows the reward function. We propose a natural and simple extension of the Upper-Confidence-Bound (UCB) family of algorithms and apply re-optimization techniques. We show that by taking advantage of re-optimization, our algorithm achieves an average regret of $O(\frac{(\ln T)^3}{T})$ where $T$ is the horizon length. Our bound significantly reduces the $O(\frac{1}{\sqrt{T}})$ bound in the literature for closely related dynamic-pricing problems that are based on re-optimization.
Contextual bandits offer a natural framework for sample-efficient personalization, but practical deployment remains difficult under sparse, biased interaction data, unreliable uncertainty estimates, and severe cold starts. We study whether pre-trained tabular foundation models with in-context learning can be turned into randomized policies for online decision making. We propose BC-ICL (Bootstrap-conditioned action selection using ICL), which at each round draws a bootstrap resample of the interaction history, conditions a frozen pre-trained ICL model on that resample, scores all actions, and selects the action with the highest sampled score. We further introduce an arm-context conditioning architecture that promotes shared statistical strength across actions and helps avoid common bootstrap failure modes of isolated-arm bandits. Empirically, this policy delivers strong early-round regret and regret performance on standard contextual bandit suites, outperforming established baselines under a strict online protocol.
Zheyuan Zhang, Manqing Mao, Hong Wang +8cs.LG cs.CL
Critic-free group-based reinforcement learning has become a scalable approach for post-training large language models. However, most existing methods allocate the same number of rollouts to every task and trajectory state, even though some rollouts provide much more useful learning signals than others. Recent work has started to treat rollout generation as an adaptive decision, but two important limitations remain. First, intervention strategies are often based on fixed heuristics and therefore cannot adjust as the policy changes during training. Second, these methods usually decide only how many rollouts to generate, without explicitly controlling where and how to intervene. To address these limitations, we propose Recoverability-Aware Intervention Learning (RAIL), a training-time framework that learns how to generate rollouts based on the improvement produced by each intervention. RAIL models intervention selection as an online contextual-bandit problem and trains a recoverability controller using intervention traces collected through a shadow-to-live procedure. This allows the controller to keep learning while the underlying policy evolves. We evaluate RAIL in terms of effectiveness, adaptivity, expressiveness, and efficiency. Across multiple settings, RAIL consistently improves performance under limited rollout budgets. These results show that recoverability-aware intervention provides a principled way to generate more informative and less redundant rollouts, leading to stronger learning signals during post-training.
This thesis studies policy learning in interactive systems where an agent observes a context, selects an action from a very large set, and receives partial feedback. The main framework is contextual bandits, with two paradigms: on-policy learning, where the agent interacts sequentially with the environment and minimizes regret, and off-policy learning, where it learns from logged data collected by a logging policy. In large action spaces, both settings face major challenges: inefficient exploration, sparse data coverage, high-variance importance weights, extrapolation bias, and difficult optimization landscapes. The first part develops structured Bayesian methods for on-policy learning. We introduce meTS, a mixed-effect extension of Thompson sampling, and dTS, which leverages diffusion-inspired priors to model dependencies between actions. These methods share information across actions and yield regret guarantees depending on an effective number of actions. The second part addresses off-policy learning. We propose sDM, a structured direct method based on latent variables, show that optimization error can dominate estimation error in large action spaces, and introduce concave, efficiently optimizable policy-weighted log-likelihood objectives. Finally, we develop differentiable pessimistic methods based on exponential smoothing and PAC-Bayesian bounds to control the bias-variance trade-off of regularized importance-sampling estimators.
Off-Policy Evaluation and Learning (OPE/L) in contextual bandits is rapidly gaining popularity in real systems because new policies can be evaluated and learned securely using only historical logged data. However, existing methods in OPE/L cannot handle many challenging but prevalent scenarios such as few-shot data, deterministic logging policies, and new actions. In many applications, such as personalized medicine, content recommendations, education, and advertising, we need to evaluate and learn new policies in the presence of these challenges. Existing methods cannot evaluate and optimize effectively in these situations due to the notorious variance issue or limited exploration in the logged data. To enable OPE/L even under these unsolved challenges, we propose a new problem setup of Cross-Domain OPE/L, where we have access not only to the logged data from the target domain in which the new policy will be implemented but also to logged datasets collected from other domains. This novel formulation is widely applicable because we can often use historical data not only from the target hospital, country, device, or user segment but also from other hospitals, countries, devices, or segments. We develop a new estimator and policy gradient method to solve OPE/L by leveraging both target and source datasets, resulting in substantially enhanced OPE/L in the previously unsolved situations in our empirical evaluations.
Joshua Spear, Matthieu Komorowski, Rebecca Pope +2cs.LG
This article presents a novel estimator for performing off-policy evaluation using only offline data for contextual bandits. The proposed estimator, Kernel-WIS is demonstrated to be asymptotically consistent and to empirically outperform strong baselines (including vanilla weighted importance sampling), particularly under complex conditions including behaviour policy miss-specification. The benefit of Kernel-WIS is derived from combining the bounded property of vanilla weighted importance sampling with the linearity of vanilla importance sampling.
We study the contextual combinatorial semi-bandit (CCSB) problem with general reward function approximation. At each round, the learner observes a context, selects a combinatorial action consisting of a subset of basic arms, and receives the reward of each selected arm; the goal is to maximize the cumulative reward over time. We propose SquareCB.Comb, a computationally efficient algorithm that, at each round, solves a convex optimization problem to sample a combinatorial action that balances exploration and exploitation. SquareCB.Comb scales to large arm sets and imposes no structural assumptions on the action set beyond a cardinality bound of $m$ on each combinatorial action. We prove that SquareCB.Comb achieves a minimax optimal regret bound of $O(\sqrt{m A T \log |\mathcal{F}|})$, where $A$ is the number of arms, $m$ is the maximum number of arms in a combinatorial action, $T$ is the time horizon, and $\mathcal{F}$ is the reward function class. In the realizable setting, this bound matches the state-of-the-art regret guarantees achieved by policy search-based algorithms in the more restricted slate recommendation settings, while simultaneously generalizing to arbitrary combinatorial action structures and general reward function approximation.
Heesang Ann, Taehyun Hwang, Min-hwan Ohstat.ML cs.LG
Existing contextual multinomial logit (MNL) bandits model relevance-driven choice but ignore the potential benefits of within-assortment diversity, while submodular/combinatorial bandits encode diversity in rewards but lack structured choice probabilities. We bridge this gap with the $\textit{diversified multinomial logit}$ (DMNL) contextual bandit, which augments MNL choice probabilities with a generally submodular diversity function, thereby formalizing the relevance--diversity trade-off within a single model. Incorporating diversity renders exact MNL assortment optimization intractable. We propose a $\textit{white-box}$ UCB-based algorithm, $\texttt{OFU-DMNL}$, that constructs assortments item-wise by maximizing optimistic marginal gains, avoids black-box optimization oracles. We show that $\texttt{OFU-DMNL}$ achieves at least a $(1-\frac{1}{e+1})$-$\textit{approximate}$ regret bound $\tilde{O}\left(d \sqrt{T/K}\right)$, where $d$ is the context dimension, $K$ the maximum assortment size, and $T$ the horizon, and attains an improved approximation factor over standard submodular baselines. Experiments demonstrate consistent gains and, relative to exhaustive enumeration, comparable regret with substantially lower runtime. Overall, DMNL bandits provide a practical foundation for diversity-aware assortment optimization under uncertainty, and $\texttt{OFU-DMNL}$ offers a statistically and computationally efficient solution.
We investigate the contextual slate bandit problem with generalized linear rewards under limited adaptivity. At each round, the learner is presented with $N$ sets of items, where each item is represented by a $d$-dimensional feature vector. The learner then constructs a slate by selecting one item per set; the resulting slate yields a scalar reward sampled from a Generalized Linear Model (GLM). We propose algorithms under two limited-adaptivity settings: (a) Batched and (b) Rarely-Switching. For the batched setting, we introduce B-SlateGLinCB, which partitions the time horizon into $\mathcal{O}(\log\log T)$ batches such that each batch's policy relies only on data from previous batches. For the rarely-switching setting, we propose RS-SlateGLinCB, which adaptively performs only $\mathcal{O}(Nd\log T)$ parameter updates. Under a diversity assumption on the item sequences, we prove that B-SlateGLinCB and RS-SlateGLinCB achieve regret bounds of $\mathcal{O}(Nd^{3/2}\sqrt{T})$ and $\mathcal{O}(Nd\sqrt{T})$, respectively. Notably, both bounds are independent of the non-linearity parameter $κ$ that is typically found to scale the regret of GLM bandit algorithms. Our algorithms are computationally efficient, requiring only $\text{poly}(N)$ time per round despite $2^{Ω(N)}$ possible slates. Simulations show our algorithms outperform existing baselines with limited adaptivity and remain competitive with Slate-GLM-OFU, a fully adaptive state-of-the-art algorithm. Notably, a slightly modified B-SlateGLinCB empirically matches this baseline. Finally, we demonstrate strong performance in a practical in-context example selection task for language models.
Contextual bandits with graph-structured arms arise in recommendation, citation retrieval, and social advertising, where arms connected on a graph tend to share reward signal. Standard dimensionality reduction ignores this structure, inflating exploration cost by a factor of $d/k$. We propose GraphDR-LinUCB, which projects arm features onto the graph's low-frequency spectral subspace and runs linear UCB in the resulting $k$-dimensional space. We prove the first $\wtO(k\sqrt{T})$ regret bound for spectral-projection-based contextual bandits, reducing dimension dependence from $d$ to $k$; a perturbation argument extends this to noisy graphs, with an explicit penalty for reward-smoothness mismatch and graph-estimation error. Our central theoretical finding is that the high-frequency reward component need not incur a worst-case linear-in-$T$ penalty: its actual cost depends on its realized impact along the played path, not on its total energy. A simple spectral comparison between subspaces ($Γ_k$) predicts which reducer wins on a given dataset, correctly calling five of six real-dataset outcomes without any fitted threshold. Across a synthetic benchmark and six real datasets (MovieLens, Amazon, LastFM, ogbn-arxiv, MIND), GraphDR-LinUCB reduces cumulative regret by $15\times$ over full-dimensional LinUCB and outperforms competing graph-aware methods on five of six; the single failure is precisely where the graph's spectral subspace is misaligned with the reward.
Tina Dongxu Li, Mouhacine Benosman, Ken Meszaros +1cs.LG eess.SY
Efficient sorter diversion control of automated material handling systems (MHS) is critical for optimizing operational efficiency in large-scale warehouse environments. In this study, we use an inbound receiving sorter at a high-volume e-commerce warehouse as our primary use case, where the sorter diversion system relies on cost functions with static weight configurations that fail to adapt to highly dynamic system contexts, such as volume mode, congestion level, equipment physical status, and upstream/downstream dependencies. To address this real-time sorter diversion optimization challenge, we conducted a comparative study of three candidate hybrid machine learning frameworks: Linear Regression with Gradient Descent Optimization (LR+GDO), XGBoost with Bayesian Optimization (XGB+BO), and Bayesian Contextual Bandits (BCB). Model training and evaluation were enabled by leveraging a high-fidelity physics-aware emulator to overcome the cold-start problem and allow a safe transition from offline to online learning. We performed comprehensive evaluations including reward model predictive accuracy, contextual sensitivity, action distribution, and projected reward uplift. Our results demonstrate that while tree-based reward models offer slightly better predictive power, the BCB framework achieved overall higher performance with 2.03% reward uplift over the heuristic baseline. Furthermore, BCB exhibits several superior characteristics, such as its decisive time-optimal policy backed by Bang-Bang control theory, continuous online learning capability, strategic balance between exploration and exploitation, and significantly shorter inference latency. These results demonstrate the potential of the BCB framework for real-time control optimization in large-scale warehouse environments, motivating further investigation toward operational deployment.
We study non-stationary linear contextual bandits where the reward model drifts over time, rendering classical contextual bandit algorithms brittle because historical data becomes systematically biased. We propose Flow-Corrected Thompson Sampling (fcTS), a Bayesian method that reuses experience by transporting past rewards to the present using an explicit drift model and incorporating each transported observation with a confidence weight that reflects transport reliability. This yields a unified template that specializes in (i) linear parameter drift via online slope estimation and reward correction, (ii) periodic variation via phase-aware reuse across cycles, and (iii) recurring regime switches via changepoint detection and regime-specific posterior memory. The resulting posterior updates remain closed-form under a linear Gaussian model and can be implemented efficiently with truncated, incrementally updated sufficient statistics. Across five controlled case studies and a semi-synthetic portfolio-selection benchmark with multiple overlapping non-stationarities, fcTS outperforms standard forgetting-based baselines (discounting, sliding windows, and periodic restarts), with the largest gains in settings exhibiting recurring temporal structure. These results demonstrate that when non-stationarity is structured, correcting and reweighting historical observations can be substantially more sample-efficient than uniformly discarding them.
We present AdaPrivate-TS, a differentially private contextual bandit algorithm that combines Thompson Sampling with batched zCDP composition. Our key insight is that differential privacy noise inflates the posterior covariance in a structured way: adding Gaussian noise $N(0,σ^2 I)$ to $b$ yields sampling covariance $v^2 A^{-1} + σ^2 A^{-2}$, which Thompson Sampling interprets as increased uncertainty rather than pure corruption. Under event-level privacy (protecting individual interactions) with stochastic contexts, we prove that the privacy cost is only $O(\sqrt{d}\,\log T/\sqrtρ)$, logarithmic in $T$, because parallel composition amortizes noise across batches. Additionally, we explore privacy amplification via Poisson subsampling, which can reduce effective noise at stringent privacy budgets. Experiments on synthetic and real-world datasets demonstrate: (1) AdaPrivate-TS achieves 93-99% of non-private performance at $\varepsilon \in [0.5, 5]$, outperforming UCB by 0.5-3.7% and up to 18% with tuned adaptive exploration at extreme $\varepsilon$; (2) privacy amplification provides additional 2-5% gains at low $\varepsilon$; (3) on MovieLens and Jester, AdaPrivate-TS achieves the best overall performance among event-level baselines, dominating at $\varepsilon \geq 2$; (4) under DP-SVD private features, TS's advantage over UCB grows to +11%, confirming noise-as-uncertainty is not limited to reward privacy. We provide rigorous proofs for privacy guarantees under interactive zCDP composition and comprehensive evaluation including convergence curves, 12-seed CIs, and DP-SVD feature ablation.
This paper considers stochastic linear contextual bandits (SLCB) with bounded reward noise. Existing works typically assume sub-Gaussian reward noise and bounded expected rewards, under which the optimal regret bound scales as $\tilde{O}(\sqrt{T})$ in terms of horizon $T$. However, in many applications, realized/observed rewards are also naturally bounded, implying bounded reward noise. Bounded noise is more informative than the sub-Gaussian condition but has not been leveraged explicitly in the SLCB literature. In this paper, we propose a novel algorithm SME-OFU by utilizing an uncertainty quantification method called set-membership estimation (SME) and applying the principle of optimism in the face of uncertainty (OFU). Our algorithm enjoys an improved regret bound $O(\log T)$. Notice that this does not contradict the existing optimal bound $\tilde{O}(\sqrt{T})$ for sub-Gaussian noise because bounded noise is a stronger condition. Finally, simulations show empirical improvements of SME-OFU over a benchmark algorithm designed for sub-Gaussian noise when the reward noise is bounded.
Yan Dai, Negin Golrezaei, Patrick Jailletcs.LG cs.AI stat.ML
Modern recommendation systems increasingly rely on dynamically routing diverse queries to multiple embedding models. Despite its practical significance, this problem remains poorly understood under realistic conditions like adversarial queries, bandit feedback, and limited observability of models. We formalize embedding model routing as an adversarial contextual linear bandit with low-rank experts, where contexts are queries, actions are items, and experts are the embedding models working on low-rank latent representation spaces. We first establish that standard regret notions suffer from structural misspecification or statistical intractability, and we identify a log-quadratic policy class that is expressive enough to capture query-dependent model routing, yet structured enough to allow efficient online learning. Second, we propose a policy gradient algorithm called Hypentropy Policy Gradient (HPG). It provably adapts to the unknown low-rank structure under incomplete information and attains $\tilde{\mathcal O}(s\sqrt{M T})$ linearized policy regret -- where $s, M$, and $T$ are the intrinsic rank of the experts, the number of models, and the number of rounds -- thus avoiding a curse of dimensionality. Finally, we also provide an computationally efficient and parameter-free implementation of HPG.
We consider a variant of the linear contextual stochastic multi-armed bandits, where the learner must provide recommendations to a group of users, each having its personalized preference vector, and in the presence of context distributions that are drifting over time. Under practitioner-friendly assumptions, we reduce this setting to linear bandit with stationary mean but heteroskedastic and non-stationary noise. We further study the case when the learner must ensure the mean reward of each decision must exceed that of a baseline strategy $\boldsymbolπ_0$ at each decision step. We introduce Dri-MED, an algorithm inspired from the linear version of the MED strategy, and carefully adapted to handle the non-stationary heteroskedastic noise. We show that the instance-dependent regret scales as $\tilde{\mathcal O}\left(\fracκ{\tildeΔ}d^2(\log(T)\right)$, where $\tildeΔ$ is the constraint-aware sub-optimality gap subject to policy $π_0$, with variance-aware multiplicative term $κ$ that we carefully handle using heteroskedastic regression. We further show Dri-MED enjoys $\tilde{\mathcal{O}}(d)$ expected constraint violations. Our numerical results suggest that Dri-MED significantly outperforms conservative baselines that ignores the drift and preference structure.
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 study KL-regularized contextual bandits and episodic reinforcement learning (RL) under general function approximation with model misspecification. Existing guarantees rely on realizability and therefore do not extend to misspecified models, where classical regret bounds may fail. This work introduces KL misspecification formulations for contextual bandits and episodic RL and analyzes regression-based algorithms with Gibbs policy updates. High-probability KL-regret guarantees with explicit misspecification terms are established, recovering the standard realizable KL-regularized setting as a special case.
Anthony GX-Chen, Ankit Anand, Gheorghe Comanici +7cs.LG cs.AI
Classical reinforcement learning (RL) typically seeks a deterministic policy that maximizes the expected sum of a scalar reward. Yet, modern applications such as language model fine-tuning or scientific discovery demand diversity. Existing remedies such as entropy regularization or diversity bonuses often require fragile trade-offs that sacrifice performance for stochasticity or rely on heuristic metrics that can misalign policy rankings. We argue that diversity is more naturally understood as the rational response to uncertainty in the reward. When the reward function is not perfectly known--as is the case with ambiguous preferences or imperfect reward models--committing to a single action can be sub-optimal. Building on this, we propose a fundamental reformulation of the RL objective by replacing the scalar reward with a distribution over reward functions, and applying a non-linear objective over sets of actions. The result is a framework in which calibrated behavioural diversity emerges naturally, remains controllable through the reward function distribution, and is obtained without sacrificing expected reward. Focusing on the contextual bandit setting, we derive a principled gradient estimator for this objective and prove that our formulation naturally generalizes both vanilla policy gradient and more recently developed action-set approaches. Our empirical results demonstrate that this framework offers a robust and theoretically grounded alternative for complex RL tasks where the traditional formulation of the problem fails to induce the desired breadth of agent behaviour.
We study offline-to-online learning in linear contextual bandits with biased offline regression data: the offline parameter need not match the online one, so history should not be treated as a single warm start. We model directional transfer with a shift certificate $(M_{\mathrm{shift}},ρ)$ and offline ridge estimation, yielding a geometry-aware confidence region for the online parameter rather than an isotropic radius. We propose \emph{Ellipsoidal-MINUCB}, which combines a standard online branch with an offline-informed pooled branch and uses offline information only when it tightens uncertainty. With high probability, regret is bounded by the minimum of a standard SupLinUCB-style fallback and a pooled term that separates statistical width from a certificate-weighted shift penalty. Under a simple alignment condition, the pooled term further simplifies to a rate governed by an effective dimension induced by the offline geometry. We also show that a purely Euclidean (scalar) shift bound, by itself, does not determine which feature directions are transferable. Beyond this fixed certificate, we show how to learn a data-driven certificate from data at finitely many refresh times and establish a high-probability regret bound for Ellipsoidal-MINUCB with epoch-wise learned certificates. Experiments match the main prediction: gains are strongest at intermediate horizons when offline coverage and transferability align, while the method otherwise tracks the safe online baseline.