We revisit the contextual cascading bandit, where a learning agent recommends an ordered list (cascade) of items, and a user scans the list sequentially, stopping at the first attractive item. Although cascading bandits underpin various applications including recommender systems and search engines, the role of the cascade length K in shaping regret has remained unclear. Contrary to prior results that regret grows with K, we prove that regret actually decreases once K is large enough. Leveraging this insight, we design a new upper-confidence-bound algorithm built on online mirror descent that attains the sharpest known regret upper bound, O min{K pK 1,1}d Tfor contextual cascading bandits. To complement this new regret upper bound, we provide a nearly matching lower bound of Ω min{KpK 1,1}d T, where 0 p p < 1. Together, these results fully characterize how regret truly scales with K, thereby closing the theoretical gap for contextual cascading bandits. Finally, comprehensive experiments validate our theoretical results and show the effectiveness of our proposed method.

We study the generalized linear bandit (GLB) problem, a contextual multi-armed bandit framework that extends the classical linear model by incorporating a nonlinear link function, thereby modeling a broad class of reward distributions such as Bernoulli and Poisson. While GLBs are widely applicable to real-world scenarios, their non-linear nature introduces significant challenges in achieving both computational and statistical efficiency. Existing methods typically trade off between two objectives, either incurring high per-round costs for optimal regret guarantees or compromising statistical efficiency to enable constant-time updates. In this paper, we propose a jointly efficient algorithm that attains a nearly optimal regret bound with O(1)time and space complexities per round. The core of our method is a tight confidence set for the online mirror descent (OMD) estimator, which is derived through a novel analysis that leverages the notion of mix loss from online prediction. The analysis shows that our OMD estimator, even with its one-pass updates, achieves statistical efficiency comparable to maximum likelihood estimation, thereby leading to a jointly efficient optimistic method.

Thompson Sampling (TS) is widely used to address the exploration/exploitation tradeoff in contextual bandits, yet recent theory shows that it does not explore aggressively enough in high-dimensional problems. Feel-Good Thompson Sampling (FG-TS) addresses this by adding an optimism bonus that biases toward high-reward models, and it achieves the asymptotically minimax-optimal regret in the linear setting when posteriors are exact. However, its performance with \emph{approximate} posteriors, common in large-scale or neural problems, has not been benchmarked. We provide the first systematic study of FG-TS and its smoothed variant (SFG-TS) across fourteen real-world and synthetic benchmarks. To evaluate their robustness, we compare performance across settings with exact posteriors (linear and logistic bandits) to approximate regimes produced by fast but coarse stochastic-gradient samplers. Ablations over preconditioning, bonus scale, and prior strength reveal a trade-off: larger bonuses help when posterior samples are accurate, but hurt when sampling noise dominates. FG-TS generally outperforms vanilla TS in linear and logistic bandits, but tends to be weaker in neural bandits. Nevertheless, because FG-TS and its variants are competitive and easy-to-use, we recommend them as baselines in modern contextual-bandit benchmarks.

This paper studies efficient online algorithms for multinomial logistic bandits (MLogB), where the feedback distribution over $K+1$ outcomes follows a multinomial logistic model of $d$-dimensional action vectors. A representative UCB-type algorithm, OFUL-MLogB, achieves a regret bound of $\tilde{\mathcal{O}}(Kd\sqrt{T})$, but still requires $\mathcal{O}(K^3d^3)$ time and $\mathcal{O}(K^2d^2)$ space per round due to parameter estimation and optimistic reward construction, which is prohibitive in high-dimensional settings. To address this limitation, we propose EOFD-MLogB, which integrates frequent directions matrix sketching into OFUL-MLogB. By maintaining a low-rank SVD sketch of the accumulated Hessian, constrained online Newton updates in parameter estimation and $Kd \times K$ spectral-norm computations in the reward bonus are reduced to one-dimensional root-finding tasks and $K \times K$ eigenvalue computations, respectively. This yields dominant per-round time complexity $\mathcal{O}(Kd(m+K)^2)$ and space complexity $\mathcal{O}(Kd(m+K))$, where $m \ll d$ is the sketch size. We further prove a regret bound of $\tilde{\mathcal{O}}(Δ_T(Kd\lnΔ_T+m)\sqrt{T})$, where the sketching error factor $Δ_T$ is controlled by the $m$-truncated spectral tail of the Hessian. Thus, when the Hessian is approximately low-rank, the regret is close to that of OFUL-MLogB. Experiments validate the computational efficiency and competitive performance.

We study reinforcement learning for episodic Markov Decision Processes (MDPs) whose transitions are modelled by a multinomial logistic (MNL) model. Existing algorithms for MNL mixture MDPs yield a regret of $\smash{\tilde{O}(dH^2\sqrt{T})}$ (Li et al., 2024), where $d$ is the feature dimension, $H$ the episode length, and $T$ the number of episodes. Inspired by the logistic bandit literature (Abeille et al., 2021; Faury et al., 2022; Boudart et al., 2026), we introduce a problem-dependent constant $\barσ\_T \leq 1/2$, measuring the normalised average variance of the optimal downstream value function along the learner's trajectory. We propose an algorithm achieving a regret of $\smash{\tilde{O}(dH^2\barσ\_T\sqrt{T})}$, which recovers the existing bound in the worst case and improves upon it for structured MDPs. For instance, for KL-constrained robust MDPs, $\barσ\_T = O(H^{-1})$, reducing the horizon dependence by a factor $H$. We further establish a matching $\smash{Ω(dH^2\barσ\_T\sqrt{T})}$ lower bound, proving minimax optimality (up to logarithmic factors) and fully characterising the regret complexity of MNL mixture MDPs for the first time.

Out of the rich family of generalized linear bandits, perhaps the most well studied ones are logistic bandits that are used in problems with binary rewards: for instance, when the learner aims to maximize the profit over a user that can select one of two possible outcomes (e.g., 'click' vs'no-click'). Despite remarkable recent progress and improved algorithms for logistic bandits, existing works do not address practical situations where the number of outcomes that can be selected by the user is larger than two (e.g., 'click', 'show me later', 'never show again', 'no click'). In this paper, we study such an extension. We use multinomial logit (MNL) to model the probability of each one of K+1 2possible outcomes (+1 stands for the'not click' outcome): we assume that for a learner's action xt, the user selects one of K +1 2outcomes, say outcome i, with a MNL probabilistic model with corresponding unknown parameter θ i. Each outcome i is also associated with a revenue parameter ρi and the goal is to maximize the expected revenue. For this problem, we present MNL-UCB, an upper confidence bound (UCB)-based algorithm, that achieves regret O(dK T) with small dependency on problemdependent constants that can otherwise be arbitrarily large and lead to loose regret bounds. We present numerical simulations that corroborate our theoretical results.

We present a unified likelihood ratio-based confidence sequence (CS) for *any* (self-concordant) generalized linear model (GLM) that is guaranteed to be convex and numerically tight. We show that this is on par or improves upon known CSs for various GLMs, including Gaussian, Bernoulli, and Poisson. In particular, for the first time, our CS for Bernoulli has a $\mathrm{poly}(S)$-free radius where $S$ is the norm of the unknown parameter. Our first technical novelty is its derivation, which utilizes a time-uniform PAC-Bayesian bound with a uniform prior/posterior, despite the latter being a rather unpopular choice for deriving CSs. As a direct application of our new CS, we propose a simple and natural optimistic algorithm called **OFUGLB**, applicable to *any* generalized linear bandits (**GLB**; Filippi et al. (2010)). Our analysis shows that the celebrated optimistic approach simultaneously attains state-of-the-art regrets for various self-concordant (not necessarily bounded) **GLB**s, and even $\mathrm{poly}(S)$-free for bounded **GLB**s, including logistic bandits. The regret analysis, our second technical novelty, follows from combining our new CS with a new proof technique that completely avoids the previously widely used self-concordant control lemma (Faury et al., 2020, Lemma 9). Numerically, **OFUGLB** outperforms or is at par with prior algorithms for logistic bandits.

OFUGLB outperforms or is at par with prior algorithms for logistic bandits.

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