Title: Generalized Linear Bandits: Almost Optimal Regret with One-Pass Update
Venue: NeurIPS 2025
Accepted: yes
Keywords: Generalized Linear Bandits; Online Mirror Descent; Computational Efficiency; Regret Analysis
Abstract: We study the generalized linear bandit (GLB) problem, a contextual multi-armed bandit framework that extends the classical linear model by incorporating a non-linear 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 $\mathcal{O}(1)$ 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.

Generalized Linear Bandits: Almost Optimal Regret with One-Pass Update

Abstract

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.

1 Introduction

Stochastic multi-armed bandits [Robbins, 1952] represent a fundamental class of sequential decisionmaking problems where a learner interacts with environments by selecting actions (or arms) and receiving feedback in the form of rewards. In this paper, we study the contextual multi-armed bandit problem under the framework of generalized linear models (GLMs). In this setting, each action is characterized by a contextual feature vector x ∈ X t ⊂ R d , where the arm set X t may vary over time. More specifically, the learning process can be seen as a T round game between the learner and environments: at each round t , the learner selects an action X t ∈ X t and then observes a stochastic reward r t ∈ R generated according to a GLM (see Definition 2.1). The goal of the learner is to maximize the cumulative expected reward obtained over the time horizon T . Under the GLM model, the expectation of the reward satisfies E [ r t | X t ] = µ ( X ⊤ t θ ∗ ) , where µ : R → R is a non-linear link determined by the GLM model and is known to the learner. The unknown part is the underlying parameter θ ∗ ∈ R d , which needs to be estimated from the observed action-reward pairs.

Compared with the classical linear case [Abbasi-Yadkori et al., 2011], the generalized linear bandit (GLB) framework allows for a richer class of reward distributions, including Gaussian, Bernoulli, and Poisson distributions. This flexibility enables the modeling of various real-world tasks, such as recommendation systems [Li et al., 2010] and personalized medicine [Tewari and Murphy, 2017], where the feedback is binary (Bernoulli) or count-based (Poisson) and inherently non-linear. Besides

its practical appeal, the study of GLB lays theoretical foundations for other sequential decisionmaking problems, such as function approximation in RL [Wang et al., 2021, Li et al., 2024], safe exploration [Wachi et al., 2021], and dynamic pricing [Chen et al., 2022, Xu and Wang, 2021].

The non-linearity of the link function raises significant concerns regarding both computational and statistical efficiency in GLBs. A canonical solution to GLB is the GLM-UCB algorithm [Filippi et al., 2010], which belongs to the family of UCB-type methods [Agrawal, 1995, Auer, 2002]. At each iteration t ∈ [ T ] , the algorithm estimates the true parameter θ ∗ using maximum likelihood estimation (MLE) based on the historical data { ( x s , r s ) } t -1 s =1 and yields an estimator θ t , which is further used for constructing the upper confidence bound for arm selection. As shown in Table 1, GLM-UCB achieves nearly optimal regret bound in terms of the dependence on T . However, its reliance on MLE incurs a computational burden: it requires storing all historical data with O ( t ) space complexity and solving an optimization problem with O ( t ) time complexity at each round t . Besides, in terms of the statistical efficiency, the non-linearity of the link function introduces a notorious constant κ = 1 / inf x ∈∪ T t =1 X t ,θ ∈ Θ µ ′ ( x ⊤ θ ) into the regret bound of GLM-UCB (see Section 2 for details), where µ ′ denotes the derivative of µ . In several applications of GLBs, such as logistic bandits ( µ ( z ) = 1 / (1 + e -z ) ) and Poisson bandits ( µ ( z ) = e z ) , the κ term can grow exponentially with the norm of the parameter ∥ θ ∗ ∥ 2 , severely affecting the theoretical performance.

Over the past decade, extensive efforts have been devoted to enhancing the computational or statistical efficiency of GLBs. However, as summarized in Table 1, how to develop a jointly efficient method that achieves both low computation cost and strong statistical guarantees remains unclear. For GLBs, Jun et al. [2017] developed computationally efficient algorithms with one-pass update, but their regret bound scales linearly with the potentially large constant κ . More recently, by leveraging the self-concordance of the loss, several works [Lee et al., 2024, Sawarni et al., 2024, Liu et al., 2024, Clerico et al., 2025] proposed statistically efficient methods that achieve improved dependence on κ ; however, these approaches are still based on the MLE, which has high computation cost.

Our Results. This paper proposes a jointly efficient algorithm for GLBs that achieves an improved regret bound in terms of κ with constant time and space complexities per round as shown in Table 1. This advance roots in the construction of a tight confidence set for the online mirror descent (OMD) estimator used for performing the UCB-based exploration. We show that the OMD estimator, even though updated in a one-pass fashion, can still match the statistical efficiency of the MLE by carefully addressing the non-linearity of the link function. Here, 'one-pass' refers to processing each data point only once, without storing past data. We also note that OMD-based online estimators have been used to develop jointly efficient algorithms in the logistic bandit setting [Faury et al., 2022, Zhang and Sugiyama, 2023, Lee and Oh, 2024], a special case of GLBs with Bernoulli rewards. However, their analyses of the confidence set rely heavily on the specific structure of the logistic link function µ ( z ) = 1 / (1 + e -z ) , which limits their applicability to the more general GLB setting.

Technical Contribution. Our main technical contribution is a new analysis of the estimation error of the OMD estimator. The analysis is based on the concept of the mix loss , which has been used in full-information online learning to achieve fast-rate regret minimization [Vovk, 2001]. Here, we show that it provides a natural way to bridge the

gap between the OMD estimator and the true parameter θ ∗ , thereby enabling the construction of tight confidence sets for bandit online learning. Our new analysis not only generalizes the OMD-based approach to the broader GLBs but also improves upon the state-of-the-art for logistic bandits. As shown in Table 2, the jointly efficient method [Faury et al., 2022] requires O (log t ) time per round and an adaptive warm-up strategy to achieve optimal regret. Zhang and Sugiyama [2023] reduces the time complexity to O (1) but incurs an extra O ( √ log T ) factor in the regret. In contrast, our refined analysis yields a tighter error bound for the OMD estimator, allowing our method to achieve improved regret and low computation cost without warm-up rounds. Details are provided in Section 4.

We were made aware that mix-loss-based analyses have been independently developed in two very recent concurrent works for constructing tight confidence sets. Specifically, Kirschner et al. [2025] developed confidence sets based on the sequential likelihood ratios mixing technique, and Clerico et al. [2025] proposed several confidence sets for GLMs. While conceptually related, these works focus on the batch setting, where all historical data are repeatedly accessed, leading to substantial computational overhead. In contrast, our confidence set is based on the OMD estimator with a one-pass update. This difference leads to a distinct formulation of the mix loss and a tailored analysis to quantify its gap relative to the time-varying OMD estimator. Details are provided in Section 4.2.

2 Preliminary

This section provides background on the GLB problem, including its formulation, underlying assumptions, and closely related previous research. In the rest of the paper, for a positive semi-definite matrix H ∈ R d × d and vector x ∈ R d , we define ∥ x ∥ H = √ x ⊤ H x and ∥ x ∥ 2 as the Euclidean norm. For the function f : R → R , its first and second derivatives are denoted by f ′ and f ′′ , respectively.

2.1 Problem Formulation and GLM-UCB [Filippi et al., 2010]

The GLB problem considers a T -round sequential interaction between a learner and the environment. At each round t ∈ [ T ] ≜ { 1 , . . . , T } , the learner selects an action X t ∈ X t ⊂ R d from the feasible domain and then receives a stochastic reward r t ∈ R . We use the notation X t to indicate that the arm set may vary over time, capturing many practical scenarios where available options change dynamically. For instance, in product recommendation systems, items can be added or removed, requiring the algorithm to adapt accordingly. Besides, we denote the learner's action by X t to emphasize its stochastic nature, which may depend on past data captured by the filtration F t = σ ( X 1 , r 1 , . . . , X t -1 , r t -1 ) . In GLBs, conditioned on F t , the reward r t follows a canonical exponential family distribution with the natural parameter given by the linear model z t = X ⊤ t θ ∗ .

Here, θ ∗ ∈ R d is a d -dimensional vector unknown to the learner. The function h ( r, τ ) is the base measure, which provides the intrinsic weighting of the variable r , while m ( z ) is twice continuously differentiable function used for normalizing the distribution. Besides, g : R → R is the dispersion function controlling the variability of the distribution and τ ∈ R is a known parameter. The expectation and variance of the exponential family distribution can be calculated as E [ r | z ] = µ ( z ) ≜ m ′ ( z ) and Var ( r | z ) = g ( τ ) m ′′ ( z ) [Wainwright and Jordan, 2008, Proposition 3.1]. Common examples of (1) include Gaussian, Bernoulli, and Poisson distributions. The goal of the learner is to maximize the cumulative expected reward, which is equivalent to minimizing the regret,

where x t, ∗ = arg min x ∈X t µ ( x ⊤ θ ∗ ) is the optimal action. Besides, we have the following standard boundness assumptions used in the GLB literature [Filippi et al., 2010].

Assumption 1 (bounded domain) . The set ⋃ t ∈ [ T ] X t is bounded such that ∥ x ∥ 2 ≤ 1 for all x ∈ X t t ∈ [ T ] and the parameter θ ∗ satisfies ∥ θ ∗ ∥ 2 ≤ S for some constant S > 0 known to the learner.

,

Assumption 2 (bounded link function) . The link function µ is twice differentiable over its feasible domain. Moreover, there exist constants c µ > 0 and L µ > 0 such that c µ ≤ µ ′ ( z ) ≤ L µ for all z ∈ [ -S, S ] . Consequently, the function m is strictly convex and µ is strictly increasing.

GLM-UCB Method and Potentially Large Constant. The canonical algorithm for the GLB problem is GLM-UCB [Filippi et al., 2010], which resolves the exploration-exploitation trade-off with an upper-confidence-bound strategy [Agrawal, 1995]. Under Assumptions 1, 2 and an additional condition that the reward r t is non-negative and almost surely bounded for all t ∈ [ T ] , GLM-UCB achieves the regret of O ( κ (log T ) 3 2 √ T ) , where the O ( · ) -notation is used to highlight the dependence on κ and time horizon T . The dependence on the horizon T matches that of the linear case, where the ˜ O ( √ T ) rate is nearly optimal [Dani et al., 2008]. The bottleneck lies in its linear dependence on the constant κ , which is defined by

where Θ = { θ ∈ R d | ∥ θ ∥ 2 ≤ S } and X [ T ] = ⋃ t ∈ [ T ] X t . In the above we also define κ ∗ to reflects the local curvature at the optimal actions. The linear dependence on κ in GLM-UCB is generally undesirable, as κ can be prohibitively large in practice. Notable examples include the Bernoulli distribution with µ ( z ) = 1 / (1 + e -z ) and the Poisson distribution with µ ( z ) = e z , for which κ = O ( e S ) , growing exponentially with the parameter-norm bound S .

2.2 New Progress with Self-Concordance

The undesirable linear dependence on κ has motivated the development of algorithms with improved theoretical guarantees. By leveraging the self-concordance of the loss, rooted in convex optimization and later used in the analysis of logistic regression [Bach, 2010], recent studies [Russac et al., 2021, Lee et al., 2024, Sawarni et al., 2024] have derived regret bounds with substantially reduced dependence on κ for GLB. Following this line, we also adopt the self-concordance assumption here. Assumption 3 (Self-Concordance) . The link function satisfies | µ ′′ ( z ) | ≤ R · µ ′ ( z ) for all z ∈ R .

Assumption 3 holds for many widely used GLMs. For GLMs where the reward is almost surely bounded in [0 , R ] , the link function satisfies Assumption 3 with coefficient R [Sawarni et al., 2024]. For example, the Bernoulli distribution is 1 -self-concordant. Many unbounded GLMs also satisfy self-concordance, including the Gaussian distribution ( R = 0 ), Poisson distribution ( R = 1 ), and Exponential distribution ( R = 0 ). Leveraging self-concordance, Lee et al. [2024] and Sawarni et al. [2024] established improved regret bounds of order ˜ O ( √ T/κ ∗ ) . In these results, the potentially large constant κ ∗ appears at the denominator, which largely improves the ˜ O ( κ √ T ) bound by Filippi et al. [2010]. However, their methods incur still O ( t ) time and space complexities per round. Our goal is to design a method with low computation cost while maintaining strong regret guarantees. Remark 1 ( Unbounded GLMs ) . Our GLM assumptions are aligned with the recent work [Lee et al., 2024], which are more general than the canonical GLM formulation introduced in Filippi et al. [2010] and later adopted in Sawarni et al. [2024]. Besides Assumptions 1 and 2, Filippi et al. [2010] further requires the rewards to be almost surely bounded, which automatically implies self-concordance and thus satisfies Assumption 3 as shown by Sawarni et al. [2024, Lemma 2.2]. Beyond bounded distributions, our GLM formulation accommodates unbounded ones, such as Gaussian or Poisson. ◆

3 Proposed Method

This section presents our method for GLBs based on the principle of optimism in the face uncertainty (OFU) [Agrawal, 1995]. The core of our approach is a tight confidence set built on an online estimator. We begin with a review of the OFU principle and then introduce our method.

3.1 OFU Principle and Computational Challenge

OFU Principle. The OFU principle provides a principle way to balance exploration and exploitation in bandits. At each iteration t , this approach maintains a confidence set C t ( δ ) ⊂ R d to account for the uncertainty arising from the stochasticity of the historical data, ensuring it contains θ ∗ with high probability. Using the confidence set, one can construct a UCB for each action x ∈ X t as UCB ( x ) = max θ ∈C t ( δ ) µ ( x ⊤ θ ) and select the arm by X t = arg max x ∈X t UCB ( x ) . A key ingredient of OFU-based methods is the design of the confidence set, as the regret bound typically scales with the 'radius' of the set. A tighter confidence set generally leads to a stronger regret guarantee.

Computational Challenge. To the best of our knowledge, most existing OFU-based methods for GLBs rely on maximum likelihood estimation (MLE) to estimate θ ∗ and construct the confidence set. Specifically, the estimator is computed as

where ℓ t ( θ ) ≜ ( m ( X ⊤ t θ ) -r t · X ⊤ t θ ) /g ( τ ) is the loss function and λ > 0 is the regularizer parameter. The MLE was first used in the classical solution [Filippi et al., 2010], yet the regret bound exhibited linear dependence on κ due to a loose analysis. Subsequent work [Lee et al., 2024, Sawarni et al., 2024, Liu et al., 2024] provided refined analyses showing that MLE is statistically efficient, with its estimation error relative to θ ∗ being independent of κ . This property enables the construction of a confidence set C t ( δ ) with a κ -free diameter, yielding the improved regret bound.

However, despite the statistical efficiency of the MLE, it has high computation cost. The existing methods mentioned above use different choices of the feasible domain Θ ′ , but in all cases, solving the optimization problem requires access to the entire historical dataset, resulting in O ( t ) space complexity. Moreover, there is generally no closed-form solution for (2); the problem is typically solved using gradient-based methods such as projected gradient descent or Newton's method, where each gradient computation requires at least O ( t ) time per iteration [Filippi et al., 2010]. Consequently, both the time and space complexities of the MLE grow linearly with the number of rounds t , making it unfavorable for online settings. In addition, Sawarni et al. [2024] set Θ ′ = R d in (2) and required an additional projection step to ensure that θ lies within the desired domain. This projection involves solving a non-convex optimization problem, which is even more time-consuming.

3.2 Jointly Efficient Method

The main contribution of this paper is a statistically efficient confidence set C OL t ( δ ) constructed based on an online estimator θ t , which has κ -free estimation error with respect to the true parameter θ ∗ and can be computed with O (1) time and space complexities per round.

Online Estimator. Drawing inspiration from the study for logistic bandits [Faury et al., 2022, Zhang and Sugiyama, 2023], we use the online mirror descent to learn the parameter θ ∗ . For t = 1 , we initialize θ 1 ∈ Θ as any point in Θ and set H 1 = λI d . For time t ≥ 1 , we update the model by

where Θ = { θ ∈ R d | ∥ θ ∥ 2 ≤ S } is a d -dimensional ball with radius S . In the above, we defined

The above two components play important roles in achieving low computation cost while maintaining statistical efficiency. The loss function ˜ ℓ t ( θ ) serves as a second-order approximation of the original loss, which preserves the curvature information of the current loss function while ensures that the resulting optimization problem can be solved efficiently. The local matrix can also be expressed as H t = λI d + ∑ t -1 s =1 µ ′ ( X ⊤ s θ s +1 ) /g ( τ ) · X s X ⊤ s , where X s ∈ R d is the action selected by the learner. The matrix explicitly captures the non-linearity of the link function by µ ′ ( X ⊤ s θ s +1 ) and retains the curvature information of historical loss functions until time t -1 . Since the optimization problem (3) is quadratic optimization over an Euclidean ball, it can be solved with a computation cost of O ( d 3 ) , independent of time t . Further details on solving (3) are provided in Appendix A.3.

Confidence Set Construction. Then, we can construct a tight confidence set based on the online estimator by carefully configuring the parameters as the following theorem.

Theorem 1. Let δ ∈ (0 , 1] . Set the step size to η = 1 + RS and the regularization parameter to λ = max { 14 dηR 2 , 6 ηRSL µ /g ( τ ) } . For each t ∈ [ T ] , define the confidence set as

Algorithm 1 GLB-OMD

Input: Self-concordant constant R , Lipchitz constant L µ , parameter radius S , confidence level δ .

1: Initialize θ 1 ∈ Θ := { θ ∈ R d | ∥ θ ∥ 2 ≤ S } and H 1 = λI d .

2: for t = 1 to T do

3: Construct the confidence set C t ( δ ) according to (5).

4: Select the arm X t according to rule (6) and receive the reward r t .

5: Update the online estimator θ t +1 by (3) and set H t +1 = H t + ∇ 2 ℓ t ( θ t +1 ) .

6: end for

where θ t is the online estimator (3) and the radius β t ( δ ) is given by

Then, under Assumptions 1, 2, and 3, we have Pr [ ∀ t ≥ 1 , θ ∗ ∈ C t ( δ ) ] ≥ 1 -δ . Besides, the time complexity for solving (3) is O ( d 3 ) , and the space complexity is O ( d 2 ) .

Theorem 1 shows that an ellipsoidal confidence set C t ( δ ) can be constructed to quantify the uncertainty of the online estimator θ t with both statistical and computational efficiency. From a computational perspective, constructing the confidence set relies only on the online estimator, which can be updated with O (1) time and space complexities. From a statistical view, the radius of the confidence set is independent of κ , which is crucial for achieving the improved regret bound, as detailed next.

Remark 2 (Comparison with Logistic Bandits Literature) . We note that OMD has been used in logistic bandits for constructing confidence sets [Faury et al., 2022, Zhang and Sugiyama, 2023, Lee and Oh, 2024]. However, existing methods are not fully jointly efficient compared with our result. Specifically, Faury et al. [2022] required optimization over the original loss at each round, incurring an additional O (log t ) computation cost and relying on a warm-up strategy to maintain statistical efficiency. In later work, Zhang and Sugiyama [2023] and Lee and Oh [2024] achieved a constant per-round cost but their regret bounds have an extra O ( √ log t ) multiplicative factor. More importantly , as we will discuss in detail in Section 4, the analyses in these works depend on the specific structure of the logistic model and do not naturally extend to the GLB setting. Our key contribution is to introduce mix-loss -based technique into the confidence set analysis, which not only enables the application of OMD to the broader GLB framework but also improves both statistical and computational efficiency over previous methods for logistic bandits. ◆

Arm Selection and Regret Bound. Based on the ellipsoidal confidence set, one can employ a variety of exploration strategies, not limited to OFU-based methods but also including randomized approaches such as Thompson sampling [Abeille and Lazaric, 2017] for action selection. Here, we adopt the classical OFU-based strategy, where the action X t is selected by solving the bilevel optimization problem X t = arg max x ∈X t max θ ∈C t ( δ ) µ ( x ⊤ θ ) . Since µ is an increasing function and C t ( δ ) is an ellipsoid, the OFU-based action selection rule is equivalent to

which allows us to avoid solving the inner optimization problem explicitly. The overall implementation of the algorithm is summarized in Algorithm 1 and we have the following regret guarantee.

Theorem 2. Let δ ∈ (0 , 1] . Under Assumptions 1, 2, and 3, with probability at least 1 -δ , Algorithm 1 with parameter η = 1 + RS and λ = max { 14 dηR 2 , 6 ηRSL µ /g ( τ ) } ensures

where ≲ is used to hide constant independence of d , κ , S , R and T .

Theorem 2 shows that our method achieves an ˜ O ( d √ T/κ ∗ ) regret, improving upon the ˜ O ( κd √ T ) bound of Filippi et al. [2010]. From a computational perspective, our OMD estimator can be updated in O (1) time and memory per round, in the same spirit as the least-squares estimator in linear bandits. Consequently, the computation cost of our algorithm matches that of LinUCB [Abbasi-Yadkori et al., 2011], indicating that the nonlinearity of the link function does not necessarily make GLBs more computationally demanding. In the finite-arm setting, our algorithm enjoys a constant per-round computational cost of O ( d 3 + d 2 |X t | ) , independent of T . In the infinite-arm case, the arm-selection step (6) could become the main computational bottleneck (once the computational issue of MLE is resolved). Our estimator remains broadly useful as a plug-in component that can be integrated into other exploration strategies, e.g., Thompson Sampling [Faury et al., 2022, Appendix D.2] where the arm-selection step reduces to a convex optimization problem for convex arm sets.

Since logistic bandits are a special case of GLBs, Theorem 2 also advances state-of-the-art results of logistic bandits by either reducing the O (log t ) time complexity of Faury et al. [2022] to O (1) , or achieving an O ( √ log T ) improvement in the regret bound over Zhang and Sugiyama [2023].

Comparison with Sawarni et al. [2024]. We note that Sawarni et al. [2024] also pursued computational efficiency, but their approach is conceptually orthogonal to ours. They reduce the computation cost by employing a rare-update strategy that limits the frequency of parameter updates. However, their method remains MLE-based and requires storing all historical data, resulting in a memory cost of O ( t ) . Moreover, although the rare-update strategy yields an amortized per-round time complexity of O ((log t ) 2 ) over T rounds, it still incurs a worst-case time complexity of O ( t ) in certain rounds. In contrast, our method performs a one-pass update with O (1) time and space complexities per round.

Discussion with Lee and Oh [2025a]. The work [Lee and Oh, 2025a] (v3 version, the latest one available before the NeurIPS submitted date) builds on the framework of Zhang and Sugiyama [2023] and also reports an O ( √ log T ) improvement in multinomial logit (MNL) bandits, a different problem that nonetheless shares certain technical connections with GLBs. Their technique could potentially be adapted to logistic bandits for an O (log T √ T/κ ∗ ) bound with O (1) cost. However, we identify potential technical issues in the analysis. The argument relies on a condition for the normalization factor of the truncated Gaussian distribution (see Eq.(C.15) of their paper, as also restated in (43)), an assumption that warrants further examination. We provide a detailed discussions in Appendix D.

3.3 More Discussions and Limitations

Dependence on T , κ and d . For the dominant term, our regret bound matches the best-known results for GLBs using the MLE [Lee et al., 2024, Sawarni et al., 2024], with respect to its dependence on T , κ , and d . In terms of the non-leading term, Sawarni et al. [2024] achieved a slightly tighter bound, as it scales with κ X = 1 / inf x ∈∪ T t =1 X t µ ′ ( x ⊤ θ ∗ ) , a quantity that can be smaller than κ . In the logistic bandit case, the non-leading term is further improved to be geometry-aware, adapting more precisely to the structure of the action set [Abeille et al., 2021]. We conjecture that similar improvements in the non-leading term, matching the κ X dependence, might be achievable by incorporating a warm-up strategy, such as Procedure 1 in [Faury et al., 2022] or Algorithm 2 in Sawarni et al. [2024] to shift the curvature term from µ ′ ( X ⊤ t θ t ) to µ ′ ( X ⊤ t θ ∗ ) . However, it remains unclear whether geometry-aware bounds, akin to those in Abeille et al. [2021], can be obtained for GLBs without using the MLE.

Dependence on S and R . The MLE-based method completely remove the dependence on S and R in the leading term [Lee et al., 2024], whereas our method still exhibits an S 2 -dependence due to the requirement of one-pass updates. For MNL bandits, Lee and Oh [2025a] showed that one can achieved improved dependence on S by incorporating an adaptive warm-up procedure. It may be possible to extend their warm-up technique to GLBs for a similar improvement on S .

4 Analysis

This section sketches the proof of Theorem 1 and highlights the key technical contributions.

4.1 A General Recipe for OMD

To prove Theorem 1, it suffices to show that the estimation error ∥ θ t +1 -θ ∗ ∥ H t ≤ β t ( δ ) . The analysis begins with the following lemma, which is commonly used for the convergence or regret analysis for

the OMD-type update [Chen and Teboulle, 1993, Orabona, 2019, Zhao et al., 2024]. Here, we show it can serve as a general recipe for analyzing the estimation error of the OMD estimator.

Lemma 1. Let f : Θ → R be a convex function on a convex set Θ and A ∈ R d × d be a symmetric positive definite matrix. Then, the update θ t +1 = arg min θ ∈ Θ f ( θ ) + 1 2 η ∥ θ -θ t ∥ 2 A satisfies

The above lemma provides a pathway to relate the estimation error to the so-called 'inverse regret'. In particular, under our configuration where f ( θ ) = ˜ ℓ t ( θ ) , A = H t , and u = θ ∗ , and with a suitable choice of parameters, Lemma 4 in Appendix A shows that the estimator by (3) satisfies

Wenote that, although the above inequality has also been shown in the logistic bandits literature [Faury et al., 2022, Zhang and Sugiyama, 2023, Lee and Oh, 2024], the proof of (8) via Lemma 7 has not been explicitly discussed. We fill this gap by explicitly establishing the connection in our analysis.

4.2 Analysis for the Inverse Regret

Main Challenge. The main challenge and technical contribution of this paper lies in upper bounding the inverse regret term. Although previous works [Faury et al., 2022, Zhang and Sugiyama, 2023] have provided valuable insights, their techniques are challenging to extend to the GLB setting and remain suboptimal even for logistic bandits. The main technical difficulty in bounding the inverse regret is that θ s +1 is itself a function of the past losses ℓ s , which prevents the direct application of standard martingale concentration inequalities. A common strategy in prior work is to introduce an intermediate term by virtually running a full-information online learning algorithm, whose estimator ˜ θ s only depends on information up to time s -1 , allowing the inverse regret to be decomposed as

Here, the intermediate term ∑ t -1 s =1 ℓ s ( ˜ θ s ) can be chosen as the cumulative loss of any online algorithm. In the case of logistic bandits, it is natural to leverage algorithms developed for online logistic regression, a well-studied problem with many established methods. For example, Faury et al. [2022] adopted the ALLIO algorithm [Jézéquel et al., 2020], while Zhang and Sugiyama [2023] built on the method proposed by Foster et al. [2018] and required the mixbaility property of the logistic loss. In contrast, for the GLB setting, the structure of the link function µ varies significantly across models, and it remains unclear how to design a unified intermediate algorithm. Moreover, even in the logistic bandit setting, existing analyses remain suboptimal. Specifically, Faury et al. [2022, Eq. (7)] required an online warm-up phase to shrink the feasible domain in order to bound term (b). Later, Zhang and Sugiyama [2023] avoided this warm-up step but instead relies on clipping the online estimator (see Eq. (35) in their paper), which incurs an additional O ( √ log T ) term in the estimation error bound.

Our Solution. Instead of introducing an intermediate term via virtually running an online algorithm, we propose an alternative decomposition based on the mix loss , which is defined as

where P s is a probability distribution over R d whose specific form will be chosen later. In general, several choices of P s are possible, and we select the one that best fits our algorithmic design. The mix loss has played a central role in the analysis of exponentially weighted methods in full-information online learning [Vovk, 2001, van der Hoeven et al., 2018]. In our analysis, the mix loss is instrumental in analyzing the inverse regret defined in (9). In particular, the decomposition based on the mix loss m s ( P s ) offers a more general and analytically versatile formulation than ℓ s ( ˜ θ s ) used in (9). The former reduces to the latter when P s is chosen as a Dirac distribution. We have the following lemma to upper bound the inverse regret under this mix-loss-based decomposition.

Lemma 2 (informal) . Let { P s } ∞ s =1 be a stochastic process such that P s is a distribution over R d and only relies on information collected until time s -1 . Then, for any δ ∈ (0 , 1] , we have

Lemma 2 follows from Ville's inequality [Ville, 1939], also known as 'no-hypercompression' inequality [Grünwald, 2007, Chapter 3]. It implies that term (a) in (9) is upper-bounded by log(1 /δ ) when the mix loss is used as an intermediate term, significantly smaller than the O ((log t ) 3 ) bound in Zhang and Sugiyama [2023, Lemma 12] that employs the online logistic regression method [Foster et al., 2018]. Notably, Lemma 2 allows flexible choices of P s , enabling us to tailor P s to closely track the OMD estimator's behavior. The formal statement is provided in Lemma 5 of Appendix A.

To bound term (b), we need to select P s = N ( θ s , 3 ηH -1 s / 2) as a Gaussian distribution centered at θ s with covariance 3 2 ηH -1 s to approximate the OMD estimator (3). We have the following lemma.

Lemma 3 (informal) . Under Assumptions 1, 2, and 3, and with a suitable choice of λ , we have

where P s = N ( θ s , 3 ηH -1 s / 2) is a Gaussian distribution.

Lemma 3 establishes a connection between the mix loss and the the cumulative loss of the 'lookahead' OMD estimator, where the loss of θ s +1 is measured over the ℓ s . A similar result for the logistic loss can be extracted from the proof of Lemma 14 in Zhang and Sugiyama [2023]. Here, we generalize this result to the GLB setting. Moreover, our proof simplifies the analysis in [Zhang and Sugiyama, 2023] by noticing that the mix loss can be interpreted as the convex conjugate of the KL divergence. The formal statement and complete proof are provided in Lemma 6 in Appendix A.

More Technical Comparisons. Our analysis of the inverse regret is closely connected to prior work that bounds the cumulative negative log likelihood L t ( θ ∗ ) = ∑ t s =1 ℓ s ( θ ∗ ) using Ville's inequality, a technique that traces back to Robbins [1970]. Most existing approaches aim to bound L t ( θ ∗ ) with the loss of the MLE estimator, whereas our analysis focuses on the OMD estimator. This fundamental difference leads to a distinct construction of the intermediate term used in the decomposition.

In the context of GLB, our Lemma 2 resembles Lemma 3.3 of Lee et al. [2024], as both apply Ville's inequality. However, their intermediate term E θ ∼ P t [ L t ( θ )] is built upon a distribution P t that is fixed across all individual functions { ℓ s } t -1 s =1 , making it naturally aligned with the MLE estimator but does not readily adapt to changing estimator. In contrast, our OMD estimator evolves with each individual function. Defining the intermediate term as the mix loss with time-varying P s thus provides the flexibility needed to track this changing comparator. Very recently, two concurrent works [Kirschner et al., 2025, Clerico et al., 2025] have also employed the mix loss to derive confidence sets. Our Lemma 2 corresponds to Proposition 2.1 of Clerico et al. [2025], and the mix loss aligns with the sequential likelihood mixing technique of Kirschner et al. [2025]. While their analyses are already applicable to GLBs, a key difference is that we use an ellipsoidal confidence set centered at the OMD estimator to ensure computational efficiency, whereas both works [Kirschner et al., 2025, Clerico et al., 2025] primarily analyze the negative log-likelihood-based set, which are tighter but substantially more computationally demanding for GLBs. This difference leads to a different specification of the mix loss and requires a tailored analysis. Specifically, in their setting, P s is defined as the output of a continuous Hedge method to track the MLE estimator (or any fixed comparator). In contrast, our analysis must dynamically track the OMD estimator, which motivates our choice of P s as a Gaussian distribution with the evolving OMD estimators as its mean. Lemma 3 is then used to quantify the gap.

5 Experiment

This section evaluates the proposed method on two representative GLB problems: logistic bandits ( µ ( z ) = 1 / (1 + e -z ) ) with bounded rewards, and Poisson bandits ( µ ( z ) = e z ), which pose a distinct challenge as an unbounded GLB setting. We also conduct experiments on real data from the Covertype dataset [Blackard, 1998], with more detailed results provided in Appendix E.

0

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Regret

GLB-OMD (Our work)

RS-GLinUCB (Sawarni et al. 2024)

ECOLog (Faury et al. 2022)

OFULog-r (Abeille et al. 2021)

GLM-UCB (Filippi et al. 2010)

(a)

Logistic Bandit (

S

= 3

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GLB-OMD (Our work)

RS-GLinUCB (Sawarni et al. 2024)

ECOLog (Faury et al. 2022)

OFULog-r (Abeille et al. 2021)

GLM-UCB (Filippi et al. 2010)

(b)

Logistic Bandit (

S

= 5

)

= 21.0 (Low)

= 137.1 (Medium)

= 963.3 (High)

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GLB-OMD (Our work)

RS-GLinUCB (Sawarni et al. 2024)

OFULog-r (Abeille et al. 2021)

GLM-UCB (Filippi et al. 2010)

ECOLog (Faury et al. 2022)

(c)

Running time for Logistic Ban-

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GLB-OMD (Our work)

RS-GLinUCB (Sawarni et al. 2024)

GLM-UCB (Filippi et al. 2010)

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GLB-OMD (Our work)

RS-GLinUCB (Sawarni et al. 2024)

GLM-UCB (Filippi et al. 2010)

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)

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= 127.0 (Medium)

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Running Time (s)

GLB-OMD (Our work)

RS-GLinUCB (Sawarni et al. 2024)

GLM-UCB (Filippi et al. 2010)

(c)

Running time for Poisson Bandit

Compared Methods. Four GLBs algorithms are compared, including GLM-UCB [Filippi et al., 2010], GLOC [Jun et al., 2017], RS-GLinCB [Sawarni et al., 2024], and OFUGLB [Lee et al., 2024]. For logistic bandits, we further include two specialized algorithms: an MLE-based method with nearly optimal regret OFULog-r [Abeille et al., 2021], and a jointly efficient method ECOLog [Faury et al., 2022]. We do not include OFUL-MLogB [Zhang and Sugiyama, 2023] since its confidence set is larger than that of ours, hence has a larger regret bound. More details of the baselines are provided in Appendix E. All experiments are conducted over 10 trials, and we report the average regret and running time.

Results on Logistic Bandits. Weconduct experiments under different configurations of S . The underlying parameter θ ∗ is sampled from a d -dimensional sphere with radius S = { 3 , 5 , 7 } , corresponding to κ = 21 , 137 , and 963 , respectively. Figure 1 reports the results. Among all methods, GLOC is the fastest but exhibits relatively large regret. OFUGLB attains the lowest regret due to its improved dependence on κ and S , but as an MLE-based method, it incurs the highest computation cost. Our method strikes a favorable balance. Compared to OFUGLB , it achieves substantial cost savings with only a modest degradation in regret. Compared with ECOLog and RS-GLinCB , our method achieves comparable and even slightly better performance with improved computation cost. Moreover, it maintains a constant per-round cost across all regimes of κ , whereas the cost of RS-GLinCB increases with κ , as its rare-update strategy results in an update frequency that scales with κ .

Results on Poisson Bandits. We set the norm of the true parameter as S ∈ { 3 , 5 , 7 } , corresponding to κ ≈ 18 , 127 and 882 . Poisson bandits have unbounded rewards, whereas GLM-UCB and RS-GLinCB require a predefined upper bound on the maximum reward as a parameter. We set it as 100, assuming rewards are effectively bounded with high probability. As shown in Figure 2, our method reduces the computational cost of OFUGLB by roughly 1000 times, with only a modest increase in regret.

6 Conclusion

This paper proposed a new method for the GLB problem that achieves a nearly optimal regret bound of O (log T √ T/κ ∗ ) with O (1) time and space complexities per round. Our approach builds on a novel analysis of the OMD estimator using the mix loss, enabling a tight confidence set construction for arm selection. A natural extension is to incorporate a warm-up strategy, as in prior work, to improve the dependence on S and obtain κ X -based bounds. It also remains open whether geometry-aware bounds for GLBs can be achieved similar to those in logistic bandits [Abeille et al., 2021]. Other directions include relaxing the self-concordance assumption toward weaker conditions [Liu et al., 2024], or improving d -dependence in the finite-arm setting [Jun et al., 2021, Mason et al., 2022].

Acknowledgments and Disclosure of Funding

Peng Zhao was supported by NSFC (62206125) and the Xiaomi Foundation. MS was supported by the Institute for AI and Beyond, UTokyo. The authors thank the reviewers for their valuable feedback and for bringing to our attention the recent works [Kirschner et al., 2025, Clerico et al., 2025].

References

Yasin Abbasi-Yadkori, Dávid Pál, and Csaba Szepesvári. Improved algorithms for linear stochastic bandits. In Advances in Neural Information Processing Systems 24 (NeurIPS) , pages 2312-2320, 2011.

Marc Abeille and Alessandro Lazaric. Linear Thompson sampling revisited. In Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) , pages 176-184, 2017.

Marc Abeille, Louis Faury, and Clément Calauzènes. Instance-wise minimax-optimal algorithms for logistic bandits. In Proceedings of the 24th International Conference on Artificial Intelligence and Statistics (AISTATS) , pages 3691-3699, 2021.

Rajeev Agrawal. Sample mean based index policies by o (log n) regret for the multi-armed bandit problem. Advances in applied probability , 27(4):1054-1078, 1995.

Peter Auer. Using confidence bounds for exploitation-exploration trade-offs. Journal of Machine Learning Research , 3:397-422, 2002.

Francis Bach. Self-concordant analysis for logistic regression. Electronic Journal of Statistics , 4: 384-414, 2010.

Jock Blackard. Covertype. UCI Machine Learning Repository, 1998.

Gong Chen and Marc Teboulle. Convergence analysis of a proximal-like minimization algorithm using bregman functions. SIAM Journal on Optimization , 3(3):538-543, 1993.

Xi Chen, David Simchi-Levi, and Yining Wang. Privacy-preserving dynamic personalized pricing with demand learning. Management Science , 68(7):4878-4898, 2022.

Eugenio Clerico, Hamish Flynn, Wojciech Kotlowski, and Gergely Neu. Confidence sequences for generalized linear models via regret analysis. ArXiv preprint , arXiv:2504.16555, 2025.

Varsha Dani, Thomas P. Hayes, and Sham M. Kakade. Stochastic linear optimization under bandit feedback. In Proceedings of the 21st Annual Conference on Learning Theory (COLT) , pages 355-366, 2008.

Louis Faury, Marc Abeille, Clément Calauzènes, and Olivier Fercoq. Improved optimistic algorithms for logistic bandits. In Proceedings of the 37th International Conference on Machine Learning (ICML) , pages 3052-3060. PMLR, 2020.

Louis Faury, Marc Abeille, Kwang-Sung Jun, and Clément Calauzènes. Jointly efficient and optimal algorithms for logistic bandits. In Proceedings of the 25th International Conference on Artificial Intelligence and Statistics (AISTATS) , pages 546-580, 2022.

Sarah Filippi, Olivier Cappé, Aurélien Garivier, and Csaba Szepesvári. Parametric bandits: The generalized linear case. In Advances in Neural Information Processing Systems 23 (NeurIPS) , pages 586-594, 2010.

Dylan J. Foster, Satyen Kale, Haipeng Luo, Mehryar Mohri, and Karthik Sridharan. Logistic regression: The importance of being improper. In Proceedings of the 31st Conference on Learning Theory (COLT) , pages 167-208, 2018.

Peter D Grünwald. The Minimum Description Length Principle . MIT press, 2007.

Shunsuke Ihara. Information Theory for Continuous Systems . World Scientific, 1993.

Rémi Jézéquel, Pierre Gaillard, and Alessandro Rudi. Efficient improper learning for online logistic regression. In Proceedings of the 33rd Conference on Learning Theory (COLT) , pages 2085-2108, 2020.

Kwang-Sung Jun, Aniruddha Bhargava, Robert D. Nowak, and Rebecca Willett. Scalable generalized linear bandits: Online computation and hashing. In Advances in Neural Information Processing Systems 30 (NeurIPS) , pages 99-109, 2017.

Kwang-Sung Jun, Lalit Jain, Blake Mason, and Houssam Nassif. Improved confidence bounds for the linear logistic model and applications to bandits. In Proceedings of the 38th International Conference on Machine Learning (ICML) , pages 5148-5157, 2021.

Johannes Kirschner, Andreas Krause, Michele Meziu, and Mojmir Mutny. Confidence estimation via sequential likelihood mixing. ArXiv preprint , arXiv:2502.14689, 2025.

Tor Lattimore and Csaba Szepesvári. Bandit Algorithms . Cambridge University Press, 2020.

Joongkyu Lee and Min-hwan Oh. Nearly minimax optimal regret for multinomial logistic bandit. In Advances in Neural Information Processing Systems 37 (NeurIPS) , 2024.

Joongkyu Lee and Min-hwan Oh. Improved online confidence bounds for multinomial logistic bandits. ArXiv preprint , arXiv: 2502.10020, 2025a. Version v3 posted 7 Mar 2025.

Joongkyu Lee and Min-hwan Oh. Improved online confidence bounds for multinomial logistic bandits. ArXiv preprint , arXiv: 2502.10020, 2025b. Version v5 posted 16 June 2025.

Junghyun Lee, Se-Young Yun, and Kwang-Sung Jun. A unified confidence sequence for generalized linear models, with applications to bandits. In Advances in Neural Information Processing Systems 37 (NeurIPS) , 2024.

Lihong Li, Wei Chu, John Langford, and Robert E. Schapire. A contextual-bandit approach to personalized news article recommendation. In Proceedings of the 19th International Conference on World Wide Web (WWW) , pages 661-670, 2010.

Long-Fei Li, Yu-Jie Zhang, Peng Zhao, and Zhi-Hua Zhou. Provably efficient reinforcement learning with multinomial logit function approximation. In Advances in Neural Information Processing Systems 37 (NeurIPS) , pages 58539-58573, 2024.

Shuai Liu, Alex Ayoub, Flore Sentenac, Xiaoqi Tan, and Csaba Szepesvári. Almost free: Selfconcordance in natural exponential families and an application to bandits. In Advances in Neural Information Processing Systems 37 (NeurIPS) , 2024.

Blake Mason, Kwang-Sung Jun, and Lalit Jain. An experimental design approach for regret minimization in logistic bandits. In Proceedings of the 36th AAAI Conference on Artificial Intelligence (AAAI) , pages 7736-7743, 2022.

Zakaria Mhammedi, Wouter M. Koolen, and Tim van Erven. Lipschitz adaptivity with multiple learning rates in online learning. In Proceedings of the 32nd Annual Conference on Learning Theory (COLT) , pages 2490-2511, 2019.

Francesco Orabona. A modern introduction to online learning. arXiv preprint , arXiv:1912.13213, 2019.

Mark D. Reid, Rafael M. Frongillo, Robert C. Williamson, and Nishant Mehta. Generalized mixability via entropic duality. In Proceedings of the 28th Conference on Learning Theory (COLT) , pages 1501-1522, 2015.

Herbert Robbins. Some aspects of the sequential design of experiments. Bulletin American Mathematical Society , 55:527-535, 1952.

Herbert Robbins. Statistical methods related to the law of the iterated logarithm. Annals of Mathematical Statistics , 41(5):1397-1409, 1970.

Yoan Russac, Louis Faury, Olivier Cappé, and Aurélien Garivier. Self-concordant analysis of generalized linear bandits with forgetting. In Proceedings of the 24th International Conference on Artificial Intelligence and Statistics (AISTATS) , pages 658-666, 2021.

Ayush Sawarni, Nirjhar Das, Siddharth Barman, and Gaurav Sinha. Generalized linear bandits with limited adaptivity. In Advances in Neural Information Processing Systems 37 (NeurIPS) , 2024.

Ambuj Tewari and Susan A. Murphy. From ads to interventions: Contextual bandits in mobile health. In Mobile Health - Sensors, Analytic Methods, and Applications , pages 495-517. 2017.

Dirk van der Hoeven, Tim van Erven, and Wojciech Kotlowski. The many faces of exponential weights in online learning. In Proceedings of the 31st Conference on Learning Theory (COLT) , pages 2067-2092, 2018.

Jean Ville. Étude critique de la notion de collectif . Number 3 in Monographies des Probabilités. Gauthier-Villars, Paris, 1939.

Vladimir Vovk. Competitive on-line statistics. International Statistical Review , 69(2):213-248, 2001.

Akifumi Wachi, Yunyue Wei, and Yanan Sui. Safe policy optimization with local generalized linear function approximations. In Advances in Neural Information Processing Systems 34 (NeurIPS) , pages 20759-20771, 2021.

Martin J. Wainwright and Michael I. Jordan. Graphical models, exponential families, and variational inference. Foundations and Trends in Machine Learning , 1(1-2):1-305, 2008.

Yining Wang, Ruosong Wang, Simon Shaolei Du, and Akshay Krishnamurthy. Optimism in reinforcement learning with generalized linear function approximation. In Proceedings of the 9th International Conference on Learning Representations (ICLR) , 2021.

Jianyu Xu and Yu-Xiang Wang. Logarithmic regret in feature-based dynamic pricing. In Advances in Neural Information Processing Systems 34 (NeurIPS) , pages 13898-13910, 2021.

Yu-Jie Zhang and Masashi Sugiyama. Online (multinomial) logistic bandit: Improved regret and constant computation cost. In Advances in Neural Information Processing Systems 36 (NeurIPS) , pages 29741-29782, 2023.

Peng Zhao, Yu-Jie Zhang, Lijun Zhang, and Zhi-Hua Zhou. Adaptivity and non-stationarity: Problemdependent dynamic regret for online convex optimization. Journal of Machine Learning Research , 25(98):1 - 52, 2024.

A Proof of Theorem 1

This section presents the proof of Theorem 1. We first provide the main proof, followed by the key lemmas used in the proof.

A.1 Main Proof

Proof of Theorem 1. This part provides the proof of Theorem 1. By Lemma 4, when setting η = 1 + RS , we can upper bound the estimation error of the online estimator by the 'inverse regret':

Then, we can further decompose the 'inverse regret term' into two parts:

In the above, we define P s = N ( θ s , αH -1 s ) as a d -dimensional multivariate Gaussian distribution with mean θ s ∈ R d and covariance matrix cH -1 s ∈ R d × d , where α > 0 is a constant to be specified latter. The function m s : P ↦→ R that maps the distribution P to a real number value is defined by

We refer to the function m s as the 'mix loss' because it mixes the loss with respect to the distribution P s . This mixing has been found useful for achieving fast rates in prediction with expert advice and online optimization problems [Vovk, 2001]. Here, we show that the mix loss plays a crucial role in obtaining a jointly efficient online confidence set.

Given P s is a Gaussian distribution with mean θ s and αH -1 s , it is F s -measurable. Then, Lemma 5 implies that for any δ ∈ (0 , 1] , we have

with probability at least 1 -δ . Next, we proceed to analyze term ( b ) . Under the condition that λ ≥ 14 dηR 2 , Lemma 6 with α = 3 2 η shows that

Plugging (12), (13), (14) into (11), we obtain

where the last inequality holds due to the condition λ ≥ 6 ηRSL µ /g ( τ ) .

A.2 Useful Lemmas

This section presents several key lemmas used in the proof of Theorem 1.

Lemma 4. Under Assumption 1 and 3 and setting η = 1 + RS , then for any λ > 0 , the online estimator returned by (3) satisfies

Furthermore, if λ ≥ 6 ηL µ RS/g ( τ ) , we can further have

Proof of Lemma 4. We begin by using the integral formulation of Taylor's expansion. Since µ is twice differentiable, we have

where ˜ h s = ∫ 1 v =0 (1 -v ) ∇ 2 ℓ s ( θ s +1 + v ( θ ∗ -θ s +1 ) ) d v. By the definition of the loss function in (4), we can further express the Hessian as

where ˜ α ( θ 1 , θ 2 , X s ) = ∫ 1 0 (1 -v ) µ ′ ( X ⊤ s θ 1 + v X ⊤ s ( θ 2 -θ 1 ) ) d v.

Next, under Assumption 3, we have | µ ′′ ( z ) | ≤ R · µ ′ ( z ) for all z ∈ [ -S, S ] . Consequently, Lemma 8 in Appendix C implies that for any θ ∗ ∈ Θ ≜ { θ ∈ R d | ∥ θ ∥ 2 ≤ S } ,

indicating that the Hessian ˜ h s is bounded from below by 1 2+2 RS ∇ 2 ℓ s ( θ s +1 ) in the positive semidefinite order. Substituting (16) into (15) yields

Since θ s +1 is the optimal solution of (3), Lemma 1 with u = θ ∗ implies

This inequality shows that

We can further express the first term on the right-hand side of (18) as

where ξ s ∈ Θ lies on the line connecting θ s and θ s +1 . The first equality follows from the definition of ∇ ˜ ℓ s ( θ s +1 ) and the Taylor expansion with Lagrange's remainder. The first inequality uses the self-concordance property of the loss function, which ensures that µ ′′ ( z ) ≤ Rµ ′ ( z ) and µ ′ ( z ) ≤ L µ for all z ∈ [ -S, S ] . The last inequality is due to ∥ X s ∥ 2 ≤ 1 and ∥ θ s -θ s +1 ∥ 2 ≤ 2 S .

Combining (18), (19) with (17), setting η = 1 + RS and taking the summation over t ∈ [ T ] yield

We complete the proof by rearranging the terms and noticing ∥ θ 1 -θ ∗ ∥ 2 H 1 ≤ 4 λS 2 .

Lemma 5. Let {F t } ∞ t =1 be a filtration defined by F t = σ ( { ( X s , r s ) } t -1 s =1 ) . Let { P t } ∞ t =1 be a stochastic process such that the random variable P t is a distribution over R d and is F t -measurable. Moreover, assume that the loss function ℓ t defined by (4) is F t +1 -measurable. For any t ≥ 1 , define

Then, for any δ ∈ (0 , 1] , we have

Proof of Lemma 5. Let M 0 = 1 and for any t ≥ 1 , we define

To prove the lemma, it suffices to show that the sequence { M t } ∞ t =1 is a non-negative (super)martingale; then, the maximum inequality Lattimore and Szepesvári [2020, Theorem 3.9] can be applied to obtain the desired result. To verify that { M t } ∞ t =1 is a (super)martingale, we begin by defining the density function of the natural exponential family distribution as follows:

where the function m , g and h share the same formulation as those in (1). Then, for each time t ≥ 1 , we can rewrite the expression of M t as

where the final equality holds because the loss function in (4) is the negative log-likelihood of an exponential family distribution; that is, for any θ ∈ R d :

Then, by taking the conditional expectation with respect to the randomness in r t given F t on both sides, we obtain:

where the first equality follows from the fact that M t -1 is F t -measurable. The second inequality holds because the reward is sampled from the exponential family distribution (1). The final equality is a consequence of the tower property of conditional expectation. We have thus shown that { M t } ∞ t =1 is a martingale, and therefore a super-martingale. Then, by applying the maximum inequality Lattimore and Szepesvári [2020, Theorem 3.9], restating as Lemma 9 in Appendix C, we have

which completes the proof.

Lemma 6. Under Assumption 1, 2 and 3, let P s = N ( θ s , αH -1 s ) be a Gaussian distribution with mean θ s and covariance matrix αH -1 s , where H s = λI d + ∑ s -1 τ =1 ∇ 2 ℓ τ ( θ τ +1 ) and α is any positive constant. We denote by θ s the model returned by (3) . Then, setting λ ≥ 64 dαR 2 / 7 we have

where the mix loss is defined as m s ( P s ) = -ln ( E θ ∼ P s [ exp ( -ℓ s ( θ ) )]) .

Proof of Lemma 6. Our analysis begin with the observation that the mix loss is a convex conjugate of the KL divergence function. Then Lemma 12 in Appendix C shows that

for any distribution Q s defined over R d , where P ∗ s ( θ ) ∝ P s ( θ ) · e -ℓ s ( θ ) for all θ ∈ R d . Here, we choose Q s = N ( θ s +1 , αH -1 s +1 ) as a Gaussian distribution with mean θ s +1 and covariance αH -1 s +1 .

Analysis of Term (a). Since Q s is symmetric around θ s +1 , we can express term (a) as

where D ℓ s ( θ, θ s +1 ) = ℓ s ( θ ) -ℓ s ( θ s +1 ) -⟨∇ ℓ s ( θ s +1 ) , θ -θ s +1 ⟩ is the Bregman divergence of ℓ s between θ and θ s +1 . In the above, the second equality follows from the definition of Q s . The last line follows from Lemma 7 in Appendix C, since ℓ s is self-concordant and the condition λ ≥ 64 dαR 2 / 7 holds.

Analysis of Term (b). Given Q s and P s are both Gaussian distributions, Lemma 13 shows that

Put All Together. Plugging (22) and (23) into (21) and summing over T rounds, we obtain

The determinant of the matrix H t +1 can be further bounded by

Then, we obtain

which completes the proof.

Lemma 7. Let ℓ s ( θ ) be the loss function of the maximum likelihood estimator and Q s = N ( θ s +1 , αH -1 s +1 ) be a Gaussian distribution with mean θ s +1 and covariance matrix αH -1 s +1 with H s = λI d + ∑ s -1 τ =1 ∇ 2 ℓ τ ( θ τ +1 ) . Under Assumption 1, 2 and 3 and setting λ ≥ 64 dαR 2 / 7 , for any constant α > 0 , we have

Proof of Lemma 7. By the the integral formulation of Taylor's expansion and the definition of ℓ s such that ∇ 2 ℓ s ( θ ) = µ ′ ( X ⊤ s θ ) /g ( τ ) · X s X ⊤ s for any θ ∈ R d , we have

where ˜ h s ( θ ) = ∫ 1 v =0 (1 -v ) µ ′ ( X ⊤ s θ s +1 + vX ⊤ s ( θ -θ s +1 ) ) d v · X s X ⊤ s /g ( τ ) . According to Lemma 8 in Appendix C and the condition ∥ X t ∥ 2 ≤ 1 by Assumption 1, we can bound the Hessian ˜ h s ( θ ) by ˜ h s ( θ ) ≤ exp( R 2 ∥ θ -θ s +1 ∥ 2 2 ) · ∇ 2 ℓ s ( θ s +1 ) . (24)

Then, the approximation error between the linearized loss g s ( θ ) and ℓ s ( θ ) is bounded by

The second inequality is due to the Cauchy-Schwarz inequality. The last inequality is due to Lemma 10 under the condition that H s ≽ λI d and the setting λ ≥ 64 dαR 2 / 7 . For the last term on the right hand side of (25), the random variable ( ∇ 2 ℓ s ( θ s +1 ) ) 1 2 ( θ -θ s +1 ) follows the same distribution as

where λ i is the i -th largest eigenvalue of the matrix ( ∇ 2 ℓ s ( θ s +1 ) ) 1 2 H -1 s +1 ( ∇ 2 ℓ s ( θ s +1 ) ) 1 2 and e i is a set of orthogonal basis. Then, we have

where Tr ( A ) denotes the trace of matrix A . In the above, the last inequality is due to the fact that E X i ,X j ∼N (0 , 1) [ X i X j ] = 3 . The last inequality is due to trace ( AB ) = trace ( BA ) for matrix A,B ∈ R d × d . Recall that H s +1 = λI d + ∑ s τ =1 ∇ 2 ℓ τ ( θ τ +1 ) .

Combining (26) and (27) with (25) yields the desired result.

A.3 Computational Cost Discussion

This part discusses the time and space complexity of solving the optimization problem (3).

Proposition 1. The time complexity for solving (3) is O ( d 3 ) , and the space complexity is O ( d 2 ) .

Proof of Proposition 1. Webegin with the analysis on the time complexity, followed by the discussion on the space complexity.

Time Complexity. According to Theorem 6.23 of Orabona [2019], the update rule of online mirror descent (3) can be equivalently expressed as

where ˜ H t = H t + η ∇ 2 ℓ t ( θ t ) . In this formulation, the first step (28a) is a gradient update, whose main computational cost lies in computing the inverse of the Hessian matrix. Since ∇ 2 ℓ t ( θ t ) = µ ′ ( X ⊤ t θ t ) /g ( τ ) · X t X ⊤ t is a rank-1 matrix, the Sherman-Morrison formula can be applied to efficiently compute the inverse of ˜ H t as

which reduces the computational complexity to O ( d 2 ) per iteration, assuming H -1 t is available. Since H t = H t -1 + ∇ 2 ℓ t -1 ( θ t ) , H -1 t can also be updated by the Sherman-Morrison formula in O ( d 2 ) time per round based on H -1 t -1 . Therefore, the total computational cost of (28a) is O ( d 2 ) . In the second step (28b), as ˜ H t is positive semi-definite, the optimization problem can be solved in O ( d 3 ) time (see Section 4.1 of [Mhammedi et al., 2019] for details). Overall, the total time complexity for solving (3) is O ( d 3 ) .

Space Complexity. Regarding space complexity, it suffices to store the current model θ t , the gradient ∇ ℓ t ( θ t ) , the inverse Hessian matrix H -1 t , and ˜ H -1 t throughout the optimization process, resulting in a total space complexity of O ( d 2 ) .

B Proof of Theorem 2

Proof of Theorem 2. Let ( X t , ˜ θ t ) = arg max x ∈X t ,θ ∈C t ( δ ) µ ( x ⊤ θ ) . We can bound the regret by

where ˜ α ( θ 1 , θ 2 , X s ) = ∫ 1 0 (1 -v ) µ ′′ ( X ⊤ s θ 1 + v X ⊤ s ( θ 2 -θ 1 ) ) d v. In the above, the first inequality is due to the arm selection rule (6) and the second equality is by the integral formulation of the Taylor's expansion. Then, we upper bound the terms respectively.

Analysis for term (a). For the first term, we have

where the first inequality is due to the Hölder's inequality and the second inequality is by the fact ∥ ˜ θ t -θ ∗ ∥ H t ≤ ∥ ˜ θ t -θ t ∥ H t + ∥ θ t -θ ∗ ∥ H t ≤ 2 β t ( δ ) . In the last inequality, we decompose the time horizon into T 1 = { t ∈ [ T ] | µ ′ ( X ⊤ t θ ∗ ) ≥ µ ′ ( X ⊤ t θ t +1 ) } and T 2 = [ T ] / T 1 .

Analysis for Term (a1) : For the time steps in t ∈ T 1 , the term µ ′ ( X ⊤ t θ ∗ ) can be further bounded by

where the first inequality is due the self-concordant property of µ . The second inequality is by Assumption 2 such that µ ′ ( z ) ≤ L µ for z ∈ [ -S, S ] and the Hölder's inequality. The last inequality is due to Theorem 1 and the fact H t +1 ≽ H t .

Then, let ˜ H t := g ( τ ) H t = λg ( τ ) I d + ∑ t -1 s =1 µ ′ ( X ⊤ s θ s +1 ) X s X ⊤ s and V t := λg ( τ ) I d + 1 κ ∑ t -1 s =1 X s X ⊤ s . We can upper term (a1) by

where the first inequality is by the condition µ ′ ( X ⊤ t θ t +1 ) ≤ µ ′ ( X ⊤ t θ ∗ ) for t ∈ T 1 . The second inequality is due to the Cauchy-Schwarz inequality and the fact that ˜ H t ≽ V t . Then, we can further bound (31) by elliptical potential lemma (Lemma 11 in Appendix C) as:

by taking z t = √ µ ′ ( X ⊤ t θ t +1 ) X t in Lemma 11. The last term in (31) can also be bounded by

For notational simplicity, we denote by

Then, plugging (32) and (33) into (31) yields

where the last inequality can be obtained following the same arguments in the proof of [Abeille et al., 2021, Theorem 1].

Analysis for Term (a2) : As for the term (a2), we have

where the last inequality holds due to the condition µ ′ ( X ⊤ t θ ∗ ) ≤ µ ′ ( X ⊤ t θ t +1 ) . Following the same arguments in bounding (31), we can obtain

Combining the upper bound for term (a1) and term (a2), we have

Analysis for term (b). As for the term by, we have

where the first inequality is due to the self-concordant property of µ . The second inequality is by Assumption 2 and Cauchy-Schwarz inequality. The third inequality is due to the fact ∥ ˜ θ t -θ ∗ ∥ 2 H t ≤ 4 β 2 T ( δ ) and H t ≽ g ( τ ) V t . The last line can be obtained following the same arguments in bounding (33).

Overall Regret Bound. Plugging (35) and (36) into (29) yields

Removing the above inequality and rearranging the terms yields

where γ 1 = γ (1) T ( δ ) and γ 2 = γ (2) T ( δ ) is defined as (34). We have completed the proof of the regret.

Computational Complexity. As shown in Proposition 1 in Appendix A.3, the time complexity for updating the online estimator θ t is O ( d 3 ) (line 5 in Algorithm 1). Additionally, the inverse Hessian matrix H -1 t can be updated in O ( d 2 ) time per round as shown in Appendix A.3. The remaining computational cost arises from the arm selection (6), which solves the optimization problem

Given θ t and H -1 t , this optimization can be performed in O ( d 2 |X t | ) time at round t . Therefore, the total per-round computational complexity is O ( d 3 + d 2 |X t | ) .

C Technical Lemmas

Lemma 8 (Lemma 9 of Faury et al. [2020]) . Let µ : R → R be a strictly increasing function satisfying | µ ′′ ( z ) | ≤ Rµ ′ ( z ) for all z ∈ Z , where R > 0 is a fixed positive constant and Z ⊂ R is a bounded interval. Then, for any z 1 , z 2 ∈ Z and z ∈ { z 1 , z 2 } , we have

and the weighted integral

We include the proof here for self-containedness.

Proof of Lemmas 8. Without loss of generality, assume z = z 1 . Let ϕ ( ν ) := µ ′ ( z 1 + ν ( z 2 -z 1 )) and ∆ := | z 2 -z 1 | . From | µ ′′ ( z ) | ≤ Rµ ′ ( z ) , we obtain the key differential inequality

The solution to (39) yields the exponential bounds

Proof of (37): Using the lower bound in (40), we have

where the last inequality uses 1 -e ≥ x/ (1 + x ) for x ≥ 0 .

Proof of LHS of (38): For the weighted integral, we have

The final inequality follows from the fact that

Proof of RHS of (38): Using the upper bound in (40), we have

where we used e x -1 -x ≤ x 2 e x 2 for x ≥ 0 .

The case for z = z 2

Lemma 9 (Ville [1939]) . Let { M t } ∞ t =0 be a supermartingale with M t ≥ 0 almost surely for all t ≥ 0 . Then, for any ε > 0 ,

Lemma 10. Let P = N ( 0 , ηH -1 ) be a Gaussian distribution with mean 0 ∈ R d and covariance ηH -1 , where H ≽ λI d is a positive definite matrix and η > 0 . Then, if λ ≥ 32 dηc/ 7 we have

Proof of Lemma 10. Let θ ∈ R d be a random variable sampled from P . One can verify that it also follows the sample distribution as

where { e i } d i =1 is a set of orthogonal base and λ i is the i -th largest eigenvalue of H -1 . Then, we have

where χ 2 denotes the chi-squared distribution with degree of freedom 1. The first inequality is because max i ∈ [ d ] λ i ≤ 1 /λ due to the condition H ≽ λI d . The last second equality is by the Jensen's inequality since x d is a convex function with respect to x . The last inequality is due to the condition cηd/λ ≤ 7 / 32 and the fact that the moment-generating function of the chi-squared distribution E [exp( tZ )] ≤ (1 -2 t ) -1 / 2 for t < 1 / 2 .

Lemma 11 (Lemma 9 of Faury et al. [2022]) . Let λ ≥ 1 and { z s } ∞ s =1 a sequence in R d such that ∥ z s ∥ 2 ≤ Z for all s ∈ N . For t ≥ 2 define V t := ∑ t -1 s =1 z s z ⊤ s + λI d . The following inequality holds

Lemma 12. Let P be a probability distribution defined over R d and ∆ be the set of all measurable distributions. For any loss function ℓ : R d → R , we have

where P ∗ = arg min P ′ ∈ ∆ E θ ∼ P ′ [ ℓ ( θ )] + 1 α KL( P ′ ∥ P ) is the optimal solution. Furthermore, for any distribution Q defined over R d , we have

Proof of Lemma 12. To prove (41), one can check that the optimal solution to the optimization problem on the right-hand side of (41) is given by

Substituting P ∗ back into the right-hand side yields the desired equality. Alternatively, (41) can be shown by noting that the mix loss is the convex conjugate of the Kullback-Leibler divergence [Reid et al., 2015]. By the definition of the convex conjugate, the equality holds.

To prove the second part of the lemma, namely (42), let Z = ∫ θ ∈ R d P ( θ ) e -αℓ ( θ ) d θ . We then have

where the second equality is due to the fact that P ∗ ( θ ) /P ( θ ) = e -αℓ ( θ ) /Z for all θ ∈ R d . Then, rearranging the above displayed equation gives us

which completes the proof by using (42).

Lemma 13 (Theorem 1.8.2 of Ihara [1993]) . The Kullback-Leibler divergence between two d -dimensional Gaussian distributions P = N ( u p , Σ P ) and Q = N ( u q , Σ q ) is given by

D More Discussions on Lee and Oh [2025a]

For MNL bandits, [Lee and Oh, 2025a] (v3 version, the latest one available before the NeurIPS submitted date) claimed an O ( √ log T ) improvement in the regret bound, which could potentially be applied to logistic bandits to achieve an O (log T √ T/κ ∗ ) bound with O (1) computational cost. However, their analysis relies on a specific upper bound on the normalization factor of a truncated Gaussian, which may not always hold. We elaborate on the main technical issue below in the context of the logistic bandit problem.

With slight abuse of notation, we define the logistic loss as ℓ s ( θ ) = r t X ⊤ t θ +log(1 + exp( X ⊤ t θ )) , where X t is the action selected by the learner and r t ∈ { 0 , 1 } is the observed reward. Specifically, building upon the framework of Zhang and Sugiyama [2023], the Lemma C.1 of Lee and Oh [2025a] shows that the estimation error of their estimator satisfies

where ¯ ℓ s ( z ) = r s z +log(1 + exp( z )) . In Lee and Oh [2025a], the intermediate term is chosen as ˜ z s = σ + ( E θ ∼ P s [ σ ( X ⊤ s θ )] ) , where σ ( z ) = 1 / (1 + e -z ) and σ + ( p ) = log( p 1 -p ) .

A key step of their analysis lies in the choice of P s , which ensures that term (a) and term (b) can be bounded separately. In contrast to the Gaussian distribution used in Zhang and Sugiyama [2023], Lee and Oh [2025a] take P s as a truncated Gaussian, thereby allowing the bound on term (a) to avoid the additional O ( √ T ) factor incurred in Zhang and Sugiyama [2023]. However, when analyzing term (b), their argument relies on a condition for the normalization factor of the truncated Gaussian distribution, which does not holds in general (see Eqn (C.15) in their paper).

For completeness, we restate it below: there exists a constant C s ≥ 1 such that

where γ, c > 0 are certain constants and H s is a symmetric positive-definite matrix. Condition (43) plays an important role in ensuring that the term log( Z s / ̂ Z s +1 ) in Eq. (C.17) of the paper remains non-positive and does not affect the final bound of term(b). However, it is not evident how to select C s ≥ 1 to guarantee this property holds throughout, as the left-hand side integrates over a strictly larger region while the integrand decays more slowly. We also mention that this issue was also concurrently addressed by the authors in the later version [Lee and Oh, 2025b] (v5 version, posted at June 2025), and the fixed technique is essentially similar to the mixability-based analysis in our work.

E Additional Experimental Details

In this section, we provide additional experimental details and results.

E.1 Experimental Setup

Implementation Details. All the experiments were conducted on Intel Xeon Gold 6242R processors (40 cores, 4.1GHz base frequency). The algorithms were implemented in Python, utilizing the scipy library for numerical computations, such as solving non-linear optimization problems and calculating vector norms, and employing np.linalg.pinv to compute the pseudo-inverse of matrices. The running time was measured using the time library. The shaded regions in the regret plots represent 99% confidence intervals, computed from 10 independent runs with different random seeds.

(a)

d

= 2

, S

= 3

, T

= 300

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= 2

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= 5

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= 300

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= 2

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= 7

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= 300

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GLB-OMD (Our work)

RS-GLinUCB (Sawarni et al. 2024)

ECOLog (Faury et al. 2022)

OFULog-r (Abeille et al. 2021)

GLM-UCB (Filippi et al. 2010)

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Logistic Bandit (

S

= 7

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GLB-OMD (Our work)

RS-GLinUCB (Sawarni et al. 2024)

ECOLog (Faury et al. 2022)

OFULog-r (Abeille et al. 2021)

GLM-UCB (Filippi et al. 2010)

(b)

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S

= 9

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GLOC

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on Poisson Bandit

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GLB-OMD (Our work)

RS-GLinUCB (Sawarni et al. 2024)

ECOLog (Faury et al. 2022)

OFULog-r (Abeille et al. 2021)

GLM-UCB (Filippi et al. 2010)

(a)

Regret for Cover Type Data

GLB-OMD

OFUGLB

RS-GLinCB

ECOLog

OFULog-r

GLOC

GLM-UCB

10 0

10 1

10 2

10 3

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0.64

4513.40

16.82

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424.42

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(b)

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GLB-OMD

OFUGLB

RS-GLinCB

ECOLog

OFULog-r

GLOC

GLM-UCB

(c)

Regret vs Running time

Algorithm Configuration. Throughout our experiments, all algorithm parameters were configured according to their theoretical derivations without additional fine-tuning, with the sole exception of the regularization parameter λ . To ensure a fair comparison, we adopted a unified approach for setting λ across different algorithm categories: we set λ = d for all efficient online algorithms (including GLB-OMD , RS-GLinCB , ECOLog , and GLOC ), while using λ = d log(1 + t ) for offline algorithms that require regularization. This distinction reflects the practical consideration that real-world scenarios often exhibit more favorable conditions than the worst-case assumptions.

E.2 More results on Synthetic Data

To visualize the accuracy of parameter estimation of the algorithms, we plot the confidence region of the parameter estimation for each algorithm in Figure 3. For illustration purposes, we only plot the confidence regions of our algorithm GLB-OMD , the theoretically optimal OFUGLB , and the classical GLM-UCB . We observe that both GLB-OMD and OFUGLB achieve a substantially smaller confidence region than GLM-UCB , indicating that our algorithm achieves an accurate parameter estimation comparable to the statistically optimal.

To further investigate the impact of parameter S on algorithm performance, we conduct additional experiments on logistic bandit tasks with larger S values (Figures 4a and 4b) and analyze the computational time scaling for Poisson bandits (Figure 4c).

The regret curves in Figures 4a and 4b consistently demonstrate that our algorithm maintains its competitive performance regardless of S variations, aligning with the trends observed in our main results. Notably, the regret does not exhibit significant growth as S increases, suggesting the robustness of our approach to parameter scaling. We also note that the performance of RS-GLinCB is very sensitive to the parameter of S . This underperformance can be attributed to the fact that the warm-up period of RS-GLinCB is heavily dependent on the constant S and κ [Sawarni et al., 2024, Lemma 4.1].

The runtime curves in Figure 4c reveals two key findings: First, our algorithm's running time remains stable (under 1 second for T = 3000 ) across different S values in Poisson bandit tasks. Second, in contrast to this consistent performance, OFUGLB exhibits a pronounced computational overhead that scales with S (requiring 2783 seconds at S = 3 compared to 7568 seconds at S = 9 ). This divergence can be attributed to the confidence radius construction in OFUGLB :

where β t ( δ ) 2 = log 1 δ +inf c t ∈ (0 , 1] { d log 1 c t +2 SL t c t } ≤ log 1 δ + d log ( e ∨ 2 eSL t d ) . For Poisson bandits specifically, L t = e S t +log( d/δ ) . This results in increasing cost in the optimization steps during the arm selection X t = arg max x ∈X t max θ ∈C t ( δ ) x ⊤ θ , as the algorithm needs to navigate a rapidly expanding nonconvex confidence region.

Overall, our algorithm demonstrates comparable statistical performance to the theoretically optimal OFUGLB while offering substantially improved computational efficiency.

E.3 Experiment on Forest Cover Type Data

In this experiment, we evaluate our proposed algorithm on the Forest Cover Type dataset from the UCI Machine Learning repository [Blackard, 1998]. This dataset comprises 581,012 labeled observations from different forest regions, with each label indicating the dominant tree species.

Following the preprocessing steps described in Filippi et al. [2010], we centered and standardized the 10 non-categorical features and appended a constant covariate. To enhance the diversity of the arm set and strengthen the experimental results, we partitioned the data into K = 60 clusters using unsupervised K -means clustering, with the cluster centroids serving as the contexts for each arm. For the logistic reward model, we binarized the rewards by assigning a reward of 1 to data points labeled as the second class ('Lodgepole Pine') and 0 otherwise. The reward for each arm was then computed as the average reward of all data points within its corresponding cluster, yielding reward values ranging from 0.103 to 0.881.

For this task, we set the horizon to T = 1000 and the confidence parameter to δ = 0 . 01 . After analyzing the data, we set S = 6 and κ = 200 . We evaluated our algorithm against the same baselines used in the logistic bandit simulation experiment, running each method over 10 independent trials and averaging the results to report the regret and the running time. The error bars in the figures denote 99% confidence intervals for both regret and runtime.

Compared to synthetic data, real-world datasets exhibit higher noise and complexity, demanding careful exploration-exploitation trade-offs. Thus, we shrank the estimated confidence set of all the algorithms in a comparable way to achieve a better balance between exploration and exploitation in this real-world dataset. Traditional GLB algorithms are particularly sensitive to noise, often leading to excessive exploration and higher regret.

Figure 5a presents the regret progression of different algorithms over time, while Figure 5b compares their computational efficiency. Figure 5c further illustrates the regret-time trade-off for our method. The results demonstrate that our algorithm achieves significantly faster runtime without compromising robustness or performance, even in noisy environments.

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Peer reviews:
comments: SUMMARY:
In this paper, the authors study the GLB problem and propose the first algorithm with $O(1)$ per-round computational complexity, achieved by leveraging the OMD estimator.
The main technical contribution is an improvement of the regret bound by a factor of $\sqrt{\log t}$ over the prior result of Zhang and Sugiyama [2023].

STRENGTHS AND WEAKNESSES:
**Strengths**

This paper is overall well-organized and easy to follow. The main technical results are supported by adequate claims, and the empirical results further strengthen the theoretical guarantees.


**Weaknesses**

- The main technical contribution seems to be the improvement by a $\sqrt{\log t}$ factor compared to Zhang and Sugiyama [2023].
Are there any additional technical challenges in adapting the OMD estimator from Zhang and Sugiyama [2023] to the GLM setting? 
Since the overall algorithm and analysis closely follow those in Zhang and Sugiyama [2023], I am not fully convinced that the $\sqrt{\log t}$ improvement alone constitutes a substantial contribution.

- No code is provided.

- In the experiments, why wasn’t the proposed algorithm compared with that of Zhang and Sugiyama [2023]?

     In Figure 1(a) and 1(b), why does RS-GLinUCB underperform in the case of $S=3$ compared to $S=5$?

     Although the values of $\kappa$ are smaller in the Poisson bandit setting than in the logistic case, warm-up-based methods like RS-GLinUCB completely fail.  Why does this happen?

QUESTIONS:
Please see the questions listed in the Weaknesses section above.

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-06-26

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose a new algorithm for the General Linearized Bandits setting which enjoys a balance of both computaional and statistical efficiency improvements over previous methods.\n\nSpecifically, they propose an OMD-based estimator for a confidence set for the latent parameter $\\theta_*$ which provides tighter confidence sets than previous approaches by leveraging self-concordance. They use this estimator to provide uncertainty sets for a UCB-style algorithm that, together with the estimator, has constant memory and time costs while achieving competitive regret performance with methods requiring orders of magnitude more compute. \n\nThey provide theoretical analysis and experiemntal results on synthetic tasks to demonstrate the gains assocaited with their approach.\n\nSTRENGTHS AND WEAKNESSES:\n*Strengths:*\n- Generalizes OMD + UCB approach to the broader GLB setting\n- Improves SotA for logistic bandits across the regret, time, memory frontier\n- Regret analysis with impressive empirical results across both statsitical and computational fronts (1000x test time computational improvement)\n- Leveraging self-concordance for inproved confidence sets is elegant and general\n\n*Weaknesses*\n\n- Although the core contributions of this paper are strong, overall the paper reads as somewhat rushed in its organsation and preparation. For example, the introduction was harder to read than necessary, with many of the core definitions deferred to the preliminaries in a way that made it hard to appreciate what exactly was being estimated and how that estimate was being used within a UCB framework. This feels unneccessary since it ultimately is quite a simple quantity to describe in one or two sentences. Generally, assumptions in the first few pages of the paper felt scattered in the prose. For example, it wasn’t clear to me if the link function $\\mu$ was known to the agent, belonged to a family etc. until the preliminaries. I found it hard to apprecaite what the setting was exactly until multiple passes. In short, the full setting only became clear to me upon reading the preliminaries; the introduction felt confusing.\n- Some technical assumptions are made in the notation which never feature in the prose of the paper, e.g. the action sets $\\mathcal{X}_t$ appear to be non-stationary given the $t$ subscripts, yet there is no motivaiton for this consideration and no regard to it in the description of the setting. This felt disracting.\n- The paper contains some instances of vague expersions such as “non-linearity is beneficial” which made for a somewhat frustrating read.\n- Phrases like “bridging the loss gap” are unclear until deeper into the proof section and made for a confusing read.\n- Some sentences seem incomplete, e.g. “We also note that OMD-based online estimators have been used to develop jointly efficient algorithms in the logistic bandit setting … their analyses of the confidence set rely heavily on the specific structure of the logistic link function, limiting their applicability to more general GLB models.” lacks a bridge word such as “but” or “however”\n- Figures lack clarity, the shaded region should be well defined.\n- There are some notation inconsistencies, such as starting wth $\\mathbf{x}$ early in the paper for actions but then switching to $X_t$ later, and mentioning that the notation is due to actions being “random variables”, which didn’t help me apprecaite why earlier notation was different; also $\\tau$ is not defined anywhere in the paper yet appears in important definitions.\n- They say “This bound is nearly optimal in terms of the horizon $T$ up to logarithmic factors” but give no specific citation for this claim or substantiate it.\n\nQUESTIONS:\nMy primary concerns with this paper are in its writing clarity and organisation. Addressing the weaknesses outlined above are sufficient grounds for me to increase my clarity score.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces a new, tight concentration bound for the online mirror descent (OMD)-based online estimator for generalized linear bandit (GLB) that takes $\\mathcal{O}(1)$ space/time complexity per round. The analysis involves appropriately using the notion of mix losses from online learning, which allows it to generalized to GLBs beyond prior works that were only applicable to logistic losses. The new concentration-based UCB algorithm is shown to achieve nearly tight (up to norm factors) regret bounds, and is validated empirically across logistic and Poisson bandits.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- Well-written\n- Technically sound, and the proposed approach to building new concentration seems distinct from prior works.\n\n\n**Weaknesses**\n- Some of the lemmas are lacking the original references. For instance, Lemma 8 is originally Lemma 7 & 8 of Abeille et al. (2021), which in turn was originally proved in Lemma 9 of Faury et al., (2020). Also, the maximal inequality is actually originally known as Ville's inequality due to Ville (1939). Also, the whole idea of utilizing self-concordance is originally due to Bach (2010).\n- At the end, the proof proceeds similarly to \"regret-to-confidence-set\" conversion-type results [1,2] as well as the recent PAC-Bayes argument for GLBs [3]. Especially the overall proof flow of Theorem 1 (martingale, Ville's inequality, Donsker-Varadhan-type characterization of KL, use of KL closed-form) resembles [3], although there are indeed key differences, such as the choice of posterior ([3] chooses uniform while this paper chooses Gaussian? please correct me if I'm way off here). Anyhow, some discussions to such relevant literature would be nice.\n- Over the 10 independent trials, some sort of confidence/error interval would be nice addition on top of the average regret! Since 10 is a bit small, maybe bootstrapped confidence interval?\n- Later, I hope that the authors would release the codes :)\n\n**Typos**\n- Line 629: o $\\Rightarrow$ To\n\n\n[1] https://proceedings.mlr.press/v238/lee24d.html\n\n[2] https://arxiv.org/abs/2504.16555\n\n[3] https://openreview.net/forum?id=MDdOQayWTA\n\nQUESTIONS:\n1. How was the runtime computed? Did you use time or timeit?\n2. Can the authors plot the confidence regions as well? Also, in line 710-711, the authors state \"as the algorithm must navigate a rapidly expanding confidence region.\" Is it really expanding, though? Because at the end, $\\beta_t(\\delta) = \\Theta(\\log t)$ and the number of terms for the log-likelihood loss on the LHS increases linearly.\n3. If the arm-set is fixed, then could one hope for slightly better guarantee by any chance, e.g., [1]?\n\n\n[1] https://arxiv.org/abs/2202.02407\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the problem of generalized linear bandits, and proposes an algorithm which is jointly sample-efficient and computationally efficient. The proposed method relies on a new confidence set construction for generalized linear models, which is the main technical contribution of the paper. The new sequence of confidence sets is a sequence of ellipsoids, where each ellipsoid is centered at the estimate returned by online mirror descent (OMD), as opposed to the usual (regularized) maximum likelihood estimator (MLE). The proposed GLB-OMD method is evaluated in several experiments. In these experiments, it is found that the run time of GLB-OMB is a lot lower than that of the state-of-the-art OFUGLB method, yet the regret of GLB-OMD is not much worse than that of OFUGLB.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\nThe paper is generally well-written and the results are presented and explained well. In particular, I think that (for the most part) Section 4 does good job of explaining how the construction of the new confidence sequence works and how it differs to previous confidence sequence constructions.\n\nThe GLB-OMD method appears to be an effective solution to the problem described in the introduction. Namely, providing a generalized linear bandit algorithm with near-optimal regret and constant (w.r.t. $t$) time and space complexity per round.\n\n**Weaknesses:**\nMy biggest concern is with the significance of the problem being solved. I’m not sure that replacing a time complexity per round of order $t$ (or $\\log t$) with constant (w.r.t. $t$) time complexity is of great important when the time complexity of running the algorithm is largely determined by the cost of maximizing the UCB, unless the action set is finite and small. Moreover, as far as I can tell, the computational challenge of maximizing the UCB is not mentioned anywhere.\n\nI think that some of the claims made are not fully supported or slightly misleading. For instance, Remark 1 fails to mention that, for instance, the OFUGLB method also accommodates unbounded GLMs such as Gaussian or Poisson. On line 221, it is stated that “self-concordant GLBs are no more difficult than linear bandits”. This is true if $T$ is large enough to make the constant term in the regret bound negligible, but this constant term may be extremely large.\n\nSome relevant prior work is not cited. In particular, Proposition 2 is not a new result. For instance, in the book by Grünwald (2007), this result appears under the name “no-hypercompression inequality”.\n\nGrünwald, Peter D. The minimum description length principle. MIT press, 2007.\n\nQUESTIONS:\nBy modifying the construction in this paper, would it be possible to construct even smaller confidence sets centered at OMD estimates which are not ellipsoids (perhaps by using a different regularizer/penalty in the OMD update)? For GLMs, restricting the confidence set to be an ellipsoid generally results in looser confidence sets.\n\nThe general technique of substituting the log-likelihood loss for the mix-loss to derive confidence sets has appeared in a couple of very recent works (Kirscher et al. 2025, Clerico et al. 2025). I don’t consider this to be a weakness, since these papers were released in close proximity to the submission deadline. Nevertheless, one could add a comment on this to the final version of the paper.\n\nKirschner, Johannes, et al. \"Confidence Estimation via Sequential Likelihood Mixing.\" arXiv preprint arXiv:2502.14689 (2025).\n\nClerico, Eugenio, et al. \"Confidence Sequences for Generalized Linear Models via Regret Analysis.\" arXiv preprint arXiv:2504.16555 (2025).\n\nLIMITATIONS:\nI understand that maximizing the upper confidence bound described in Section 3.1 is in general NP-hard. Indeed, in the linear case, the UCB is a convex function of the action, and convex maximization is in general NP-hard. Given that one of the main claims of the paper is about computational efficiency, it is a bit odd that this is never mentioned.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-06"}