Title: Linear Bandits with Non-i.i.d. Noise
Venue: NeurIPS 2025
Accepted: no
Keywords: linear bandits

non-i.i.d

online learning
Abstract: We study the linear stochastic bandit problem, relaxing the standard i.i.d. assumption on the observation noise. 
As an alternative to this restrictive assumption, we allow the noise terms across rounds to be sub-Gaussian but 
interdependent, with dependencies that decay over time. To address this setting, we develop new confidence sequences 
using a recently introduced reduction scheme to sequential probability assignment, and use these to derive a bandit 
algorithm based on the principle of optimism in the face of uncertainty. We provide regret bounds for the 
resulting algorithm, expressed in terms of the decay rate of the strength of dependence between observations. Among 
other results, we show that our bounds recover the standard rates up to a factor of the mixing time for geometrically 
mixing observation noise.

Abstract

We study the linear stochastic bandit problem, relaxing the standard i.i.d. assumption on the observation noise. As an alternative to this restrictive assumption, we allow the noise terms across rounds to be sub-Gaussian but interdependent, with dependencies that decay over time. To address this setting, we develop new confidence sequences using a recently introduced reduction scheme to sequential probability assignment, and use these to derive a bandit algorithm based on the principle of optimism in the face of uncertainty. We provide regret bounds for the resulting algorithm, expressed in terms of the decay rate of the strength of dependence between observations. Among other results, we show that our bounds recover the standard rates up to a factor of the mixing time for geometrically mixing observation noise.

1 Introduction

The linear bandit problem (Abe and Long, 1999; Auer, 2003) is an instance of a multi-armed bandit 13 framework, where the expected reward is linear in the feature vector representing the chosen arm. 14 More concretely, it is a sequential decision-making problem, where an agent each round picks an arm 15 X t , and receives a reward Y t = ⟨ θ ⋆ , X t ⟩ + ε t , with θ ⋆ a fixed parameter unknown to the agent, and 16 ε t zero-mean random noise. This framework has gained significant attention in the literature as it 17 yields analytic tools that can be applied to several concrete applications, such as online advertising 18 (Abe et al., 2003), recommendation systems (Li et al., 2010; Korkut and Li, 2021), and dynamic 19 pricing (Cohen et al., 2020). 20

Apopular strategy to tackle linear bandits leverages the principle of optimism in the face of uncertainty , 21 via upper confidence bound (UCB) algorithms. The idea of optimism can be traced back to Lai and 22 Robbins (1985), and its application to linear bandits was already advanced by Auer (2003). Since 23 then, this approach has been improved and analysed by several works (Abbasi-Yadkori et al., 2011; 24 Lattimore and Szepesvári, 2020; Flynn et al., 2023). This class of methods requires constructing an 25 adaptive sequence of confidence sets that, with high probability, contain the true parameter θ ⋆ . Each 26 round, the agent selects the arm maximising the expected reward under the most optimistic parameter 27 (in terms of reward) in the current confidence set. UCB-based algorithms have become popular as 28 they are often easy to implement and come with tight worst-case regret guarantees. 29

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For a UCB algorithm to perform well, it is necessary that the confidence sets are tight, which can be ensured by taking advantage of the structure of the problem. In this paper, our focus is on studying various assumptions on the observation noise. A commonly studied situation is when ( ε t ) t ≥ 0 consists

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of a sequence of i.i.d. realisations of some bounded or sub-Gaussian random variable (see Lattimore

and Szepesvári, 2020, Chapter 20). Often, the standard analysis can be extended to the case in which

the realisation are not independent, but conditionally centred and sub-Gaussian (Abbasi-Yadkori 35

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et al., 2011). Yet, in real-world settings, this assumption is often unrealistic, as one can expect the

presence of interdependencies among the noise at different rounds. For instance, in the context 37

of advertisement selection, the noise models the ensemble of external factors that influence the 38

user's choice on whether to click or not an ad. The i.i.d. assumption implies that across different 39 rounds these external factors are completely independent. In practice, the user choice will be affected 40 by temporally correlated events, such as recent browsing history or exposure to similar content. 41 Therefore, a more realistic assumption is to allow the dependencies to decay with time, rather than 42 being completely absent. This way to model dependencies, often referred to as mixing , is common to 43 study concentration for sums of non-i.i.d. random variables, with applications to machine learning 44 (Bradley, 2005; Mohri and Rostamizadeh, 2008; Abélès et al., 2025). 45

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In the present paper we relax the assumption that the noise is conditionally zero-mean in the bandit problem, and we allow for the presence of dependencies. Concretely, we replacethe standard conditionally sub-Gaussian setting with a more general formulation that accounts for conditional dependence of the noise on the past, by introducing a natural notion of mixing sub-Gaussianity . Within this context, we introduce a UCB algorithm for which we rigorously establish regret guarantees. There are two key challenges for our approach: constructing a valid confidence sequence under dependent noise, and deriving a regret upper bound for the UCB algorithm that we propose.

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We derive the confidence sequence by adapting the online-to-confidence-sets technique to accommodate temporal dependencies in the noise. This approach, originally introduced by Abbasi-Yadkori et al. (2011) and recently extended and improved (Jun et al., 2017; Lee et al., 2024; Clerico et al., 2025), involves constructing an abstract online learning game whose regret guarantees can be turned into a confidence sequence. To deal with the dependencies in the noise, we modify the standard online-to-confidence-sets framework by introducing delays in the feedback received within the abstract online game. This approach is inspired by the recent work of Abélès et al. (2025) on extending online-to-PAC conversions to non-i.i.d. mixing data sets in the context of deriving generalisation bounds for statistical learning. There, a delayed-feedback trick similar to ours is employed to derive statistical guarantees (generalisation bounds) from an abstract online learning game.

For the regret analysis of the bandit algorithm, we also need to face some challenges due to the correlated observation noise. We address these by introducing delays into the decision-making policy as well. This makes our approach superficially similar to algorithms used in the rich literature on bandits with delayed feedback (see, e.g., Vernade et al., 2020a; Howson et al., 2023). These works consider delay as part of the problem statement and not part of the solution concept, and are thus orthogonal to our work. In particular, a simple adaptation of results from this literature would not suffice for dealing with dependent observations, which we tackle by developing new concentration inequalities. Another line of work that is conceptually related to ours is that of non-stationary bandits (Garivier and Moulines, 2008; Russac et al., 2019). In that setting, the parameter vector θ ⋆ t evolves in time according to a nonstationary stochastic process, and the observation noise remains i.i.d., once again making for a rather different problem with its own challenges. Namely, the main obstacle to overcome is that comparing with the optimal sequence of actions becomes impossible unless strong assumptions are made about the sequence of parameter vectors. A typical trick to deal with these nonstationarities is to discard old observations (which may have been generated by a very different reward function), and use only recent rewards for decision-making. This is the polar opposite of our approach that is explicitly disallowed to use recent rewards, which clearly highlights how different these problems are. That said, there exists an intersection between the worlds of delayed and nontationary bandits (Vernade et al., 2020b), and thus we would not discard the possibility of eventually building a bridge between bandits with nonstationary reward functions and bandits with nonstationary observation noise. For simplicity, we focus on the second of these two components in this paper.

Notation. Throughout the paper, we will often use the following notations. For u and v in R p , we let ⟨ u, v ⟩ denote their dot product. ∥ u ∥ 2 = √ ⟨ u, u ⟩ is the Euclidean norm, while for a non-negative definite ( p × p ) -matrix A , ∥ u ∥ A = √ ⟨ u, Au ⟩ is a semi-norm (a norm if the matrix is strictly positive definite). For r > 0 , B ( r ) denotes the closed centred Euclidean ball in R p with radius r . Given a non-empty set U ⊆ R p , we let ∆ U denote the space of (Borel) probability measures on R p whose support in U . Finally, ( u t ) t ≥ t 0 denotes a sequence indexed on the integers, with t 0 its smallest index.

2 Preliminaries on linear bandits

We consider a version of the classic problem of regret minimisation in stochastic linear bandits, where 91 an agent needs to make a sequence of decisions (or pick an arm ) from a given contextual decision set 92

that may change over the sequence of rounds. We assume that the environment is oblivious to the 93 actions of the agent, in the sense that the decision sets are determined in advance, and do not depend 94 neither on the realisations of the noise nor on the agent's arm-selection strategy. 95

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Concretely, we define the problem as follows. Let θ ⋆ ∈ R p be a parameter vector that is unknown to the learning agent. We assume as known an upper bound B > 0 on its euclidean norm (namely, θ ⋆ ∈ B ( B ) ). Fix a sequence of decision sets ( X t ) t ≥ 1 in R p . We assume that for all t we have X t ⊆ B (1) . At each round t , the agent is required to pick an arm X t ∈ X t , and receives the reward Y t = ⟨ θ ⋆ , X t ⟩ + ε t . The sequence ( ε t ) t ≥ 1 represents the random feedback noise. The noise across different rounds is typically assumed to be conditionally centred and to have well behaved tails. For instance, a common assumption is to ask that E [ ε t |F t -1 ] is centred and sub-Gaussian, where F t = σ ( ε 1 , . . . , ε t ) is the σ -field generated by the noise. 1 This is the assumption this work relaxes.

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The agent aims to find a good strategy to pick arms X t that lead to a high expected T -round reward ∑ T t =1 ⟨ X t , θ ⋆ ⟩ . To compare their performance to that of an agent playing each round the best available arm (in expectation), we define the regret after T rounds as

A common approach to tackle the linear bandit problem is to follow an upper confidence bound (UCB) strategy. This involves the following protocol. At each round t , we first derive a confidence set C t -1 , based on the arm-reward pairs ( X s , Y s ) s ≤ t -1 . This is a random set (as it depends on the past noise realisations), which must be constructed ensuring that θ ⋆ ∈ C t -1 with high probability. More precisely, the regret can be effectively controlled if one can ensure that θ ⋆ uniformly belongs to every set ( C t ) t ≥ 1 , with high probability (a property often referred to as anytime validity ). Then, for every available arm x , we let

By definition, this is a high-probability upper bound on ⟨ x, θ ⋆ ⟩ , which justifies the name 'upper 107 confidence bound'. The idea is then to optimistically pick as X t ∈ X t the arm maximising UCB C t -1 . 108

A key technical challenge in designing a UCB algorithm is to construct the anytime valid confidence sequence ( C t ) t ≥ 1 . Typically, under sub-Gaussian assumptions on the noise, these sets take the form of an ellipsoid, centred on a (regularised) maximum likelihood estimator. Explicitly, we often have

where ̂ θ t is the least-squares estimator of θ ⋆ , V t is the feature-covariance matrix and β t is a radius carefully chosen so that the high-probability coverage requirement is satisfied. In this work, to construct the confidence sets we will leverage an online-to-confidence-set-conversion approach, a method that reduces the problem of proving statistical concentration bounds to proving existence of well-performing algorithms for an associated game of sequential probability assignment . We refer to Section 4 for more details on our technique to construct the confidence sequence.

3 Linear bandits with non-i.i.d. observation noise

We study a variant of the standard linear stochastic bandit problem where the observation-noise variables feature dependencies across different rounds. We focus on the case of weakly stationary noise, meaning we assume all the ε t to have the same marginal distribution. However, the core assumption we make is what we call mixing sub-Gaussianity . This provides a way to control how dependencies decay as the time between two observations increases. It is defined in terms of a sequence of mixing coefficients ϕ d , which quantify this decay.

Assumption 1 (Mixing sub-Gaussianity) . Fix σ > 0 and let ϕ = ( ϕ d ) d ≥ 0 be a non-negative and non-increasing sequence. We say that the random sequence ( ϵ t ) t ≥ 1 is ( σ, ϕ ) -mixing sub-Gaussian if

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ε t is centred and σ -sub-Gaussian for every t , and, for all d ≥ 0 and all t > d , we have 124

and

Clearly, the above assumption generalises the standard conditionally sub-Gaussian assumption (that can be recovered by setting ϕ d = 0 for all t ), sometimes considered in the bandit literature. Although this might look like an unusual mixing assumption, it is very natural for our problem at hand, and can be weaker than standard mixing hypotheses. For instance, if the noise sequence is φ -mixing (see Bradley, 2005) and each ε t is centred and bounded in [ -a, b ] , it is straightforward to check that | E [ ε t |F t -d ] | ≤ ( a + b ) ϕ d , and so Assumption 1 is satisfied since the boundedness automatically implies sub-Gaussianity. In the rest of the paper we assume σ = 1 for simplicity.

Under Assumption 1, we can build the confidence sequence needed for our UCB algorithm. We state this result below, but defer the explicit derivation to Section 4 (see Corollary 1 there).

Proposition 1. For some given ϕ , let the noise satisfy Assumption 1 with σ = 1 . Fix δ ∈ (0 , 1) , λ > 0 , and d ≥ 1 . For t ≥ 1 let

where V t = ∑ t s =1 X t X ⊤ t + λ Id , and ̂ θ t = arg min θ ∈B ( B ) ∑ n s =1 ( ⟨ θ, X t ⟩ -Y t ) 2 . Then, ( C t ) t ≥ 1 is an anytime valid confidence sequence, in the sense that

Leveraging the confidence sequence above, we can define a UCB approach for our problem (Algo135 rithm 1). At a high level, the algorithm operates by taking the confidence sets defined in Proposition 136 1, and selecting the arm optimistically, as in the standard UCB. A key point is that a delay d is 137 introduced, which at round t restricts the agent to use only the information available from the first 138 t -d rounds. Although the actual technical reason behind this restriction will become fully clear only 139 with the analysis of the coming sections, one can intuitively think of it as a way to prevent overfitting 140 to recent noise, which might be highly correlated. If d is sufficiently large, the noise observed in 141 each round t will be sufficiently decorrelated from the previous observations, which allows accurate 142 estimation and uncertainty quantification of the true parameter θ ⋆ and the associated rewards. 143

Algorithm 1 Mixing-LinUCB

In Section 5 we provide a detailed analysis of the regret of the algorithm that we proposed. For 144 instance, assuming that the mixing coefficients decay exponentially as ϕ d = Ce -d/τ ( geometric 145 mixing ), we show that the regret can be upper bounded in high probability as 146

We refer to Theorem 2 and Corollary 2 in Section 5 for more details.

4 Constructing the confidence sequence

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In this section we derive a confidence sequence for linear models with non-i.i.d. noise. First, we 149 briefly describe the online-to-confidence-set conversion scheme from Clerico et al. (2025), which 150 serves as our starting point. We then extend this technique to handle mixing noise. 151

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4.1 Online-to-confidence set conversion for i.i.d. data

Before proceeding for the analysis of mixing sub-Gaussian noise, which is the focus of this work, we start by describing how to derive a confidence sequence when the noise is independent (or conditionally) centred and sub-Gaussian across different rounds, as in Clerico et al. (2025). The online-to-confidence sets framework that we consider instantiates an abstract game played between an online learner and an environment . We define the squared loss ℓ s ( θ ) = 1 2 ( ⟨ θ, X s ⟩ -Y s ) 2 . For each round s = 1 , . . . , t , the following steps are repeated:

the environment reveals X s to the learner;

the learner plays a distribution Q s ∈ ∆ R p ;

the environment reveals Y s to the learner;

the learner suffers the log loss L s ( Q s ) = -log ∫ R p exp( -ℓ s ( θ ))d Q s ( θ ) .

This game is a special case of a well-studied problem called sequential probability assignment (Cesa-Bianchi and Lugosi, 2006). The learner can use any strategy to choose Q 1 , . . . , Q t , as long as each Q s depends only on X 1 , Y 1 , . . . , X s -1 , Y s -1 , X s . We define the regret of the learner against a (possibly data-dependent) comparator ¯ θ ∈ R p as

Clerico et al. (2025) provide a regret bound upper bound (Proposition 3.1 there) for when the learner's strategy is from an exponential weighted average (EWA) forecaster with a centred Gaussian prior Q 1 . However, to account for the presence of dependencies in our analysis, we will need the prior's support to be bounded. We hence state here a regret bound (whose proof is deferred to Appendix A.2) for the regret of an EWA forecaster with a uniform prior.

Proposition 2. Fix B > 0 and consider the EWA forecaster with as prior the uniform distribution on B ( B +1) . Then, for all ¯ θ ∈ B ( B ) and any t ≥ 1 ,

We remark that, by adding and subtracting the total log loss of the learner, the excess loss of θ ⋆ (relative to ¯ θ ) can be rewritten as

This simple decomposition is the key idea in the online-to-confidence sets scheme. 177

Since the noise is conditionally sub-Gaussian and the distributions played by the online learner are predictable ( Q s cannot depend on Y s ), ∑ t s =1 ℓ s ( θ ⋆ ) -∑ t s =1 L s ( Q s ) is the logarithm of a non-negative super-martingale (cf. the no-hypercompression inequality in Grünwald, 2007 or Proposition 2.1 in Clerico et al., 2025) with respect to the noise filtration ( F t ) t ≥ 1 . 2 Henceforth, from Ville's inequality (a classical anytime valid Markov-like inequality that holds for non-negative super-martingales) one can easily derive that θ ⋆ ∈ C t (uniformly for all t ) with probability at least 1 -δ , where

This result can be relaxed by replacing Regret t ( ¯ θ ) by any known regret upper bound for the online 178 algorithm used in the abstract game ( e.g. , the bound of Proposition 2 for the EWA forecaster). 179

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4.2 Confidence sequence under mixing sub-Gaussian noise

The standard online-to-confidence sets scheme relies on the fact that ∑ t s =1 ℓ s ( θ ⋆ ) -∑ t s =1 L s ( Q s ) is the logarithm of a non-negative super-martingale, whose fluctuations can be controlled uniformly in time thanks to Ville's inequality. However, this property hinges on the fact that the noise is assumed to be conditionally centred and sub-Gaussian, which now is not anymore the case. Yet, thanks to our mixing assumption, if we restrict our focus on rounds that are sufficiently far apart, the mutual dependencies get weaker, and the exponential of the sum behaves almost like a martingale. This insight suggests to partition the rounds into blocks, whose elements are mutually far apart, then apply concentration results to each block, and finally use a union bound to recover the desired confidence sequence spanning all rounds. We remark that this is a classical approach to derive concentration results for mixing processes, often referred to as the blocking technique (Yu, 1994).

In order for the online-to-confidence sets scheme to leverage the blocking strategy outlined above, the abstract online game used for the analysis must be designed in a way that is compatible with the block structure. To address this point, we adopt an approach inspired by Abélès et al. (2025), who introduced delays in the feedbacks received by the online learner in order to address a similar challenge. More precisely, we will now consider the following delayed-feedback version of the online game. Fix a delay d > 0 . For each round s = 1 , . . . , t , the following steps are repeated:

the environment reveals to the learner X s , which is assumed to be F s -d -measurable;

the learner plays a distribution Q s ∈ ∆ R p ;

if s > d , the environment reveals Y s -d +1 to the learner;

the learner suffers the log loss L s ( Q s ) = -log ∫ R p exp( -ℓ s ( θ ))d Q s ( θ ) .

Note that the delay d only applies for the rewards, while Q s can still depend on X s . Indeed, the choice of X s in our mixing UCB algorithm is already 'delayed', as it depends on C t -d (see Algorithm 1).

Of course, in this setting the decomposition of (3) is still valid. We now want to deal with the concentration of ∑ t s =1 ℓ s ( θ ⋆ ) -∑ t s =1 L s ( Q s ) via the blocking technique. For convenience, let us write D t = ℓ t ( θ ⋆ ) - L t ( Q t ) . We denote as S ( i ) = ( S ( i ) k ) k ≥ 1 the subsequence defined as S ( i ) k = ∑ k j =1 D i +( j -1) d . The key idea is now that each of these S ( i ) behaves as the log of a martingale, up to a cumulative remainder that accounts for the conditional mean shift in the mixing sub-Gaussianity assumption. In particular, Ville's inequality and a union bound yield the following.

Lemma 1. Fix a delay d > 0 and δ ∈ (0 , 1) . We have that

Now that we have a concentration result to control S t , we only need to be able to upper bound the regret of an algorithm for the 'delayed' online game that we are considering. To this purpose, we propose the following approach. We run d independent EWA forecaster (with uniform prior), each one only making prediction and receiving the feedback once every d rounds. More explicitly, the first forecaster acts at rounds 1 , d +1 , 2 d +1 ..., the second at round 2 , d +2 , 2 d +2 ..., and so on. As a direct consequence of Proposition 2, by summing the individual regret upper bounds we get a regret bound for the joint forecaster, which at each round returns the distribution predicted by the currently active forecaster. This technique of partitioning rounds into blocks for the regret analysis of online learning is common in the literature ( e.g. , see Weinberger and Ordentlich, 2002).

Lemma 2. Fix B > 0 , d > 0 , and consider a strategy with d independent EWA forecasters outlined above, all initialised with the uniform distribution on B ( B +1) as prior. For all ¯ θ ∈ B ( B ) and t ≥ 1 ,

Putting together what we have, we get a confidence sequence suitable for our mixing UCB algorithm. Theorem 1. Consider the setting introduced above. Fix δ ∈ (0 , 1) and a delay d > 0 . Assume as known that θ ⋆ ∈ B ( B ) . Let ̂ θ t = arg min θ ∈B ( B ) { ∑ t s =1 ℓ s ( θ ) } and Λ t = ∑ t s =1 X s X ⊤ s . Define

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Then, ( C t ) t ≥ 1 is an anytime valid confidence sequence for θ ⋆ , namely

Proof. The optimality of ̂ θ t implies ∑ t s =1 ⟨ θ -̂ θ t , ∇ ℓ s ( ̂ θ t ) ⟩ ≥ 0 , for all θ ∈ B ( B ) . As ∑ t s =1 ℓ s is quadratic, it equals its second order Taylor expansion around ̂ θ t and its Hessian is everywhere Λ t . So,

for any θ ∈ B ( B ) . This, together with (3), Lemma 1, and Lemma 2, yields the conclusion.

We remark that the confidence sets of Theorem 1 take the form of the intersection between the ball B ( B ) and the 'ellipsoid' { θ : ∥ θ -̂ θ t ∥ Λ t ≤ β t } , for a suitable radius β t . In order to implement and analyse the bandit algorithm, it will be more convenient to work with a relaxation of these sets, a pure ellipsoid not intersected with B ( B ) . We make this explicit in the following corollary.

Corollary 1. Fix λ > 0 , d > 0 , and δ ∈ (0 , 1) . For t ≥ 1 , let V t = Λ t + λ Id . Assuming that θ ⋆ ∈ B ( B ) , the following compact ellipsoids define an anytime valid confidence sequence for θ ⋆ :

Proof. Let β 2 t = dp log ( B +1) 2 e max( dp,t + d ) dp + 2 tϕ d ( B + 1) + 2 d log d δ . From Theorem 1, with probability at least 1 -δ , uniformly for every t , ∥ θ ⋆ -̂ θ t ∥ 2 Λ t ≤ β 2 t . Adding to both sides of this inequality λ 2 ∥ θ ⋆ -̂ θ t ∥ 2 2 , and relaxing the RHS using that ∥ θ ⋆ -̂ θ t ∥ 2 2 ≤ 4 B 2 , we conclude.

5 Regret bounds for Mixing-LinUCB

In this section, we establish worst-case and gap-dependent cumulative regret bounds for mixing UCB algorithm (Mixing Lin-UCB). However, to account for the fact that Mixing-LinUCB selects actions with delays, the standard elliptical potential arguments must be modified. Throughout this section, we let R t = ⟨ θ ⋆ , X ⋆ t -X t ⟩ (where X ⋆ t = arg max x ∈X t ⟨ θ ⋆ , x ⟩ ) denote the regret in round t , and β 2 t = dp log ( B +1) 2 e max( dp,t + d ) dp +4 λB 2 +2 tϕ d ( B +1)+2 d log d δ denote the squared radius of the ellipsoid C t in Corollary 1.

5.1 Worst-case regret bounds

First, following the regret analysis in Abbasi-Yadkori et al. (2011) (see also Section 19.3 in Lattimore and Szepesvári, 2020), we upper bound the instantaneous regret. From our boundedness assumptions ( θ ⋆ ∈ B ( B ) and X t ⊆ B (1) ), we easily deduce that R t ≤ 2 B . Under the event that our confidence sequence contains θ ⋆ at every step t , we have another bound on R t . If we define ˜ θ t -d ∈ C t -d to be the point at which ⟨ ˜ θ t -d , X t ⟩ = UCB C t -d ( X t ) , then from the definition of X t we have

Recall that, for all s , V s = Λ s + λ Id , which is invertible as λ > 0 . Thus, by Cauchy-Schwarz,

This means that the instantaneous regret satisfies the bound 243

Next, we separate the regret suffered in the first d rounds and the remaining T -d rounds. We then 244 use Cauchy-Schwarz once more, and the fact that β t is increasing in t , to obtain 245

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At this point, we must depart from the standard linear UCB analysis (Abbasi-Yadkori et al., 2011; Lattimore and Szepesvári, 2020). We bound the sum of the elliptical potentials ∑ T t = d +1 min(1 , ∥ X t ∥ 2 V -1 t -d ) using the following variant of the well-known 'elliptical potential lemma' (see Appendix), which accounts for the fact that the feature covariance matrix V t -d is updated with a delay of d steps.

Lemma 3. For all T ≥ 1 ,

We can now state a worst-case regret upper bound for Mixing-LinUCB.

Theorem 2. Fix λ = 1 /B 2 , d > 0 and δ ∈ (0 , 1) . With probability at least 1 -δ , for all T > d , the regret of Mixing-LinUCB satisfies

From the definition of β T , we see that this regret bound is of the order

For any fixed ( i.e. , not depending on T ) delay d , this regret bound is linear in T . To obtain meaningful regret bounds, it is therefore crucial to set d as a function of T and the rate at which the mixing coefficients decay to zero 3 . Under the assumption that the noise variables are either geometrically or algebraically mixing, we obtain the following worst-case regret bounds.

Corollary 2. Suppose that the noise satisfies Assumption 1 with ϕ d = Ce -d τ for some C, τ > 0 ( geometric mixing ), and set d = ⌈ τ log BCT p ⌉ . Then, the regret of Mixing-LinUCB satisfies

Corollary 3. Suppose that the noise satisfies Assumption 1 with ϕ d = Cd -r for some C > 0 and r > 0 ( algebraic mixing ), and set d = ⌈ CT 1 / (1+ r ) ⌉ . Then, the regret of Mixing-LinUCB satisfies

Up to a factor of τ log T , the bound for geometrically mixing noise matches the regret bound for linear UCB with i.i.d. noise. This bound is trivial for r ≤ 1 , however for r > 1 we get sublinear regret, and in particular we recover standard rates up to logarithmic factors in the limit where r →∞ .

5.2 Gap-dependent regret bounds

Under the assumption that, each round, the gap between the expected reward of the optimal arm and the expected reward of any other arm is at least ∆ > 0 , we get regret bounds with better dependence

In our worst-case analysis, we showed that 274

Combined with the previous inequality, we obtain the following gap-dependent regret bound.

Theorem 3. Fix λ = 1 /B 2 , d > 0 , and δ ∈ (0 , 1) . With probability at least 1 -δ , for all T > d , the regret of Mixing-LinUCB satisfies

Similarly to the worst-case bound in Theorem 2, for any fixed d > 0 , this regret bound is linear in T . By setting d as a suitable function of T , we obtain the following gap-dependent regret bounds under geometrically or algebraically mixing noise.

Corollary 4. Suppose that the noise variables are geometrically mixing and set d = ⌈ τ log BCT p ⌉ . Then the regret of Mixing-LinUCB satisfies

Corollary 5. Suppose that the noise variables are algebraically mixing and set d = ⌈ CT 1 / (1+ r ) ⌉ . Then the regret of Mixing-LinUCB satisfies

6 Conclusion

We leave several interesting questions open for future research. Some of these are listed below.

An important limitation of our algorithm is that it requires the knowledge of the mixing coefficients (or at least an upper-bound on them). It would be interesting to explore the possibility of relaxing this assumption and to design an algorithm which infers the mixing coefficients while minimizing the regret. We note that the problem of estimating mixing coefficients is already a hard problem on its own right, with tight sample-complexity results only available in special cases such as Markov chains (Hsu et al., 2019; Wolfer, 2020). We also note that in order to recover the standard rate for the regret bound, the delay d introduced in our algorithm need to be chosen as a function of the horizon T . We believe that this could be fixed at little conceptual expense by using time-varying delay in the analysis, but we did not attempt to work out the (potentially non-trivial) details here.

Another limitation is that our analysis assumed throughout that the adversary picking the decision sets X t is oblivious, which is typically not required in linear bandit problems. For us, this was necessary to avoid potential statistical dependence between decision sets and the nonstationary observations. We believe that this issue can be handled at least for some classes of adversaries. For instance, it is easy to see that our analysis would carry through under the assumption that the decision sets be selected based on delayed information only. We leave the investigation of this question under more realistic assumptions open for future work.

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on T . More precisely, define the minimum gap ∆ = min t ∈ [ T ] min x ∈X t : x = X ⋆ t ⟨ X ⋆ t -x, θ ⋆ ⟩ , and 272 assume that ∆ > 0 . Since we either have R t = 0 or R t ≥ ∆ > 0 , it follows that 273

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NeurIPS Paper Checklist

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The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

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Justification: This article is purely theoretical and addresses a mathematical problem which it attempts to solve.

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The answer NA means that the paper poses no such risks.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the linear stochastic bandit problem by relaxing the standard i.i.d. assumption for observation noise, instead allowing for sub-Gaussian noise with dependencies that decay over time. The authors introduce a concept called \"mixing sub-Gaussianity\" to formalize this setting and develop a new UCB-style algorithm, Mixing-LinUCB, which introduces a delay to manage these dependencies. By adapting the online-to-confidence-set conversion scheme with a blocking technique, the paper derives regret bounds for the algorithm that are expressed in terms of the dependency decay rate. A key result is that for geometrically mixing noise, these bounds can recover standard regret rates, with an additional factor related to the mixing time.\n\nSTRENGTHS AND WEAKNESSES:\n- Strengths\n\t- Significant and Well-Motivated Problem: The paper tackles a crucial limitation of most bandit literature. The standard i.i.d. noise assumption is often unrealistic in practical applications such as online advertising or recommendation systems, where external factors can introduce temporal correlations in user feedback. This work provides a principled approach to model and handle such dependencies, significantly increasing the practical relevance of the linear bandit framework.\n\t- Novel Technical Methodology with Strong Theoretical Guarantees: The core strength of this paper lies in its innovative and sound theoretical analysis. The authors successfully adapt the modern online-to-confidence-set conversion framework to handle dependent data by introducing a novel blocking technique within a delayed abstract game. This powerful machinery leads to compelling and rigorous regret bounds that explicitly depend on the noise's mixing rate. Crucially, the analysis demonstrates that for geometrically mixing noise, the algorithm achieves nearly optimal regret, recovering the standard rates up to logarithmic factors and the mixing time.\n\n- Weaknesses\n\t- I find Assumption 1 somewhat unintuitive. It seems more natural to assume that each $\\epsilon_t$ satisfies a delay-dependent variance bound $\\sigma_d$, rather than a magnitude bound $\\phi_d$. Could you elaborate on the motivation behind the current assumption and clarify why it is preferred in your analysis?\n\t- Required Knowledge of Noise Structure: A related and arguably stronger assumption is that selecting the optimal delay $d$ requires a priori knowledge of the mixing coefficients that characterize the noise process (e.g., the decay rate $\\tau$ and constant $C$ for geometric mixing, or $r$ for algebraic mixing). The authors themselves acknowledge this limitation, noting that estimating these coefficients is a hard problem in its own right. This creates a methodological tension: the paper solves the problem of unknown noise correlation by assuming the structure of that correlation is known, which can be an equally restrictive assumption in practice.\n\nQUESTIONS:\n1. Novelty: While the paper addresses the important problem of linear bandits with non-i.i.d. noise, the core framework appears to be a novel and effective synthesis of several existing techniques. Specifically, it combines the online-to-confidence-set conversion scheme , the classic \"blocking technique\" for mixing processes , and the use of \"delayed feedback\" in an abstract online game to accommodate the block structure, which has been explored in other contexts. Could the authors elaborate on the specific technical novelty required to make these components work together for the linear bandit problem?\n\n2. Sensitivity to $d$: The derived regret bounds depend critically on the choice of the delay parameter $d$. Could the authors provide more insight into the algorithm's sensitivity to the choice of d? A simple numerical experiment, even on synthetic data, would be highly valuable to address this.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the classical linear contextual bandits problem, but relaxes the usual assumption that the noise​ is conditionally i.i.d. sub-Gaussian. Instead, the authors proposes a new assumption that the noise sequence to be mixing sub-Gaussian, and shows that this assumption is not stronger than standard mixing assumptions. Under this assumption, the author proposes mixing-LinUCB algorithm and shows regret bounds for it. The author shows that mixing-LinUCB matches the regret bound up to logarithmic factors in the standard setting.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n- The paper is theoretically solid. Most lemmas and theorems have proofs and the author introduces and explains each result in detail. \n- The paper is generally clear. It explains all the concepts well, starting from the definition of linear bandits to the idea of optimism under uncertainty. I think it is decently accessible to general readers who are not expert in bandits. \n\nWeaknesses: \n- There is no experiment. Given the description of the algorithm it does not seem too hard to construct at least a simulation (I could be wrong). Also, it would be good to demonstrate that the algorithm works empirically to further show the validity of it. For example, one could simulate a usual linear bandit instance (e.g., in \"On Elimination Strategies for Bandit Fixed-Confidence Identification\" by Tirinzoni and Degenne) where the noise follows an AR(1) process or some other dependency structures that satisfy the mixing sub-Gaussianity assumption in this paper. One can adjust the dependency level and see how much Mixing-LinUCB outperforms other benchmarks when the noise is very dependent or almost independent. Although I'm not sure what the result would be, seeing big improvements with dependent noise demonstrates that we need new algorithms in this setting instead of just using current ones like LinUCB and LinTS. \n- The paper seems hastily written for some parts. For example, Proposition 1 is almost the same as Theorem 1, so I’m not sure why the author named this a separate proposition. Also, the proof of Proposition 1 is missing, and this proposition actually seems quite critical since the algorithm builds on this. \n- I’m a bit struggled to find the technical novelty in the proof. In particular, I feel like a lot of it is a reproof of LinUCB and EWA forecaster, which both seem present in existing works. The main novelty I found might be the new notion of mixing sub-Gaussianity. I think the authors could make it more clear about their technical contributions especially given that this is a pure theoretical paper.\n\nQUESTIONS:\n- As I mentioned in the above section, I feel like Proposition 1 is almost the same as Theorem 1, so it might be good to rename them the same instead of duplicating. \n- Also as I mentioned, I think it would be good to see at least a simulation study to show that your algorithm empirically is comparable to existing approaches, e.g. LinTS, under i.i.d. noise settings. \n- Related to the above point, I'd love to see if there are any benchmarks for linear bandits under non-i.i.d. noise and how your algorithm compares to them.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces the linear stochastic bandits under non-i.i.d. noise modeled by a mixing process.\nIt loosens standard assumptions used in previous bandit studies, such as conditionally sub-Gaussian noise. \nTo handle this setting, the authors develop new confidence sequences that remain valid over time despite the dependence in the noise. They propose a delay-based decision-making strategy that avoids using the most recent observations. The resulting algorithm achieves regret upper bounds that adapt to the degree of noise dependence.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\nPromising paper! Though I am not deeply familiar with the literature on non-i.i.d. noise in bandits, but from my perspective, this work is well-structured. \n\nS1. The paper tackles the linear bandit problem under dependent (non-i.i.d.) noise. This noble setting more closely reflects real-world scenarios. \n\nS2. They introduce a new confidence set construction tailored for dependent noise, leveraging recent advances in sequential probability assignment. \n\nS3. They provide regret upper bound of their algorithm in terms of the decay rate of mixing coefficient. \n\nS4. Clear writing and intuitive description of their claims.\n\n**Weaknesses**\nOverall, I do not have major concerns. Potential concerns, such as the reliance on prior knowledge of mixing coefficient and the assumption of an oblivious environment.—are already acknowledged by the authors as limitations of the work.\n\nW1. While the theoretical contributions are clear, the lack of empirical validation raises concerns about the practical applicability. Why you exempt the numerical experiments?\n\nTypo\n1. In line 54-55, Abbasi-Yadkori et al. (2011) -> Abbasi-Yadkori et al. (2012)\n2. In line 106, Reg(T) = \\sum_{t=1}^{T} \\sup ...\n3. In Proposition 1. (line 134), $ V_t = \\sum_{s=1}^{t} X_s X_s^\\top + \\lambda I_d$\n4. In Proposition 1. (line 134), $\\sum_{s=1}^{t} ( < \\theta, X_s> - Y_s)^2\n\nAbbasi-Yadkori, Yasin, David Pal, and Csaba Szepesvari. \"Online-to-confidence-set conversions and application to sparse stochastic bandits.\" Artificial Intelligence and Statistics. PMLR, 2012.\n\nQUESTIONS:\nQ1. Out of academic curiosity, I was wondering whether there are alternative ways to model noise dependence beyond mixing processes.\n\nQ2. To handle the dependency of the noise, the proposed method deliberately discards recent observations. Then, could applying this method to real-world scenarios lead to any unexpected difficulties?\n\nQ3. Is it possible to establish a lower bound in this setting, or at least gain some intuition about how it might behave?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper revisits the classical stochastic linear bandit problem under weaker assumptions on the exogenous noise process than studied in prior work. In particular, prior work typically assumes the additive noise process in the observation model to be i.i.d. sub-Gaussian. In contrast, this work allows the noise process to be temporally correlated. To model correlations, the authors introduce the notion of a “mixing sub-Guassian” condition. Under this condition, confidence sets are derived for the unknown parameter $\\theta$. Relative to the standard confidence set width, the confidence sets under correlated noise are inflated by a delay term that captures the range of dependencies. Once the confidence sets are derived, the rest of the algorithm follows the standard principle of optimism in the face of uncertainty. \nThe authors derive regret bounds under different types of assumptions on mixing (geometric mixing and polynomial mixing).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n-\tThis appears to be the first work to provide an analysis of linear bandit algorithms under correlated noise. \n-\tThe strategy of using the “online-to-confidence” set conversion to construct the confidence sets is interesting. \n-\tThe derived regret bounds are consistent with what one would intuitively expect. For instance, under geometric mixing (i.e., when the temporal dependencies are weak), the regret bounds are only a factor of the mixing time away from the corresponding bounds under i.i.d. noise. \nWeaknesses and Comments: \n-\tMy main comment and concern is that while the aspect of considering correlated noise might be relatively new for the bandit problem, this issue has been extensively studied in the context of stochastic approximation algorithms for RL. See, for instance, references [R1] and [R2] below that employ the so-called blocking technique (similar to this paper) to deal with correlations.  \n\nAt a high level, samples that are sufficiently spaced out over time are weakly correlated. As such, one can simply sub-sample the data-stream to first generate a sequence of nearly independent observations and then play the standard algorithm on this sub-sampled sequence. Doing so would recover the same guarantees as in the i.i.d. setting, up to a factor of ‘d’, where ‘d’ is the block length/delay/gap between samples that scales with the mixing time (up to log factors). This basic idea is formalized via a coupling argument in references [R1] and [R2] below. Given that this idea is fairly well-known (both for stochastic approximation and for supervised learning), I am not entirely convinced by the significance of the contribution (despite the differences in the elliptical potential argument to account for delays). \n\n[R1] Adapting to mixing time in stochastic optimization with Markovian data, Dorfman and Levy, ICML 2022. \n\n[R2] Finite time analysis of temporal difference learning with linear function approximation: Tail averaging and regularization, Patil et al., AISTATS 2023. \n\n-\tThe paper only provides upper bounds, and as such, the tightness of the derived bounds as a function of the rate of mixing remains unclear. In this context, I would encourage the authors to take a look at [R3] below to see if similar ideas can be employed to establish lower bounds for the bandit setting under correlated noise. \n\n[R3] Least squares regression with markovian data: Fundamental limits and algorithms, Nagaraj et al., NeuRIPS 2020. \n\n-\tIt would be good if the authors can provide a more detailed discussion of how the radius of the confidence set under correlated noise compares with that under i.i.d. noise?\n\nQUESTIONS:\nMy main question/comment, as discussed above, is that the result in this paper, while novel, seems to be a rather straight\tforward application of the blocking technique that has appeared before in stochastic approximation algorithms for RL.\n\nLIMITATIONS:\nYES", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}