Title: An Adaptive Approach for Infinitely Many-armed Bandits under Generalized Rotting Constraints
Venue: NeurIPS 2024
Accepted: yes
Keywords: bandits

rotting rewards

infinitely many arms
Abstract: In this study, we consider the infinitely many-armed bandit problems in a rested rotting setting, where the mean reward of an arm may decrease with each pull, while otherwise, it remains unchanged. We explore two scenarios regarding the rotting of rewards: one in which the cumulative amount of rotting is bounded by $V_T$, referred to as the slow-rotting case, and the other in which the cumulative number of rotting instances is bounded by $S_T$, referred to as the abrupt-rotting case. To address the challenge posed by rotting rewards, we introduce an algorithm that utilizes UCB with an adaptive sliding window, designed to manage the bias and variance trade-off arising due to rotting rewards. Our proposed algorithm achieves tight regret bounds for both slow and abrupt rotting scenarios. Lastly, we demonstrate the performance of our algorithm using numerical experiments.

An Adaptive Approach for Infinitely Many-armed Bandits under Generalized Rotting Constraints

Abstract

In this study, we consider the infinitely many-armed bandit problems in a rested rotting setting, where the mean reward of an arm may decrease with each pull, while otherwise, it remains unchanged. We explore two scenarios regarding the rotting of rewards: one in which the cumulative amount of rotting is bounded by V T , referred to as the slow-rotting case, and the other in which the cumulative number of rotting instances is bounded by S T , referred to as the abrupt-rotting case. To address the challenge posed by rotting rewards, we introduce an algorithm that utilizes UCB with an adaptive sliding window, designed to manage the bias and variance trade-off arising due to rotting rewards. Our proposed algorithm achieves tight regret bounds for both slow and abrupt rotting scenarios. Lastly, we demonstrate the performance of our algorithm using numerical experiments.

1 Introduction

We consider multi-armed bandit problems [15], which are fundamental sequential learning problems where an agent plays an arm at each time and receives a corresponding reward. The core challenge lies in balancing the exploration-exploitation trade-off. Bandit problems have significant implications across diverse real-world applications, such as recommendation systems [17] and clinical trials [23]. In a recommendation system, each arm could represent an item, and the objective is to maximize the click-through rate by making effective recommendations.

In practice, the mean rewards associated with arms may decrease over repeated plays. For instance, in content recommendation systems, the click rates for each item (arm) may diminish due to user boredom with repeated exposure to the same content. Another example is evident in clinical trials, where the efficacy of a medication can decline over time due to drug tolerance induced by repeated administration. The decline in mean rewards resulting from playing arms, referred to as (rested) rotting bandits , has been studied by Levine et al. [16], Seznec et al. [20, 21]. The previous work focuses on finite K arms, in which Seznec et al. [20] proposed algorithms achieving ˜ O ( √ KT ) regret. This suggests that rotting bandits with a finite number of arms are no harder than the stationary case.

However, in real-world scenarios like recommendation systems, where the content items such as movies or articles are numerous, prior methods encounter limitations as the parameter K becomes large, resulting in trivial regret. This emphasizes the necessity of studying rotting scenarios with infinitely many arms, particularly when there is a lack of information about the features of each item. The consideration of infinitely many arms for rested rotting bandits fundamentally distinguishes these problems from those with a finite number of arms, as we will explain later.

The study of multi-armed bandit problems with an infinite number of arms has been extensively conducted in the context of stationary rewards [6, 24, 8, 10, 5], where the agent has no chance to play all the arms at least once until horizon time T . Initially, the distribution of the mean rewards for

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the arms was assumed to be uniform over the interval [0 , 1] [6, 8]. This assumption was expanded to include a much wider range of distributions satisfying P ( µ ( a ) > µ ∗ -x ) = Θ( x β ) , for a parameter β > 0 , where µ ( a ) represents the mean reward of arm a and µ ∗ is the mean reward of the best-performing arm [24, 10, 5]. Additionally, feature information for each arm is not required for multi-armed bandit problems with infinitely many arms, which differs from linear bandits [1] or continuum-armed bandits [3, 14], where feature information for each arm, either for the Lipschitz or linear structure, is involved. While Kim et al. [13], as the closest work, explores the concept of diminishing rewards in the context of bandits with infinitely many arms, their focus is restricted to the case of the maximum rotting rate constraint, where the amount of rotting at each time step is bounded by ρ (= o (1)) . This naturally directs focus towards regret regarding the maximum rotting rate rather than the total rotting rate over the time horizon. Furthermore, their focus is limited to the case where the initial mean rewards are uniformly distributed ( β = 1 ).

In our study, we explore rotting bandits with infinitely many arms, subject to generalized initial mean reward distribution with β > 0 and, importantly, generalized constraints on the rate at which the mean reward of an arm declines. Our investigation into diminishing, or 'rotting,' rewards encompasses two scenarios: one with the total amount of rotting bounded by V T , and the other with the total number of rotting instances bounded by S T . This allows us to capture characteristics of entire rotting rates over the time horizon. Similar constraints of V T or S T regarding nonstationarity have been explored in the context of nonstationary finite K -armed bandit problems [7, 4, 19], where the reward distribution changes over time independently of the agent. Following established terminology for nonstationary bandits, we denote the environment with a bounded total amount of rotting as the slow rotting ( V T ) case and the one with a bounded total number of rotting instances as the abrupt rotting ( S T ) case.

Here we discuss why (rested) rotting bandits for infinitely many arms are fundamentally different from those for finite arms. In the case of finite arms, rested rotting is known to be no harder than stationary case [20, 21]. This result arises from the confinement of mean rewards of optimal arms and played arms within confidence bounds, even under rested rotting (as demonstrated in Lemma 1 of Seznec et al. [20, 21]). However, in the case of infinite arms under distribution for initial mean reward that allows for an infinite number of near-optimal arms, there always exist near-optimal arms outside of explored arms. Therefore, the mean reward gap may not be confined within confidence bounds. This fundamental difference from finite-armed rotting bandits introduces additional challenges. In our setting of infinite arms, there exists an additional cost for exploring new (unexplored) arms to find near-optimal arms while eliminating explored suboptimal arms. If the total rotting effect on explored arms is significant, then the frequency at which new near-optimal arms must be sought increases substantially, resulting in a large regret. This is why the rested rotting significantly affects the exploration cost regarding V T or S T in our setting, which differs from the case of finite arms.

To solve our problem, we introduce algorithms that employ an adaptive sliding window mechanism, effectively managing the tradeoff between bias and variance stemming from rotting rewards. Notably, to the best of our knowledge, this is the first work to consider slow and abrupt rotting scenarios, in the context of infinitely many-armed bandits. Furthermore, it is the first work to consider the generalized initial mean reward distribution for rotting bandits with infinitely many arms.

Summary of our Contributions. The key contributions of this study are summarized in the following points. Please refer to Table 1 for a summary of our regret bounds.

· To address the slow and abrupt rotting scenarios, we propose a UCB-based algorithm using an adaptive sliding window and a threshold parameter. This algorithm allows for effectively managing the bias and variance trade-off arising from rotting rewards.

In the context of slow rotting ( V T ) or abrupt rotting ( S T ), for any β > 0 , we present regret upper bounds achieved by our algorithm with an appropriately tuned threshold parameter. It is noteworthy that V T , S T , and β are being considered for the first time in the context of rotting bandits with infinitely many arms.

We establish regret lower bounds for both slow rotting and abrupt rotting scenarios. These regret lower bounds imply the tightness of our upper bounds when β ≥ 1 . In the other case, when 0 < β < 1 , there is a gap between our upper bounds and the corresponding lower bounds, similar to what can be found in related literature, which is discussed in the paper.

Lastly, we demonstrate the performance of our algorithm through numerical experiments on synthetic datasets, validating our theoretical results.

2 Problem Statement

We consider rotting bandits with infinitely many arms where the mean reward of an arm may decrease when the agent pulls the arm. Let A be the set of infinitely many arms and let µ t ( a ) denote the unknown mean reward of arm a ∈ A at time t . At each time t , an agent pulls arm a π t ∈ A according to policy π and observes stochastic reward r t given by r t = µ t ( a π t ) + η t , where η t is a noise term following a 1 -sub-Gaussian distribution. To simplify, we use a t for a π t when there is no confusion about the policy. We assume that initial mean rewards { µ 1 ( a ) } a ∈A are i.i.d. random variables on [0 , 1] , a widely accepted assumption in the context of infinitely many-armed bandits [8, 6, 24, 10, 5, 13].

As in Wang et al. [24], Carpentier and Valko [10], Bayati et al. [5], we consider, to our best knowledge, the most general condition on the distribution of the initial mean reward of an arm, satisfying the following condition: there exists a constant β > 0 such that for every a ∈ A and all x ∈ [0 , 1] ,

where ∆ 1 ( a ) = 1 -µ 1 ( a ) is the initial sub-optimality gap. As noted in [24, 10, 5], Eq.(1) is a non-trivial condition only when x approaches 0 , as for any constant x ∈ (0 , 1] , it becomes P (∆ 1 ( a ) < x ) = Θ(1) , which may accommodate a wide range of distributions. It is noteworthy that the larger the value of β , the smaller the probability of sampling a good arm. Furthermore, the uniform distribution is a special case when β = 1 . Importantly, our work allows for a wider range of distributions satisfying (1) for any constant β > 0 than the uniform distribution ( β = 1) considered in Kim et al. [13]. Additional discussion is deferred to Appendix A.2.

The rotting of arms is defined as follows. At each time t ≥ 1 , the mean rewards of arms are updated as

where ρ t ( a t ) ≥ 0 for the pulled arm a t and ρ t ( a ) = 0 for every a ∈ A / { a t } , which implies that the rotting may occur only for the pulled arm at each time. Note that, for every a ∈ A and t ≥ 2 , it holds µ t ( a ) = µ 1 ( a ) -∑ t -1 s =1 ρ s ( a ) , allowing µ t ( a ) to take negative values. For notation simplicity, in what follows, we write ρ t for ρ t ( a t ) when there is no confusion. We refer to ρ 1 , ρ 2 , . . . as rotting rates. We also use the notation [ m ] := { 1 , . . . , m } , for any integer m ≥ 1 .

̸

Assumption 2.1 (Adaptive Adversary) . At each time t ∈ [ T ] , the value of the rotting rate ρ t ≥ 0 is arbitrarily determined immediately after the agent pulls a t , subject to the constraint of either slow rotting for a given V T or abrupt rotting for a given S T .

We consider two cases for rotting rates: (a) slow rotting case where, for given V T ≥ 0 , the cumulative amount of rotting is required to satisfy the slow rotting constraint ∑ T -1 t =1 ρ t ≤ V T , and (b) abrupt rotting case where, for given S T ∈ [ T ] , the cumulative number of rotting instances (plus one) is required to satisfy the abrupt rotting constraint 1 + ∑ T -1 t =1 ✶ ( ρ t = 0) ≤ S T . The values of rotting rates of pulled arms, { ρ t } t ∈ [ T -1] , are assumed to be determined by an adversary, described as follows.

Remark 2.2 . The adaptive adversary under the slow rotting constraint ( V T ) is more general than that in Kim et al. [13], in which the adversary is under a maximum rotting rate constraint ; that is, for given ρ = o (1) , ρ t ≤ ρ for all t ∈ [ T -1] . This is because our adversary is under a weaker constraint bounding the total sum of the rotting rates rather than each individual rotting rate. Additionally, the abrupt rotting constraint ( S T ) is fundamentally different from the maximum rotting constraint [13] because the adversary for abrupt rotting is under a constraint on the total number of rotting instances rather than the magnitude of rotting rates.

Our problem's objective is to find a policy that minimizes the expected cumulative regret over a time horizon of T time steps. For a given policy π , the regret is defined as E [ R π ( T )] = E [ ∑ T t =1 (1 -µ t ( a π t ))] . The use of 1 in the regret definition for the optimal mean reward is justified because among the infinite arms with initial mean rewards following the distribution specified in (1), there always exists an arm whose mean reward is sufficiently close to 1 . 1

We note that while we have S T ≤ T because the number of rotting instances is at most T -1 , the upper bound for V T may not exist due to the lack of a constraint on the values of ρ t 's. Here we discuss an assumption for the cumulative amount of rotting. In the case of ∑ T -1 t =1 ρ t > T , the problem becomes trivial as shown in the following proposition.

Proposition 2.3. In the case of ∑ T -1 t =1 ρ t > T , there always exists a rotting adversary that incurs regret of Ω( T ) and a simple policy that samples a new arm every round achieves the optimal regret of Θ( T ) .

Proof. The proof is provided in Appendix A.3

From the above proposition, when ∑ T -1 t =1 ρ t > T , the regret lower bound of this problem is Ω( T ) , which can be achieved by a simple policy. Therefore, we consider the following assumption for the region of non-trivial problems.

Assumption 2.4. ∑ T -1 t =1 ρ t ≤ T .

Notably, from the above assumption, we consider V T ≤ T for the slow rotting case. We also note that the assumption is not strong, as it frequently arises in real-world scenarios and is more general than the assumption made in prior work, as described in the following remarks.

Remark 2.5 . The assumption of ∑ T -1 t =1 ρ t ≤ T is satisfied if mean rewards are under the constraint of 0 ≤ µ t ( a t ) ≤ 1 for all t ∈ [ T ] , because this condition implies ρ t ≤ 1 for all t ∈ [ T ] . Such a scenario is frequently encountered in real-world applications, where reward is represented by metrics like click rates or (normalized) ratings in content recommendation systems.

Remark 2.6 . Our rotting scenario with ∑ T -1 t =1 ρ t ≤ T is more general in scope than the one with the maximum rotting rate constraint where ρ t ≤ ρ = o (1) for all t ∈ [ T -1] , which was explored in Kim et al. [13]. This is because for our setting, ρ t is not necessarily bounded by o (1) , and under the maximum rotting constraint, the condition ∑ T -1 t =1 ρ t ≤ T is always satisfied.

3 Algorithms and Regret Analysis

We propose an algorithm (Algorithm 1) utilizing an adaptive sliding window for delicately controlling bias and variance tradeoff of the mean reward estimator from rotting rewards, drawing on insights from [4, 21]. This is why our algorithm can adapt to varying rotting rates ρ t and achieve tight regret bounds with respect to V T or even S T . Furthermore, our algorithm accommodates the general mean reward distribution with β > 0 by employing a carefully optimized threshold parameter.

Here we describe our proposed algorithm in detail. We define ̂ µ [ t 1 ,t 2 ] ( a ) = ∑ t 2 t = t 1 r t ✶ ( a t = a ) /n [ t 1 ,t 2 ] ( a ) where n [ t 1 ,t 2 ] ( a ) = ∑ t 2 t = t 1 ✶ ( a t = a ) for t 1 ≤ t 2 . Then for window-UCB index of the algorithm, we define WUCB ( a, t 1 , t 2 , T ) = ̂ µ [ t 1 ,t 2 ] ( a ) + √ 12 log( T ) /n [ t 1 ,t 2 ] ( a ) . In what follows, 'selecting an arm' means that a policy chooses an arm before pulling it. In Algorithm 1, we first select an arbitrary new arm a ∈ A ′ without prior knowledge regarding the arms in A ′ , denoting the corresponding time as t ( a ) . We define T t ( a ) as the set of starting times for sliding windows of doubling lengths, defined as T t ( a ) = { s ∈ [ T ] : t ( a ) ≤ s ≤ t -1 and s = t -2 i -1 for some i ∈ N } . Then the algorithm pulls the arm consecutively until the following threshold condition is satisfied: min s ∈T t ( a ) WUCB ( a, s, t -1 , T ) < 1 -δ, in which the sliding window having minimized windowUCB is utilized for adapting nonstationarity. If the threshold condition holds, then the algorithm considers the arm to be a sub-optimal (bad) arm and withdraws the arm. Then it selects a new arm and repeats this procedure.

Algorithm 1 UCB-Threshold with Adaptive Sliding Window

Meanreward(arm a)

Small

Large

Firsttimearma

μt(a)(a)

armpulls

windowcase

windowcase

is pulled

WUCB

t(a)

t-1

1 -8

ttingwith

WUCB

Time step

μt-1(a)

Y

Arm a

Largebias

Largevariance

Smallvariance

Slidingwindowcandidates

Utilizing the adaptive sliding window having minimized window UCB index enhances the algorithm's ability to dynamically identify poorly-performing arms across varying rotting rates. This adaptability is achieved by managing the tradeoff between bias and variance. The concept is depicted in Figure 1 (left), where an arm a undergoes multiple rotting events. WUCB with a smaller window exhibits minimal bias with the arm's most recent mean reward but introduces higher variance. Conversely, WUCB with a larger window displays increased bias but reduced variance. In this visual representation, the value of WUCB with a small window reaches a minimum, enabling the algorithm to compare this value with 1 -δ to identify the suboptimal arm. Moreover, as illustrated in Figure 1 (right), by taking into account the constraint of s = t -2 i -1 for the size of the adaptive windows, we can reduce the computation time for determining the appropriate window and reduce the required memory from O ( t ) to O (log t ) , respectively, for each time t .

Having introduced our algorithm, we compare it with the previously proposed algorithm UCB-TP [13], which is tailored for the maximum rotting rate constraint ρ t ≤ ρ (= o (1)) for all t > 0 and the uniform initial mean reward distribution ( β = 1 ). The mean reward estimator in UCB-TP considers the worst-case scenario with the maximum rotting rate ρ as ˜ µ o t ( a ) -ρn t ( a ) where ˜ µ o t is an estimator for the initial mean reward, n t ( a ) is the number of times arm a is pulled until time t -1 , and ρn t ( a ) is for reducing the bias from the worst-case rotting, which leads to achieving a regret bound of ˜ O (max { ρ 1 / 3 T, √ T } ) . This estimator is not appropriate for dealing with our generalized rotting constraints because it aims to attain the regret bound regarding the maximum rotting rate ρ without adequately addressing individual ρ t values. Our algorithm resolves this by using an adaptive sliding window estimator, which can handle rotting rates carefully. Furthermore, it can accommodate any constant β > 0 by using a carefully optimized δ , as shown below.

Slow Rotting ( V T ). Here we consider the case of slow rotting, where, recall, the adaptive adversary is constrained such that the total amount of rotting is bounded by V T . We analyze the regret of Algorithm 1 with tuned δ using β and V T . Wedefine δ V ( β ) = c 1 max { ( V T /T ) 1 / ( β +2) , 1 /T 1 / ( β +1) } when β ≥ 1 and δ V ( β ) = c 1 max { ( V T /T ) 1 / 3 , 1 / √ T } when 0 < β < 1 for some constant 0 < c 1 < 1 . The algorithm with δ V ( β ) achieves a regret bound in the following theorem.

Theorem 3.1. The policy π of Algorithm 1 with δ = δ V ( β ) achieves:

We observe that when β increases above 1 , the regret bound becomes worse because the likelihood of sampling a good arm decreases. However, when β decreases below 1 , the regret bound remains the same due to the inability to avoid a certain level of regret arising from estimating the mean reward. Further discussion will be provided later. Also, we observe that when V T = O (max { 1 /T 1 / ( β +1) , 1 / √ T } ) where the problem becomes near-stationary, the regret bound in Theorem 3.1 matches the previously known regret bound for stationary infinitely many-armed bandits, ˜ O (max { T β/ ( β +1) , √ T } ) , as shown in Wang et al. [24], Bayati et al. [5].

Proof sketch. The full proof is provided in Appendix A.4. Here we outline the main ideas of the proof. There are several technical challenges involved in regret analysis, such as dealing with varying ρ t individually with respect to the total rotting budget of V T , adaptive estimation in our algorithm, and the generalized distributions of initial mean rewards of arms with parameter β > 0 , none of which appear in Kim et al. [13].

We separate the regret into two components: one associated with pulling initially good arms and another with pulling initially bad arms. An arm a is said to be good if µ 1 ( a ) ≥ 1 -2 δ and, otherwise, it is said to be bad. The reason why the separation is required is that our adaptive algorithm has different behaviors depending on the category of arms. Good arms may be pulled repeatedly when rotting rates are sufficiently small but bad arms are not. We write R π ( T ) = R G ( T ) + R B ( T ) , where R G ( T ) is the regret from good arms and R B ( T ) is the regret from bad arms.

We first provide a bound for E [ R G ( T )] . For analyzing regret from good arms, we analyze the cumulative amount of rotting while pulling a selected good arm before withdrawing the arm by the algorithm. Let A G T be a set of distinct good arms selected until T , t 1 ( a ) be the initial time step at which arm a is pulled, and t 2 ( a ) be the final time step at which the arm is pulled by the algorithm so that the threshold condition holds when t = t 2 ( a ) + 1 . For simplicity, we use t 1 and t 2 for t 1 ( a ) and t 2 ( a ) , when there is no confusion. For any time steps n ≤ m , we define V [ n,m ] ( a ) = ∑ m t = n ρ t ( a ) and ρ [ n,m ] ( a ) = V [ n,m ] ( a ) /n [ n,m ] ( a ) . We show that the regret is decomposed as

which consists of regret from the initial mean reward and the cumulative amount of rotting for each arm. For the first term of ∑ a ∈A G T ∆ 1 ( a ) n [ t 1 ,t 2 ] ( a ) in (2), since ∆ 1 ( a ) = O ( δ ) from the definition of good arms a ∈ A G T , we have E [ ∑ a ∈A G T ∆ 1 ( a ) n [ t 1 ,t 2 ] ( a )] = O ( δT ) .

The main difficulty in (2) lies in dealing with the second term, ∑ a ∈A G T ∑ t 2 t = t 1 +1 V [ t 1 ,t -1] ( a ) , where we need to analyze the amount of cumulative rotting until the arm is eliminated by using the adaptive threshold condition. A careful analysis of the adaptive threshold policy is required to limit the total variation of rotting. By examining the estimation errors arising from variance and bias due to the adaptive threshold condition, we can establish an upper bound for the cumulative amount of rotting as

Therefore, from δ = δ V ( β ) , V T ≤ T , and Eqs. (2) and (3), using Hölder's inequality, we have

Next, we provide a bound for E [ R B ( T )] . We employ episodic regret analysis, defining an episode as the time steps between consecutively selected distinct good arms by the algorithm. By analyzing

bad arms within each episode, we can derive an upper bound for the overall regret arising from bad arms. We define the regret from bad arms over m G episodes as R B m G . We first consider the case of V T > max { 1 / √ T, 1 /T 1 / ( β +1) } . In this case, by setting m G = ⌈ 2 V T /δ ⌉ , we can show that R B ( T ) ≤ R B m G with a high probability. By analyzing R B m G with the episodic analysis, we can show that E [ R B ( T )] ≤ E [ R B m G ] = ˜ O (max { T β +1 β +2 V 1 β +2 T , T 2 3 V 1 3 T } ) . As in the similar manner, when V T ≤ max { 1 / √ T, 1 /T 1 / ( β +1) } , by setting m G = C 3 for some constant C 3 > 0 , we can show that E [ R B ( T )] ≤ E [ R B m G ] = ˜ O (max { T β β +1 , √ T } ) . From the above two inequalities, we have

Finally, from (4) and (5), we can conclude the proof from E [ R π ( T )] = E [ R G ( T )] + E [ R B ( T )] .

Remark 3.2 . We compare our result in Theorem 3.1 with that in Kim et al. [13], which, recall, is under the maximum rotting rate constraint ρ t ≤ ρ = o (1) for all t and uniform distribution of initial mean rewards ( β = 1 ). For a fair comparison, we consider an oblivious adversary for rotting rates where the values of ρ t 's are determined before an algorithm is run, which may imply V T = ∑ T -1 t =1 ρ t and ρ = max t ∈ [ T -1] ρ t . Then with β = 1 , from V T ≤ Tρ , we can observe that the regret bound of Algorithm 1 is tighter than that of UCB-TP [13] as ˜ O (max { V 1 3 T T 2 3 , √ T } ) ≤ ˜ O (max { ρ 1 3 T, √ T } )

, where the latter is the regret bound of UCB-TP . We will demonstrate this in our numerical results.

Abrupt Rotting ( S T ). Here we consider abruptly rotting reward distribution under the constraint of S T . We consider Algorithm 1 with δ newly tuned by S T and β . We define δ S ( β ) = c 1 ( S T /T ) 1 / ( β +1) when β ≥ 1 and δ S ( β ) = c 1 ( S T /T ) 1 / 2 when 0 < β ≤ 1 for some constant 0 < c 1 < 1 . We also define ¯ V T = ∑ T -1 t =1 E [ ρ t ] . In the following theorem, we present a regret upper bound for Algorithm 1 with δ S ( β ) .

Theorem 3.3. The policy π of Algorithm 1 with δ = δ S ( β ) achieves:

As in the slow rotting case, for the abrupt rotting case ( S T ), we observe that when β increases above 1 , the regret bound in the above theorem worsens as the likelihood of sampling a good arm decreases. When β decreases below 1 , the regret bound remains the same because we cannot avoid a certain level of regret arising from estimating the mean reward of an arm. Additionally, we observe that the regret bound is linearly bounded by ¯ V T , which is attributed to the algorithm's necessity to pull a rotted arm at least once to determine its status as bad. Later, in the analysis of regret lower bounds, we will establish the impossibility of avoiding ¯ V T regret in the worst-case. Notably, in the typical cases where 0 ≤ ρ t ≤ 1 for all t > 0 , as discussed in Remark 2.5, ¯ V T is negligible in the regret bound from ¯ V T ≤ S T ≤ T . Furthermore, we observe that for the case of S T = 1 , where the problem becomes stationary (implying ¯ V T = 0 ), the regret bound matches the previously known regret bound of ˜ O (max { T β/ ( β +1) , √ T } ) for the stationary infinitely many-armed bandits [24, 5].

Proof sketch. The full proof is provided in Appendix A.5. Here we provide a proof outline. We follow the proof framework of Theorem 3.1 but the main difference lies in carefully dealing with substantially rotted arms. For the ease of presentation, we consider each arm that experiences abrupt rotting as if it were newly selected by the algorithm, treating the arm before and after abrupt rotting as distinct arms. The definition of a good arm and a bad arm is based on the mean reward at the time when it is newly selected. Then we divide the regret into regret from good and bad arms as R π ( T ) = R G ( T ) + R B ( T ) . From the definition of good arms, we can easily show that

For dealing with R B ( T ) , we partition the regret into two scenarios: one where the bad arm is initially bad sampled from the distribution of (1) and another where it becomes bad after rotting. This can be

Abrupt rotting event

t

Timestep

1

Appropriate slidingwindow

expressed as R B ( T ) = R B , 1 ( T ) + R B , 2 ( T ) . Then for the former regret, R B , 1 ( T ) , as in the proof of Theorem 3.1, by using the episodic analysis with m G = S T , we can show that

For the regret from rotted bad arms, R B , 2 ( T ) , it is critical to analyze significant rotting instances to obtain a tight bound with respect to S T , a factor not addressed in the regret analysis of slow rotting ( V T ) in Theorem 3.1. We analyze that when there exists significant rotting, then the algorithm can efficiently detect it as a bad arm and eliminate it by pulling it at once. From this analysis, we have

Putting all the results together with E [ R π ( T )] = E [ R G ( T )]+ E [ R B , 1 ( T )]+ E [ R B , 2 ( T )] and S T ≤ T , we can conclude the proof.

Remarkably, our proposed method, utilizing an adaptive sliding window, yields a tight bound (lower bounds will be presented later) not only for slow rotting but also for abrupt rotting ( S T ) scenarios characterized by a limited number of rotting instances. The rationale behind the effectiveness of the adaptive sliding window in controlling the bias and variance tradeoff with respect to abrupt rotting is as follows. It can be observed that the adaptive threshold condition of min s ∈T t ( a ) WUCB ( a, s, t -1 , T ) < 1 -δ is equivalent to the condition of WUCB ( a, s, t -1 , T ) < 1 -δ for some s such that t 1 ( a ) ≤ s ≤ t -1 (ignoring the computational reduction trick). The latter expression represents the threshold condition tested for every time step before t , encompassing the time step immediately following an abrupt rotting event. Consequently, as illustrated in Figure 2, this adaptive threshold condition can identify substantially rotted arms by mitigating bias and variance using the window starting from the time step following the occurrence of rotting.

Slow rotting ( V T ) and abrupt rotting ( S T ). In what follows, we study the case of rotting under both slow rotting and abrupt rotting constraints. In this case, Algorithm 1, with δ = min { δ V ( β ) , δ S ( β ) } , can achieve a tighter regret bound as noted in the following corollary, which can be obtained from Theorems 3.1 and 3.3.

Corollary 3.4. Let R V and R S be defined as

The policy π of Algorithm 1 with δ = min { δ V ( β ) , δ S ( β ) } achieves the regret bound of E [ R π ( T )] = ˜ O (min { R V , R S } ) .

Case without Prior Knowledge of V T , S T , and β . Here we study the case when the algorithm does not have prior information about the values of V T , S T , and β under the constraints of V T and S T . These parameters play a crucial role in determining the optimal threshold parameter δ in Algorithm 1. We propose an algorithm based on estimating the optimal threshold parameter δ directly (Algorithm 2), rather than estimating each unknown parameter separately, employing the Bandit-overBandit ( BoB ) approach [11]. Under assumptions concerning the bounds for the cumulative amount of

rotting and a constrained version of the adaptive adversary for rotting rates, which are less general than Assumptions 2.1 and 2.4 but still more general than those in Kim et al. [13], the algorithm achieves a regret bound of E [ R π ( T )] = ˜ O (min { R V , R S } + max { T (2 β +1) / (2 β +2) , T 3 / 4 } ) . The additional cost arises from learning δ compared to the regret bound of Corollary 3.4. Further details of the algorithm and regret analysis are provided in Appendix A.6.

4 Regret Lower Bounds

In this section, we present regret lower bounds for our problem under Assumptions 2.1 and 2.4 to provide guidance on the tightness of our regret upper bounds. For the regret lower bounds, we consider worst-case instances of rotting rates. In the following theorems, we provide regret lower bounds for slow rotting ( V T ) and abrupt rotting ( S T ), respectively.

Theorem 4.1. For the slow rotting case with the constraint V T and β > 0 , for any policy π , there always exists a rotting rate adversary such that the regret of π satisfies

Proof.

Theorem 4.2. For the abrupt rotting case with the constraint S T and β > 0 , for any policy π , there always exists a rotting rate adversary such that the regret of π satisfies

Proof.

For the abrupt rotting ( S T ) case, it is unavoidable to incur a Ω( ¯ V T ) regret because an arm may only be rotted once and any algorithm pulls this rotted arm at least once in the worst case. From Table 1, we can observe that Algorithm 1 achieves near-optimal regret when β ≥ 1 . The optimality proven only for β ≥ 1 has also been observed for stationary infinitely many-armed bandits [5, 24]. We believe that our regret upper bounds are near-optimal across the entire range of β . Achieving tighter regret lower bounds when β < 1 is left for future research; see Appendix A.1 for further discussion.

5 Experiments

0

1

2

3

4

5

T

1e6

0

1

2

3

E[

R

(

T

)]

1e6

SSUCB

UCB-TP

Algorithm 2

Algorithm 1

In this section, we present numerical results validating some claims of our theoretical analysis. 2 We use randomly generated datasets under a uniform distribution for initial mean rewards ( β = 1 ).

We first compare the performance of our Algorithms 1 and 2 with UCB-TP [13], the state-of-the-art algorithm for the rotting setting, and SSUCB [5], a near-optimal algorithm for stationary infinitely

many-armed bandits. For comparison with UCB-TP , recall our discussion in Remark 3.2. We set the rotting rates such that ρ t = 1 / ( t log( T )) for all t , for which ρ = ρ 1 = 1 / log( T ) = o (1) , V T = O (1) , and S T = T . In Figure 3, we can observe that Algorithms 1 and 2 perform better than UCB-TP and SSUCB (and Algorithm 1 outperforms Algorithm 2), which is in agreement with our theoretical analysis for the case β = 1 . In this case, the regret bounds for Algorithms 1 and 2 are ˜ O ( T 2 / 3 ) and ˜ O ( T 3 / 4 ) from Corollary 3.4 and Theorem A.15, respectively, which are tighter than the regret bound of ˜ O ( T/ log( T ) 1 / 3 ) for UCB-TP . Additional experiments can be found in Appendix A.10.

6 Conclusion

We explore the challenges of infinitely many-armed bandit problems with rotting rewards, focusing on slow rotting ( V T ) and abrupt rotting ( S T ) scenarios. To address these challenges, we propose an algorithm incorporating an adaptive sliding window, which achieves tight regret bounds for both cases. We also provide regret lower bounds for both slow rotting and abrupt rotting cases. Lastly, we demonstrate our algorithm using synthetic datasets.

References

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Peter Auer, Nicoló Cesa-Bianchi, Yoav Freund, and Robert E. Schapire. The nonstochastic multiarmed bandit problem. SIAM Journal on Computing , 32(1):48-77, 2002.

Peter Auer, Ronald Ortner, and Csaba Szepesvári. Improved rates for the stochastic continuumarmed bandit problem. In International Conference on Computational Learning Theory , pages 454-468. Springer, 2007.

Peter Auer, Pratik Gajane, and Ronald Ortner. Adaptively tracking the best bandit arm with an unknown number of distribution changes. In Conference on Learning Theory , pages 138-158. PMLR, 2019.

Mohsen Bayati, Nima Hamidi, Ramesh Johari, and Khashayar Khosravi. The unreasonable effectiveness of greedy algorithms in multi-armed bandit with many arms. arXiv preprint arXiv:2002.10121 , 2020.

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Omar Besbes, Yonatan Gur, and Assaf Zeevi. Stochastic multi-armed-bandit problem with non-stationary rewards. Advances in neural information processing systems , 27, 2014.

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Wang Chi Cheung, David Simchi-Levi, and Ruihao Zhu. Learning to optimize under nonstationarity. In The 22nd International Conference on Artificial Intelligence and Statistics , pages 1079-1087. PMLR, 2019.

Alexander Rakhlin Dylan J. Foster. Statistical reinforcement learning and decision making: Course notes. MIT Lecture notes for course 9.S915 Fall 2022 , 2022.

Jung-hun Kim, Milan Vojnovic, and Se-Young Yun. Rotting infinitely many-armed bandits. In International Conference on Machine Learning , pages 11229-11254. PMLR, 2022.

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

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Phillippe Rigollet and Jan-Christian Hütter. High dimensional statistics. Lecture notes for course 18S997 , 813:814, 2015.

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Julien Seznec, Andrea Locatelli, Alexandra Carpentier, Alessandro Lazaric, and Michal Valko. Rotting bandits are no harder than stochastic ones. In The 22nd International Conference on Artificial Intelligence and Statistics , pages 2564-2572. PMLR, 2019.

Julien Seznec, Pierre Menard, Alessandro Lazaric, and Michal Valko. A single algorithm for both restless and rested rotting bandits. In Silvia Chiappa and Roberto Calandra, editors, Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics , volume 108 of Proceedings of Machine Learning Research , pages 3784-3794. PMLR, 26-28 Aug 2020.

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A Appendix

A.1 Limitations & Discussion

As we summarize our results in Table 1, Algorithm 1 achieves near-optimal regret only when β ≥ 1 . Here, we discuss the discrepancies between lower and upper bounds when 0 < β < 1 . From (1), we can observe that as β decreases below 1 , the probability to sample good arms may increase, which appears to be beneficial with respect to regret. However, the regret upper bounds for 0 < β < 1 in Theorems 3.1 and 3.3 remain the same as the case when β = 1 while the regret lower bounds in Theorems 4.1 and 4.2 decrease as β decreases, resulting in a gap between the regret upper and lower bounds. The phenomenon that the regret upper bound remains the same when β decreases has also been observed in previous literature on infinitely many-armed bandits [5, 24, 10]. As mentioned in Carpentier and Valko [10], although there are likely to be many good arms when β is small, it is not possible to avoid a certain amount of regret from estimating mean rewards to distinguish arms under sub-Gaussian reward noise. Therefore, we believe that our regret upper bounds are near-optimal across the entire range of β , and achieving tighter regret lower bounds when β < 1 is left for future research. Notably, the optimality proven only for β ≥ 1 has also been observed for stationary infinitely many-armed bandits [5, 24].

A.2 Additional Explanations for Eq. (1)

1

1

𝛽 = 1

𝛽 > 1

𝛽 < 1

𝑥

ℙ Δ

!

𝑎

< 𝑥

(Uniform distribution)

(Good arm

¯

)

(Good arm

­

)

To discuss the effect of β on the distribution of ∆ 1 ( a ) and the probability of sampling a good arm (having small ∆ 1 ( a )) , we consider the case when P (∆ 1 ( a ) < x ) = x β , which is shown in Figure 4 for some values of β . It is noteworthy that the uniform distribution is a special case when β = 1 . Importantly, the larger the value of β , the smaller the probability of sampling a good arm.

A.3 Proof of Proposition 2.3

Recall ∆ 1 ( a ) = 1 -µ 1 ( a ) . We first show that E [ µ 1 ( a )] = Θ(1) . For any randomly sampled a ∈ A , we have E [ µ 1 ( a )] ≥ y P ( µ 1 ( a ) ≥ y ) = y P (∆ 1 ( a ) < 1 -y ) for y ∈ [0 , 1] . With y = 1 / 2 , we have E [ µ 1 ( a )] ≥ (1 / 2) P (∆ 1 ( a ) < (1 / 2)) = Θ(1) from constant β > 0 and (1). Then with E [ µ 1 ( a )] ≤ 1 , we can conclude E [ µ 1 ( a )] = Θ(1) (Especially when P (∆( a ) < x ) = x β , we have E [∆ 1 ( a )] = ∫ 1 0 P (∆ 1 ( a ) ≥ x ) dx = 1 -∫ 1 0 P (∆ 1 ( a ) < x ) dx = 1 -∫ 1 0 x β dx = 1 -1 β +1 , which implies E [ µ 1 ( a )] = Θ(1) with constant β > 0 ). We then think of a policy π ′ that randomly samples a new arm and pulls it only once every round. Since E [ µ 1 ( a )] = Θ(1) for any randomly sampled a , we have E [ R π ′ ( T )] = Θ( T ) .

Next we show that the policy π ′ is optimal for the worst case of ∑ T -1 t =1 ρ t > T . We think of any policy π ′′ except π ′ . For any policy π ′′ , there always exists an arm a such that the policy must pull arm a at least twice. Let t ′ and t ′′ be the rounds when the policy pulls arm a . If we consider ρ t ′ > 0 and ρ t = 0 for t ∈ [ T -1] / { t ′ } such that ρ t ′ = ∑ T -1 t =1 ρ t then such policy has Ω( ∑ T -1 t =1 ρ t ) regret bound. Since ∑ T -1 t =1 ρ t > T , for any algorithm π ′′ except π ′ , there always exist a rotting rate adversary such that E [ R π ′′ ( T )] = Ω( ∑ T -1 t =1 E [ ρ t ]) = Ω( T ) . Therefore we can conclude that π ′ is the optimal algorithm for achieving the optimal regret of Θ( T ) .

A.4 Proof of Theorem 3.1: Regret Upper Bound of Algorithm 1 for Slow Rotting with V T

Let ∆ t ( a ) = 1 -µ t ( a ) . Using a threshold parameter δ , we classify an arm a as good if ∆ 1 ( a ) ≤ δ/ 2 , near-good if δ/ 2 < ∆ 1 ( a ) ≤ 2 δ , and otherwise, we classify a as a bad arm. In A , let ¯ a 1 , ¯ a 2 , . . . , be a sequence of arms, which have i.i.d. mean rewards with uniform distribution on [0 , 1] . Without loss of generality, we assume that the policy samples arms, which are pulled at least once, according to the sequence of ¯ a 1 , ¯ a 2 , . . . , . Let A T be the set of sampled arms over the horizon of T time steps, which satisfies |A T | ≤ T . Let A G T be a set of good or near good arms in A T . WLOG, the following proofs proceed under the given A T , since the proofs hold for any A T .

Let µ [ s 1 ,s 2 ] ( a ) = ∑ s 2 t = s 1 µ t ( a ) /n [ s 1 ,s 2 ] ( a ) for the time steps 0 < s 1 ≤ s 2 . We define event E 1 = {| ̂ µ [ s 1 ,s 2 ] ( a ) -µ [ s 1 ,s 2 ] ( a ) | ≤ √ 12 log( T ) /n [ s 1 ,s 2 ] ( a ) for all 1 ≤ s 1 ≤ s 2 ≤ T, a ∈ A T } . By following the proof of Lemma 35 in Dylan J. Foster [12], from Lemma A.30 we have

where X i = r τ i -µ τ i ( a ) and τ i is the i -th time that the policy pulls arm a starting from s 1 . We note that even though X i 's seem to depend on each other from τ i 's, each value of X i is independent of each other. Then using union bound for s 1 , s 2 , and a ∈ A T , we have P ( E c 1 ) ≤ 2 /T 2 . From the cumulative amount of rotting V T , we note that ∆ t ( a ) = O ( V T + 1) for any a and t , which implies E [ R π ( T ) | E c 1 ] = O ( T 2 ) from V T ≤ T . For the case where E 1 does not hold, the regret is E [ R π ( T ) | E c 1 ] P ( E c 1 ) = O (1) , which is negligible compared to the regret when E 1 holds, which we show later. Therefore, for the rest of the proof, we assume that E 1 holds.

For regret analysis, we divide R π ( T ) into two parts, R G ( T ) and R B ( T ) corresponding to regret of good or near-good arms, and bad arms over time T , respectively, such that R π ( T ) = R G ( T )+ R B ( T ) . We first provide a bound of R G ( T ) in the following lemma.

Lemma A.1. Under E 1 and policy π , we have

Proof. Here we consider arms a ∈ A G T . Let V [ n,m ] ( a ) = ∑ m l = n ρ l ( a ) and ρ [ n,m ] ( a ) = ∑ m l = n ρ l ( a ) /n [ n,m ] ( a ) for time steps n ≤ m . For ease of presentation, for time steps r > q , we define V [ r,q ] ( a ) = n [ r,q ] ( a ) = ρ [ r,q ] ( a ) = ∑ q t = r x ( t ) = 0 for x ( t ) ∈ R and 1 / 0 = ∞ . Then, for any s such that n ≤ s ≤ m , under E 1 we have

Therefore, from µ n ( a ) ≤ 1 we obtain

Let t 1 ( a ) be the initial time when the arm a is sampled and pulled and t 2 ( a ) be the final time when the policy pulls the arm. For simplicity, we use t 1 and t 2 instead of t 1 ( a ) and t 2 ( a ) , respectively, when there is no confusion. We define A 0 as a set of arms a ∈ A G T such that t 2 ( a ) = t 1 ( a ) and define A 1 as a set of arms a ∈ A G T such that t 2 ( a ) = t 1 ( a ) + 1 . We also define a set of arms A G T = { a ∈ A G T / {A 0 ∪ A 1 } : n [ t 1 ,t 2 -1] ( a ) > ⌈ (log T ) 1 / 3 /ρ [ t 1 ,t 2 -2] ( a ) 2 / 3 ⌉} . Let w ( a ) = ⌈ (log T ) 1 / 3 /ρ [ t 1 ,t 2 -2] ( a ) 2 / 3 ⌉ . For simplicity, we use w for w ( a ) when there is no confusion. Then with the fact that µ t ( a ) = µ t 1 ( a ) -∑ t -1 t = t 1 ( a ) ρ t ( a ) = µ t 1 ( a ) -V [ t 1 ,t -1] ( a ) for t 1 ( a ) ≤ t ≤ t 2 ( a ) , we have

where the first inequality comes from ∆ 1 ( a ) ≤ 2 δ for any a ∈ A G T . For the second term in the right hand side of the last inequality (8),

For the third term in (8), from the fact that n [ t 1 +1 ,t 2 ] ( a ) = n [ t 1 ,t 2 -1] ( a ) < w ( a ) for any a ∈ A G T / A G T from the definition of A G T , we have

Now, we focus on the fourth term in (8). From t 1 ( a ) + w ( a ) + 1 ≤ t 2 ( a ) for a ∈ A G T from the definition of A G T and (10), we first have

Now we focus on ∑ a ∈ A G T ∑ t 2 ( a ) t = t 1 ( a )+ w ( a )+1 V [ t 1 ,t -1] ( a ) in (8). From the definition of t 2 and the threshold condition in the algorithm with (7), for any t 1 ≤ t ≤ t 2 and any t 1 ≤ s ≤ t -1 s.t. s = t -2 l -1 for l ∈ Z + , we have

For t ≥ t 1 + w ( a )+1 , there always exists t 1 ≤ s ( t ) ≤ t -1 such that w ( a ) / 2 ≤ n [ s ( t ) ,t -1] ( a ) ≤ w ( a ) and s ( t ) = t -2 l -1 for l ∈ Z + . Then from (12) with s = s ( t ) , we have

Using the facts that n [ s ( t ) ,t -2] ( a ) ≥ n [ s ( t ) ,t -1] ( a ) / 2 ≥ w ( a ) / 4 and t -s ( t ) ≤ w ( a ) from n [ s ( t ) ,t -1] ( a ) ≤ w ( a ) , we can obtain that

where the second inequality is obtained from the fact that the number of times that ρ t is duplicated for each t ∈ [ t 1 ( a ) + 1 , t 2 ( a ) -2] in the expression ∑ t 2 ( a ) t = t 1 ( a )+1+ w ( a ) ∑ t -2 k = t -w ( a ) ρ k is at most w ( a ) . Then with (13) and (14), using the fact that

we have

Then putting the results from (8),(10),(11), and (15) altogether, we have

where the last equality comes from Hölder's inequality and V T ≤ T . This concludes the proof.

Now, we provide a bound for R B ( T ) . We note that the initially bad arms can be defined only when 2 δ < 1 . Otherwise when 2 δ ≥ 1 , we have R ( T ) = R G ( T ) , which completes the proof. Therefore, for the regret from bad arms, we consider the case of 2 δ < 1 . We adopt the episodic approach in Kim et al. [13] for the remaining regret analysis. The episodic approach is reformulated using the cumulative amount of rotting instead of the maximum rotting rate. In the following, we define some notation.

Given a policy sampling arms in the sequence order, let m G be the number of samples of distinct good arms and m B i be the number of consecutive samples of distinct bad arms between the i -1 -st and i -th sample of a good arm among m G good arms. We refer to the period starting from sampling the i -1 -st good arm before sampling the i -th good arm as the i -th episode . Observe that m B 1 , . . . , m B m G are i.i.d. random variables with geometric distribution with parameter 2 δ , given a fixed value of m G . Therefore, with some constant C > 0 , for non-negative integer k we have P ( m B i = k ) = (1 -C (2 δ ) β ) k C (2 δ ) β , for i = 1 , . . . , m G . Define ˜ m T to be the number of episodes from the policy π over the horizon T , ˜ m G T to be the total number of samples of a good arm by the policy π over the horizon T such that ˜ m G T = ˜ m T or ˜ m G T = ˜ m -1 , and ˜ m B i,T to be the number of samples of a bad arm in the i -th episode by the policy π over the horizon T .

Under a policy π , let R B i,j be the regret (summation of mean reward gaps) contributed by pulling the j -th bad arm in the i -th episode. Then let R B m G = ∑ m G i =1 ∑ j ∈ [ m B i ] R B i,j , which is the regret from initially bad arms over the period of m G episodes.

Let a ( i ) be a good arm in the i -th episode and a ( i, j ) be a j -th bad arm in the i -th episode. We define V T ( a ) = ∑ T t =1 ρ t ✶ ( a t = a ) . Then excluding the last episode ˜ m T over T , we provide lower bounds of the total rotting variation over T for a ( i ) , denoted by V T ( a ( i )) , in the following lemma. Lemma A.2. Under E 1 , given ˜ m T , for any i ∈ [ ˜ m G T ] / { ˜ m T } we have

Proof. Suppose that V T ( a ( i )) < δ/ 2 , then we have

where the first inequality is obtained from E 1 , and the last inequality is from V T ( a ( i )) < δ/ 2 and µ 1 ( a ( i )) ≥ 1 -δ/ 2 . Therefore, from the threshold condition, policy π must pull arm a ( i ) until its total rotting amount is greater than (or equal to) δ/ 2 , which implies V T ( a ( i )) ≥ δ/ 2 .

In the following, we consider two different cases with respect to V T ; large and small V T .

Case 1: We consider V T > max { 1 / √ T, 1 /T 1 / ( β +1) } in the following.

In this case, we have δ = δ V ( β ) = c 1 max { ( V T /T ) 1 / ( β +2) , ( V T /T ) 1 / 3 } . Here, we define the policy π after time T such that it pulls a good arm until its total rotting variation is equal to or greater than δ/ 2 and does not pull a sampled bad arm. We note that defining how π works after T is only for the proof to get a regret bound over time horizon T . For the last arm ˜ a over the horizon T , it pulls the arm until its total variation becomes max { δ/ 2 , V T (˜ a ) } if ˜ a is a good arm. For i ∈ [ m G ] , j ∈ [ m B i ] let V G i and V B i,j be the total rotting variation of pulling the good arm in i -th episode and j -th bad arm in i -th episode from the policy, respectively. Here we define V G i 's and V B i,j 's as follows:

If ˜ a is a good arm,

Otherwise,

For i ∈ [ m G ] , j ∈ [ m B i ] let n B i,j be the number of pulling the j -th bad arm in i -th episode from the policy. We define n T ( a ) be the total amount of pulling arm a over T . Here we define n B i,j 's as follows:

Then we provide m G such that R B ( T ) ≤ R B m G in the following lemma.

Lemma A.3. Under E 1 , when m G = ⌈ 2 V T /δ ⌉ we have

Proof. From Lemma A.2, we have which implies that R B ( T ) ≤ R B m G .

From the result of Lemma A.3, we set m G = ⌈ 2 V T /δ ⌉ . We analyze R B m G for obtaining a bound for R B ( T ) in the following.

Lemma A.4. Under E 1 and policy π , we have

Proof. Let a ( i, j ) be a sampled arm for j -th bad arm in the i -th episode and ˜ m T be the number of episodes from the policy π over the horizon T . Suppose that the algorithm samples arm a ( i, j ) at time t 1 ( a ( i, j )) . Then the algorithm stops pulling arm a ( i, j ) at time t 2 ( a ( i, j )) + 1 if ̂ µ [ s,t 2 ( a ( i,j ))] ( a ) + √ 12 log( T ) /n [ s,t 2 ( a ( i,j ))] ( a ) < 1 -δ for some s such that t 1 ( a ( i, j )) ≤ s ≤ t 2 ( a ( i, j )) and s = t 2 ( a ( i, j )) + 1 -2 l -1 for l ∈ Z + . For simplicity, we use t 1 and t 2 instead of t 1 ( a ( i, j )) and t 2 ( a ( i, j )) when there is no confusion. We first consider the case where the algorithm stops pulling arm a ( i, j ) because the threshold condition is satisfied. For the regret analysis, we consider that for t > t 2 , arm a is virtually pulled. We note that under E 1 , we have

Then we assume that ˜ t 2 ( ≥ t 2 ) is the smallest time that there exists t 1 ≤ s ≤ ˜ t 2 with s = ˜ t 2 +1 -2 l -1 for l ∈ Z + such that the following threshold condition is met:

From the definition of ˜ t 2 , we observe that for given ˜ t 2 , the time step s = s ′ which satisfying (17) equals to t 1 (i.e. s ′ = t 1 ). Then, we can observe that n [ s ′ , ˜ t 2 ] ( a ( i, j )) = n [ t 1 , ˜ t 2 ] ( a ( i, j )) = ⌈ C 2 log( T ) / (∆ t 1 ( a ( i, j )) -δ ) 2 ⌉ for some constant C 2 > 0 , which satisfies (17). Then from n [ t 1 ,t 2 ] ( a ( i, j )) ≤ n [ t 1 , ˜ t 2 ] ( a ( i, j )) , for all i ∈ [ ˜ m T ] , j ∈ [ ˜ m B i,T ] we have n B i,j = ˜ O (1 / (∆ 1 ( a ( i, j )) -δ ) 2 ) . Then with the facts that n B i,j = 0 for i ∈ [ m G ] / [ ˜ m G T ] , j ∈ [ m B i ] / [ ˜ m B i,T ] , we have, for any i ∈ [ m G ] and j ∈ [ m B i ] ,

For 2 δ < x ≤ 1 , let b ( x ) = P (∆ 1 ( a ) = x | a is a bad arm ) . Then we have

where C (2 δ ) β < 1 with small enough positive constant c 1 < 1 for δ . We note that 2 δ < ∆ t 1 ( a ( i, j )) = ∆ 1 ( a ( i, j )) ≤ 1 . Since n B i,j = ˜ O (1 / (∆ t 1 ( a ( i, j )) -δ ) 2 ) = ˜ O (1 /δ 2 ) , we have

Recall that we consider 2 δ < 1 for regret from bad arms. We adopt some techniques introduced in Appendix D of Bayati et al. [5] to deal with the generalized mean reward distribution with β . Let K = (1 -2 δ ) /δ , a j = 2 jδ , and p j = ∫ ( j +1) δ jδ b ( t + δ ) dt . Then for obtaining a bound of the last equality in (18) we have

We note that ∑ j i =1 p i ≤ C 0 ( jδ ) β for all j ∈ [ K ] for some constant C 0 > 0 . Then for getting a bound of the last equality in (19), we have

Now we analyze the term in the last equality in (20) according to the criteria for β . For β = 1 , we can obtain

For β > 1 , we have j β -( j -1) β ≤ βj β -1 using the mean value theorem. Therefore, we obtain the following.

For β < 1 , when j > 1 we have j β -( j -1) β ≤ β ( j -1) β -1 using the mean value theorem. Therefore, we obtain

From (19),(20),(21),(22), and (23), we have

Then for any i ∈ [ m G ] , j ∈ [ m B i ] , we have

Recall that R B m G = ∑ m G i =1 ∑ j ∈ [ m B i ] R B i,j . With δ = c 1 max { ( V T /T ) 1 / ( β +2) , ( V T /T ) 1 / 3 } and m G = ⌈ 2 V T /δ ⌉ , from the fact that m B i 's are i.i.d. random variables with geometric distribution with E [ m B i ] = (1 /C (2 δ ) β ) -1 for some constant C > 0 , we have

From R π ( T ) = R G ( T ) + R B ( T ) and Lemmas A.1, A.3, A.4, with δ = max { ( V T /T ) 1 / ( β +2) , ( V T /T ) 1 / 3 } we have

Case 2: Now we consider V T ≤ max { 1 / √ T, 1 /T 1 / ( β +1) } in the following. In this case, we have δ = c 1 max { 1 /T 1 β +1 , 1 / √ T } . For getting R B m G , here we define the policy π after time T such that it pulls V T amount of rotting variation for a good arm and 0 for a bad arm. We note that defining how π works after T is only for the proof to get a regret bound over time horizon T . For the last arm ˜ a over the horizon T , it pulls the arm up to V T amount of rotting variation if ˜ a is a good arm. For i ∈ [ m G ] , j ∈ [ m B i ] let V G i and V B i,j be the amount of rotting variation from pulling the good arm in i -th episode and j -th bad arm in i -th episode from the policy, respectively. Here we define V G i 's and V B i,j 's as follows:

If ˜ a is a good arm,

Otherwise,

For i ∈ [ m G ] , j ∈ [ m B i ] let n B i,j be the number of pulling the j -th bad arm in i -th episode from the policy. We define n T ( a ) be the total amount of pulling arm a over T . Here we define n B i,j 's as follows:

Then we provide m G such that R B ( T ) ≤ R B m G in the following lemma.

Lemma A.5. Under E 1 , when m G = C 3 for some constant C 3 > 0 , we have R B ( T ) ≤ R B m G .

Proof. From Lemma A.2, under E 1 we can find that V G i ≥ min { δ/ 2 , V T } for i ∈ [ m G ] . Then if m G = C 3 for large enough C 3 > 0 , then with δ = c 1 max { 1 /T 1 / ( β +1) , 1 / √ T } and V T ≤ max { 1 /T 1 / ( β +1) , 1 / √ T } , we have

which implies R B ( T ) ≤ R B m G .

We analyze R B m G for obtaining a bound for R B ( T ) in the following. Lemma A.6. Under E 1 and policy π , we have

Proof. From (24), for any i ∈ [ m G ] , j ∈ [ m B i ] , we have

Recall that R B m G = ∑ m G i =1 ∑ j ∈ [ m B i ] R B i,j . With δ = c 1 max { (1 /T ) 1 / ( β +1) , 1 /T 1 / 2 } and m G = C 3 , from the fact that m B i 's are i.i.d. random variables with geometric distribution with E [ m B i ] = (1 /C (2 δ ) β ) -1 for some constant C > 0 , we have

From Lemma A.1, with δ = c 1 max { 1 /T 1 β +1 , 1 / √ T } we have

From R π ( T ) = R G ( T ) + R B ( T ) and Lemmas A.1, A.5, A.6 with δ = c 1 max { 1 /T 1 β +1 , 1 / √ T } we have

Conclusion: Overall, from (26) and (27), we have

A.5 Proof of Theorem 3.3: Regret Upper Bound of Algorithm 1 for Abrupt Rotting ( S T )

Using the threshold parameter δ in the algorithm, we define an arm a as a good arm if ∆ t ( a ) ≤ δ/ 2 , a near-good arm if δ/ 2 < ∆ t ( a ) ≤ 2 δ , and otherwise, a is a bad arm at time t . For analysis, we consider abrupt change as sampling a new arm. In other words, if a sudden change occurs to an arm a by pulling the arm a , then the arm is considered to be two different arms; before and after the change. The type of abruptly rotted arms (good, near-good, or bad) after the change is determined by the current value of rotted mean reward. Without loss of generality, we assume that the policy samples arms, which are pulled at least once, in the sequence of ¯ a 1 , ¯ a 2 , . . . , . Let A T be the set of sampled arms, which are pulled at least once, over the horizon of T time steps, which satisfies |A T | ≤ T . We also define A S as a set of arms that have been rotted and pulled at least once, which satisfies |A S | ≤ S T . To better understand the definitions, we provide an example. If an arm a suffers abrupt rotting at first, then the arm a is considered to be a different arm a ′ after the rotting. If the arm a ′ again suffers abrupt rotting, then it is considered to be a ′′ after the rotting. If arms a, a ′ , a ′′ are pulled at least once, then { a, a ′ , a ′′ } ∈ A T and { a ′ , a ′′ } ∈ A S but a / ∈ A S . If arm a ′′ is not pulled at least once but a and a ′ are pulled at least once, then { a, a ′ } ∈ A T and a ′ ∈ A S but a ′′ / ∈ A S .

WLOG, the following proofs proceed under the given A T , since the proofs hold for any A T . Let µ [ t 1 ,t 2 ] ( a ) = ∑ t 2 t = t 1 µ t ( a ) ✶ ( a t = a ) /n [ t 1 ,t 2 ] ( a ) . We define the event E 1 = {| ̂ µ [ s 1 ,s 2 ] ( a ) -µ [ s 1 ,s 2 ] ( a ) | ≤ √ 12 log( T ) /n [ s 1 ,s 2 ] ( a ) for all 1 ≤ s 1 ≤ s 2 ≤ T, a ∈ A T } . By following the proof of Lemma 35 in Dylan J. Foster [12], from Lemma A.30 we have

where X i = r τ i -µ τ i ( a ) and τ i is the i -th time that the policy pulls arm a starting from s 1 . We note that even though X i 's seem to depend on each other from τ i 's, each value of X i is independent of each other. Then using union bound for s 1 , s 2 , and a ∈ A T , we have

Let t ( s ) be the time when s -th abrupt rotting occurs with ρ t ( s ) for s ∈ [ S T ] . Then we have ∆ t ( a ) = O (1 + ∑ S T s =1 ρ t ( s ) ) = O (1 + V T ) for any a and t , which implies E [ R π ( T ) | E c 1 ] = O ( T + TV T ) . For the case that E 1 does not hold, the regret is E [ R π ( T ) | E c 1 ] P ( E c 1 ) = O ((1 + V T ) /T ) , which is negligible comparing with the regret when E 1 holds true which we show later. Therefore, in the rest of the proof we assume that E 1 holds true.

Recall that R π ( T ) = ∑ T t =1 (1 -µ t ( a t )) . For regret analysis, we divide R π ( T ) into two parts, R G ( T ) and R B ( T ) corresponding to regret of good or near-good arms, and bad arms over time T , respectively, such that R π ( T ) = R G ( T ) + R B ( T ) . Recall that we consider abrupt change as sampling a new arm in this analysis. Then, from ∆ t ( a ) ≤ 2 δ for any good or near-good arms a at time t , we can easily obtain that

Now we analyze R B ( T ) . We divide regret R B ( T ) into two regret from bad arms in A T / A S , denoted by R B , 1 ( T ) , and regret from bad arms in A S , denoted by R B , 2 ( T ) such that R B ( T ) = R B , 1 ( T ) + R B , 2 ( T ) . We denote bad arms in A S by A B S . We first analyze R B , 1 ( T ) in the following. For regret analysis, we adopt the episodic approach suggested in Kim et al. [13]. The main difference lies in analyzing our adaptive window UCB and a more generalized mean-reward distribution with

β . In the following, we introduce some notation. Here we only consider arms in A T / A S so that the following notation is defined without considering (rotted) arms in A S . We note that from the definition of A T , arms a before having undergone rotting are contained in A T / A S . Here we consider the case of 2 δ S ( β ) < 1 since otherwise when 2 δ S ( β ) ≥ 1 , bad arms are not defined in A T / A S .

Given a policy sampling arms in the sequence order, let m G be the number of samples of distinct good arms and m B i be the number of consecutive samples of distinct bad arms between the i -1 -st and i -th sample of a good arm among m G good arms. We refer to the period starting from sampling the i -1 -st good arm before sampling the i -th good arm as the i -th episode . Observe that m B 1 , . . . , m B m G are i.i.d. random variables with geometric distribution with parameter C (2 δ ) β for some constant C > 0 , given a fixed value of m G . Therefore, for non-negative integer k we have P ( m B i = k ) = (1 -C (2 δ ) β ) k C (2 δ ) β , for i = 1 , . . . , m G .

Define ˜ m G T to be the total number of samples of a good arm by the policy π over the horizon T and ˜ m B i,T to be the number of samples of a bad arm in the i -th episode by the policy π over the horizon T . For i ∈ [ ˜ m G T ] , j ∈ [ ˜ m B i,T ] , let ˜ n G i be the number of pulls of the good arm in the i -th episode and ˜ n B i,j be the number of pulls of the j -th bad arm in the i -th episode by the policy π over the horizon T . Let ˜ a be the last sampled arm over time horizon T by π .

With a slight abuse of notation, we use π for a modified strategy after T . Under a policy π , let R B i,j be the regret (summation of mean reward gaps) contributed by pulling the j -th bad arm in the i -th episode. Then let R B m G = ∑ m G i =1 ∑ j ∈ [ m B i ] R B i,j , which is the regret from initially bad arms over the period of m G episodes. For getting R B m G , here we define the policy π after T such that it pulls T amounts for a good arm and zero for a bad arm. After T we can assume that there are no abrupt changes. For the last arm ˜ a over the horizon T , it pulls the arm up to T amounts if ˜ a is a good arm and ˜ n G ˜ m G T < T . For i ∈ [ m G ] , j ∈ [ m B i ] let n G i and n B i,j be the number of pulling the good arm in i -th episode and j -th bad arm in i -th episode under π , respectively. Here we define n G i 's and n B i,j 's as follows:

If ˜ a is a good arm,

Otherwise,

Using the above notation and newly defined π after T , we show that if m G = S T + 1 , then R B ( T ) ≤ R B m G in the following.

Lemma A.7. Under E 1 , when m G = S T we have

Proof. There are S T -1 number of abrupt changes over T . We consider two cases; there are S T abrupt changes before sampling S T -th good arm or there are not. For the former case, if π samples the S T -th good arm and there are S T -1 number of abrupt changes before sampling the good arm, then it continues to pull the good arm until T . This is because when the algorithm samples a good arm a at time t ′ , from E 1 and the stationary period, we have

This implies that from the threshold condition, the algorithm does not stop pulling the good arm a . After T , from the definition of π for the case when ˜ a is a good arm, n G ˜ m G T = T . Therefore, the algorithm pulls the good arm for T rounds.

Now we consider the latter case, such that π samples the S T -th good arm before the S T -1 -st abrupt change over T . Before sampling the S T -th good arm, there must exist two consecutive good arms such that there is no abrupt change between the two sampled good arms. This is a contraction because π must pull the first good arm among the two up to T under E 1 and S T -1 -st abrupt change must occur after T .

Therefore, it is enough to consider the former case. When m G = S T , we have

which implies R B , 1 ( T ) ≤ R B m G .

From the above lemma, we set m G = S T . We analyze R B m G to get a bound for R B , 1 ( T ) in the following lemma.

Lemma A.8. Under E 1 and policy π , we have

Proof. Recall that we consider arms in A T / A S . Let a ( i, j ) be a sampled arm for j -th bad arm in the i -th episode and ˜ m T be the number of episodes from the policy π over the horizon T . Suppose that the algorithm samples arm a ( i, j ) at time t 1 ( a ( i, j )) . Then the algorithm stops pulling arm a ( i, j ) at time t 2 ( a ( i, j )) + 1 if ̂ µ [ s,t 2 ( a ( i,j ))] ( a ) + √ 12 log( T ) /n [ s,t 2 ( a ( i,j ))] ( a ) < 1 -δ for some s such that t 1 ( a ( i, j )) ≤ s ≤ t 2 ( a ( i, j )) and s = t 2 ( a ( i, j )) + 1 -2 l -1 for l ∈ Z + . For simplicity, we use t 1 and t 2 instead of t 1 ( a ( i, j )) and t 2 ( a ( i, j )) when there is no confusion. For the regret analysis, we consider that for t > t 2 , arm a is virtually pulled. With E 1 , we assume that ˜ t 2 ( ≥ t 2 ) is the smallest time that there exists t 1 ≤ s ≤ ˜ t 2 with s = ˜ t 2 +1 -2 l -1 for l ∈ Z + such that the following condition is met:

From the definition of ˜ t 2 , we observe that for given ˜ t 2 , the time step s = s ′ satisfying (30) equals to t 1 (i.e. s ′ = t 1 ). Then, we can observe that n [ s ′ , ˜ t 2 ] ( a ( i, j )) = n [ t 1 , ˜ t 2 ] ( a ( i, j )) = ⌈ C 2 log( T ) / (∆ t 1 ( a ( i, j )) -δ ) 2 ⌉ for some constant C 2 > 0 , which satisfies (30). Then from n [ t 1 ,t 2 ] ( a ( i, j )) ≤ n [ t 1 , ˜ t 2 ] ( a ( i, j )) , for all i ∈ [ ˜ m T ] , j ∈ [ ˜ m B i,T ] we have n B i,j = ˜ O (1 / (∆ t 1 ( a ( i, j )) -δ ) 2 ) . We note that this bound for the number of pulling an arm holds for not only the case where the arm stops being pulled from the threshold condition but also the case where the arm stops being pulled from meeting an abrupt change (recall that abrupt changes are considered as sampling a new arm) or T . Then with the facts that n B i,j = 0 for i ∈ [ m G ] / [ ˜ m T ] , j ∈ [ m B i ] / [ ˜ m B i,T ] , we have, for any i ∈ [ m G ] and j ∈ [ m B i ] ,

For 2 δ < x ≤ 1 , let b ( x ) = P (∆ t 1 ( a ) = x | a is a bad arm ) . Then we have P (∆ t 1 ( a ) = x | a is a bad arm ) = P (∆ t 1 ( a ) = x | ∆ t 1 ( a ) > 2 δ ) = P (∆ t 1 ( a ) = x ) / P (∆ 1 ( a ) > 2 δ ) = P (∆ t 1 ( a ) = x ) / (1 -C (2 δ ) β ) = O ( P (∆ t 1 ( a ) = x )) , where the last equality comes from small δ with small enough c 1 < 1 . For any i ∈ [ m G ] , j ∈ [ m B i ] , we have

From the above results in (31),(19),(20),(21),(22),(23), for β > 0 we have

Recall that R B m G = ∑ m G i =1 ∑ j ∈ [ m B i ] R B i,j . With δ = c 1 max { ( S T /T ) 1 / ( β +1) , ( S T /T ) 1 / 2 } and m G = S T , from Lemma A.7 and the fact that m B i 's are i.i.d. random variables following geometric

distribution with E [ m B i ] = (1 /C (2 δ ) β ) -1 for some constant C > 0 , we have

From Lemma A.8, we have E [ R B , 1 ( T )] = E [ R B m G ] = ˜ O ( max { S 1 / ( β +1) T T β/ ( β +1) , √ S T T } ) .

Now we analyze R B , 2 ( T ) in the following lemma. Here, we consider arms in A B S , which is allowed to have negative mean rewards.

Lemma A.9. Under E 1 and policy π , we have

Proof. Recall that we consider arms a ∈ A B S so that ∆ t 1 ( a ) > 2 δ from definition. Suppose that the arm a is sampled and pulled for the first time at time t 1 ( a ) . Then the algorithm stops pulling arm a at time t 2 ( a ) + 1 if ̂ µ [ s,t 2 ( a )] ( a ) + √ 12 log( T ) /n [ s,t 2 ( a )] ( a ) < 1 -δ for some s such that s ≤ t 2 ( a ) and s = t 2 ( a ) + 1 -2 l -1 for l ∈ Z + . For simplicity, we use t 1 and t 2 instead of t 1 ( a ) and t 2 ( a ) when there is no confusion. For regret analysis, we consider that for t > t 2 , arm a is virtually pulled. With E 1 , we assume that ˜ t 2 ( ≥ t 2 ) is the smallest time that there exists t 1 ≤ s ≤ ˜ t 2 with s = ˜ t 2 +1 -2 l -1 for l ∈ Z + such that the following condition is met:

From the definition of ˜ t 2 , we observe that for given ˜ t 2 , the time step s , which satisfies (32), equals to t 1 . Then, we can observe that n [ t 1 , ˜ t 2 ] ( a ) = max {⌈ C 2 log( T ) / (∆ t 1 ( a ) -δ ) 2 ⌉ , 1 } for some constant C 2 > 0 , which satisfies (32). From the above, for any a ∈ A B S satifying ∆ t 1 ( a ) ≥ √ C 2 log( T ) + δ , we have n [ t 1 , ˜ t 2 ] ( a ) = 1 . This implies that after pulling the arm a once, the arm is eliminated and after that, the arm is not pulled anymore. Therefore, for any arm a ′ which was rotted to a , we have ∆ t 1 ( a ′ ) ( a ′ ) < √ C 2 log( T ) + δ . This is because otherwise such that ∆ t 1 ( a ′ ) ( a ′ ) ≥ √ C 2 log( T ) + δ , the arm a ′ is eliminated and a cannot be pulled which means a / ∈ A B S , which is a contradiction. Then for any arm a ∈ A B S , we have ∆ t 1 ( a ) ≤ √ C 2 log( T ) + δ + ρ t 1 ( a ) -1 . Recall that we consider abrupt rotting of an arm as sampling a new arm. Let t ( s ) be the time step when the s -th abrupt rotting occurs. Then we note that ρ t 1 ( a ) -1 = ρ t ( s ) when arm a is a sampled arm from s -th abrupt rotting for s ∈ [ S T ] .

From n [ t 1 ,t 2 ] ( a ) ≤ n [ t 1 , ˜ t 2 ] ( a ) , we have n [ t 1 ,t 2 ] ( a ) = ˜ O (max { 1 / (∆ t 1 ( a ) -δ ) 2 , 1 } ) . We note that this bound for number of pulling an arm holds for not only the case where the arm stops to be pulled from the threshold condition, but also the case where the arm stops to be pulled from meeting an abrupt change (recall that abrupt changes are considered as sampling a new arm) or T . From the definition of bad arms, we have ∆ t 1 ( a ) ≥ 2 δ . Then the regret from arm a , denoted by R ( a ) , is bounded as follows: R ( a ) = ∆ t 1 ( a ) n [ t 1 ,t 2 ] ( a ) = ˜ O (max { ∆ t 1 ( a ) / (∆ t 1 ( a ) -δ ) 2 , ∆ t 1 ( a ) } ) . Since x/ ( x -δ ) 2 ≤ 2 /δ for any x ≥ 2 δ , we have R ( a ) = ˜ O (max { 1 /δ, ∆ t 1 ( a ) } ) . Therefore, with the fact that ∆ t 1 ( a ) ≤ √ C 2 log( T ) + δ + ρ t ( s ) for the corresponding s ∈ [ S T ] such that ρ t 1 ( a ) -1 = ρ t ( s ) ,

we have

where the second last equality comes from S T /δ ≥ S T .

Finally, from R π ( T ) = R G ( T ) + R B ( T ) , (29), and Lemmas A.8, A.9, we have

A.6 Details for the Case of Unknown Parameters

Algorithm 2 Adaptive UCB-Threshold with Adaptive Sliding Window

Here, we consider the case where there are constraints for both S T and V T , and parameters of V T , S T , and β are unknown to the agent. We note that ¯ V T ≤ V T from the constraint. The parameters of β , V T , and S T are used to set the optimal threshold parameter δ in Algorithm 1. Therefore, when the parameters are not given, the procedure to find the optimal value δ is required. We adopt the Bandit-over-Bandit ( BoB ) approach in Cheung et al. [11], Kim et al. [13] by additionally considering adaptive window. In Algorithm 2, the algorithm consists of a master and several base algorithms with B . For the master, we use EXP3 [2] to find a nearly best base in B . Each base represents Algorithm 1 with a candidate threshold δ ′ ∈ B . The algorithm divides the time horizon into several blocks of length H . At each block, the algorithm samples a base in B from the EXP3 strategy and runs the base over the time steps of the block. Using the feedback from the block, the algorithm updates EXP3 and samples a new base for the next block. By block time passes, the master is likely to find an optimized δ in B . Let B = |B| . Then for Algorithm 2, we set α = min { 1 , √ B log B/ (( e -1) ⌈ T/H ⌉ ) } and C > 0 to be a large enough constant.

We define δ † V = c 1 max { ( V T /T ) 1 / ( β +2) , ( V T /T ) 1 / 3 , 1 /H 1 / ( β +1) , 1 / √ H } and δ † S = c 1 max { ( S T /T ) 1 / ( β +1) , ( S T /T ) 1 / 2 , 1 /H 1 / ( β +1) , 1 / √ H } for some constant 0 < c 1 < 1 . Then the optimized threshold parameter is δ † V S = min { δ † S , δ † V } . The optimized threshold parameter can be derived from the theoretical analysis in Appendix A.7. The target of the master is to find the parameter. From the above, we can observe that c 1 / √ H ≤ δ † V S ≤ 1 . Therefore, we set B = { 1 / 2 , . . . , 1 / 2 log 2 √ H/c 1 } which is the candidate values for unknown δ † .

̸

Assumption A.10 (Constrained Adaptive Adversary) . At each time t > 0 , the value of rotting rate ρ t is determined arbitrarily immediately after the agent pulls an arm a t under the constraint of 0 ≤ ρ t ≤ ϱ t for given ϱ t .

The regret is composed of two factors from the master and bases. To ensure that the regret bound from each base concerning V T and S T remains guaranteed irrespective of the bases chosen, we consider a constrained adaptive adversary. For the following, we consider that ϱ t for all t > 0 are arbitrarily determined before an algorithm is run, under the constraints of V T and S T where ∑ T -1 t =1 ϱ t ≤ V T and 1 + ∑ T -1 t =1 ✶ ( ϱ t = 0) ≤ S T . Then under ϱ t for all t > 0 , we consider the following adversary for rotting rates.

Remark A.11 . Assumption A.10 is still more general than that for the maximum rotting rate constraint in Kim et al. [13] where ϱ t = ρ for all t > 0 . We also observe that the constrained adaptive adversary in Assumption A.10 is milder than the adaptive adversary in Assumption 2.1. Additionally, we note that a special case of constraint ρ t = ϱ t for all t > 0 in Assumption A.10 represents an oblivious adversary because ϱ t 's are determined before an algorithm is run.

With a time block size of H (where H = ⌈ √ T ⌉ ), the algorithm operates over ⌈ T/H ⌉ blocks. Denote by T i the set of time steps in the i -th block containing time steps of ( i -1) H +1 ≤ t ≤ iH ∧ T . It is possible to encounter large rotting for some i block, potentially resulting in an arm's mean reward having a significantly low negative value, leading to suboptimal behavior by the master incurring large regret from the master. To address this, we introduce the assumption of equally distributed cumulative rotting for blocks, stated as follows:

Assumption A.12. ∑ t ∈T i ρ t ≤ H for all i ∈ [ ⌈ T/H ⌉ ]

Remark A.13 . As similarly highlighted in Remark 2.5, this assumption is satisfied when mean rewards are under the constraint of 0 ≤ µ t ( a t ) ≤ 1 for all t ∈ [ T ] , which is frequently encountered in real-world applications where reward is represented by metrics like click rates or (normalized) ratings in content recommendation systems.

Remark A.14 . Our rotting scenario with ∑ t ∈T i ρ t ≤ H for all i ∈ [ ⌈ T/H ⌉ ] is more general in scope than the one with a maximum rotting rate constraint where ρ t ≤ ρ = o (1) for all t ∈ [ T -1] , which was explored in Kim et al. [13]. This is because for our setting, ρ t is not necessarily bounded by o (1) , and for the maximum rotting constraint setting with ρ t ≤ ρ = o (1) , the condition of ∑ t ∈T i ρ t ≤ H for all i ∈ [ ⌈ T/H ⌉ ] is always satisfied. It is noteworthy that Assumption A.12 implies Assumption 2.4.

We provide a regret bound of Algorithm 2 under Assumption A.10 and Assumption A.12 in the following.

Theorem A.15. Let R ′ V and R ′ S be defined as

Then, the policy π of Algorithm 2 with H = ⌈ √ T ⌉ achieves the following regret bound:

Proof. The proof is provided in Appendix A.7.

We can observe that there is the additional regret cost of T (2 β +1) / (2 β +2) for β ≥ 1 or T 3 / 4 for 0 < β < 1 compared to Algorithm 1. This additional cost originates from the additional procedure to learn the optimal value of δ in Algorithm 2, which is negligible when V T and S T are large enough.

Remark A.16 . In the case where the value of β is known, setting H = ⌈ max { T ( β +1) / ( β +3) , √ T }⌉ reduces the additional regret cost of Algorithm 2 to max { T ( β +2) / ( β +3) , T 3 / 4 } .

Attaining the optimal regret bound under a parameter-free algorithm remains an open problem.

A.7 Proof of Theorem A.15: Regret Upper Bound of Algorithm 2

In the following, we deal with the cases of (a) δ † V ≤ δ † S so that δ † V S = δ † V and (b) δ † V > δ † S so that δ † V S = δ † S , separately.

A.7.1 Case of δ † V ≤ δ † S

Let π i ( δ ′ ) for δ ′ ∈ B denote the base policy for time steps between ( i -1) H +1 and i · H ∧ T in Algorithm 2 using 1 -δ ′ as a threshold. Denote by a π i ( δ ′ ) t the pulled arm at time step t by policy π i ( δ ′ ) . Then, for δ † ∈ B , which is set later for a near-optimal policy, we have

where and

Note that R π 1 ( T ) accounts for the regret caused by the near-optimal base algorithm π i ( δ † ) 's against the optimal mean reward and R π 2 ( T ) accounts for the regret caused by the master algorithm by selecting a base with δ ∈ B at every block against the base with δ † . In what follows, we provide upper bounds for each regret component. We first provide an upper bound for E [ R π 1 ( T )] by following the proof steps in Theorem 3.1. Then we provide an upper bound for E [ R π 2 ( T )] . We set H = ⌈ T 1 / 2 ⌉ and δ † to be a smallest value in B which is larger than δ † V = c 1 max { ( V T /T ) 1 / ( β +2) , ( V T /T ) 1 / 3 , 1 /H 1 / ( β +1) , 1 /H 1 / 2 } .

Remark A.17 . One might wonder whether R π 1 ( T ) , regret from the near-optimal base of δ † , satisfies the constraint of V T and S T even though the master may not select the near-optimal base in the algorithm for each block. From Assumption A.10, we can guarantee the constraints of V T and S T for rotting rates from each base regardless of the selected bases from the master for each block because the rotting upper bound ϱ t 's are determined before staring the game regardless of the behavior of the master. Therefore, we can utilize V T and S T for bounding R π 1 ( T ) , which is the regret from the near-optimal base of δ † .

Upper Bounding E [ R π 1 ( T )] . We refer to the period starting from time step ( i -1) H +1 to time step i · H ∧ T as the i -th block . For any i ∈ ⌈ ( T/H ) -1 ⌉ , policy π i ( δ † ) runs over H time steps independent to other blocks so that each block has the same expected regret and the last block has a smaller or equal expected regret than other blocks. Therefore, we focus on finding a bound on the regret from the first block equal to ∑ H t =1 1 -µ t ( a π 1 ( δ † ) t ) . We define an arm a as a good arm if ∆( a ) ≤ δ † / 2 , a near-good arm if δ † / 2 < ∆( a ) ≤ 2 δ † , and otherwise, a is a bad arm. In A , let ¯ a 1 , ¯ a 2 , . . . , be a sequence of arms, which have i.i.d. mean rewards following (1). Without loss of generality, we assume that the policy samples arms in the sequence of ¯ a 1 , ¯ a 2 , . . . , .

Denote by A ( i ) the set of selected (explored) arms in the i -th block, which satisfies |A ( i ) | ≤ H . WLOG, we consider the case of given A ( i ) for the following because the proof can be applied to any given A ( i ) . Let µ [ t 1 ,t 2 ] ( a ) = ∑ t 2 t = t 1 µ t ( a ) /n [ t 1 ,t 2 ] ( a ) . We define the event E 1 = {| ̂ µ [ s 1 ,s 2 ] ( a ) -µ [ s 1 ,s 2 ] ( a ) | ≤ √ 12 log( H ) /n [ s 1 ,s 2 ] ( a ) for all 1 ≤ s 1 ≤ s 2 ≤ H,a ∈ A ( i ) } . As in (6), we have

We denote by V H,i = ∑ t ∈T i ρ t the cumulative amount of rotting in the time steps in the i -th block. From the cumulative amount of rotting, we note that ∆ t ( a ) = O ( V H,i +1) for any a and t in i -th block, which implies E [ R π ( T ) | E c 1 ] = O ( H 2 ) from V H,i ≤ H under Assumption A.12. For the case where E 1 does not hold, the regret is E [ R π ( T ) | E c 1 ] P ( E c 1 ) = O (1) , which is negligible compared to the regret when E 1 holds, which we show later. For the case that E 1 does not hold, the regret is E [ R π ( H ) | E c 1 ] P ( E c 1 ) = O (1) , which is negligible compared with the regret when E 1 holds true which we show later. Therefore, in the rest of the proof we assume that E 1 holds true.

In the following, we first provide a regret bound over the first block.

For regret analysis, we divide R π 1 ( δ † ) ( H ) into two parts, R G ( H ) and R B ( H ) corresponding to regret of good or near-good arms, and bad arms over time H , respectively, such that R π 1 ( δ † ) ( H ) = R G ( H ) + R B ( H ) . We denote by V H,i the cumulative amount of rotting in the time steps in the i -th block. We first provide a bound of R G ( H ) in the following lemma.

Lemma A.18. Under E 1 and policy π , we have

Proof. We can easily prove the theorem by following the proof steps in Lemma A.1

Now, we provide a regret bound for R B ( H ) . We note that the initially bad arms can be defined only when 2 δ † < 1 . Otherwise when 2 δ † ≥ 1 , we have R ( T ) = R G ( T ) , which completes the proof. Therefore, for the regret from bad arms, we consider the case of 2 δ † < 1 . For the proof, we adopt the episodic approach in Kim et al. [13] for regret analysis.

Given a policy sampling arms in the sequence order, let m G be the number of samples of distinct good arms and m B i be the number of consecutive samples of distinct bad arms between the i -1 -st and i -th sample of a good arm among m G good arms. We refer to the period starting from sampling the i -1 -st good arm before sampling the i -th good arm as the i -th episode . Observe that m B 1 , . . . , m B m G are i.i.d. random variables with geometric distribution with parameter 2 δ , given a fixed value of m G . Therefore, for non-negative integer k we have P ( m B i = k ) = (1 -C (2 δ † ) β ) k C (2 δ † ) β for some constant C > 0 , for i = 1 , . . . , m G . Define ˜ m H to be the number of episodes from the policy π over the horizon H , ˜ m G H to be the total number of samples of a good arm by the policy π over the horizon H such that ˜ m G H = ˜ m H or ˜ m G H = ˜ m -1 , and ˜ m B i,H to be the number of samples of a bad arm in the i -th episode by the policy π 1 ( δ † ) over the horizon H .

Under a policy π 1 ( δ † ) , let R B i,j be the regret (summation of mean reward gaps) contributed by pulling the j -th bad arm in the i -th episode. Then let R B m G = ∑ m G i =1 ∑ j ∈ [ m B i ] R B i,j , which is the regret from initially bad arms over the period of m G episodes.

For obtaining a regret bound, we first focus on finding a required number of episodes, m G , such that R B ( T ) ≤ R B m G . Then we provide regret bounds for each bad arm and good arm in an episode. Lastly, we obtain a regret bound for E [ R B ( T )] using the episodic regret bound.

Let a ( i ) be a good arm in the i -th episode and a ( i, j ) be a j -th bad arm in the i -th episode. We define V H ( a ) = ∑ H t =1 ρ t ✶ ( a t = a ) . Then excluding the last episode ˜ m H over H , we provide lower bounds of the total rotting variation over H for a ( i ) , denoted by V H ( a ( i )) , in the following lemma. Lemma A.19. Under E 1 , given ˜ m H , for any i ∈ [ ˜ m G H ] / { ˜ m H } we have

Proof. We can easily prove the theorem by following the proof steps in Lemma A.2

We first consider the case where V T > max { T/H 3 / 2 , T/H ( β +2) / ( β +1) } . In this case, we have δ † = c 1 max { ( V T /T ) 1 / ( β +2) , ( V T /T ) 1 / 3 } . Here, we define the policy π after time H such that it pulls a good arm until its total rotting variation is equal to or greater than δ † / 2 and does not pull a sampled bad arm. We note that defining how π works after H is only for the proof to get a regret

bound over time horizon H . For the last arm ˜ a over the horizon H , it pulls the arm until its total variation becomes max { δ † / 2 , V H (˜ a ) } if ˜ a is a good arm. For i ∈ [ m G ] , j ∈ [ m B i ] let V G i and V B i,j be the total rotting variation of pulling the good arm in i -th episode and j -th bad arm in i -th episode from the policy, respectively. Here we define V G i 's and V B i,j 's as follows:

If ˜ a is a good arm,

Otherwise,

For i ∈ [ m G ] , j ∈ [ m B i ] let n B i,j be the number of pulling the good arm in i -th episode and j -th bad arm in i -th episode from the policy, respectively. We define n H ( a ) be the total amount of pulling arm a over H . Here we define n B i,j 's as follows:

Then we provide m G such that R B ( H ) ≤ R B m G in the following lemma.

Proof. We can easily show the theorem by following the proof steps of Lemma A.3

From the result of Lemma A.20, we set m G = ⌈ 2 V H, 1 /δ † ⌉ . In the following, we anlayze R B m G for obtaining a regret bound for R B ( H ) .

Lemma A.21. Under E 1 and policy π , we have

Proof. We can easily prove the theorem by following proof steps in Lemma A.4. From (24), for any i ∈ [ m G ] , j ∈ [ m B i ] , we have

Recall that R B m G = ∑ m G i =1 ∑ j ∈ [ m B i ] R B i,j . With δ † = c 1 max { ( V T /T ) 1 / ( β +2) , ( V T /T ) 1 / 3 } and m G = ⌈ 2 V H, 1 /δ † ⌉ , from the fact that m B i 's are i.i.d. random variables with geometric distribution with E [ m B i ] = 1 / (2 δ † ) β -1 , we have

From R π 1 ( δ † ) ( H ) = R G ( H ) + R B ( H ) and Lemmas A.18, A.20, A.21, with δ † = max { ( V T /T ) 1 / ( β +2) , ( V T /T ) 1 / 3 } we have

The above regret bound is for the first block. Therefore, by summing regrets from ⌈ T/H ⌉ number of blocks, from V T > max { T/H ( β +2) / ( β +1) , T/H 3 / 2 } , H = ⌈ T 1 / 2 ⌉ and the fact that E [ ∑ T -1 t =1 ρ t ] ≤ V T , using Hölder's inequality we have shown that

Now, we consider the case where V T ≤ max { T/H 3 / 2 , T/H ( β +2) / ( β +1) } . In this case, we have δ † = c 1 max { 1 / √ H, 1 /H 1 β +1 } . From the result of Lemma A.20, by setting m G = ⌈ 2 V H, 1 /δ † ⌉ we have R B ( H ) ≤ R B m G .

Lemma A.22. Under E 1 and policy π , we have

Proof. We can easily prove the theorem by following proof steps in Lemma A.4. From (24), for any i ∈ [ m G ] , j ∈ [ m B i ] , we have

Recall that R B m G = ∑ m G i =1 ∑ j ∈ [ m B i ] R B i,j . With δ † = c 1 max { 1 /H 1 / 2 , 1 /H 1 / ( β +1) } and m G = ⌈ 2 V H, 1 /δ † ⌉ , from the fact that m B i 's are i.i.d. random variables with geometric distribution with E [ m B i ] = (1 /C (2 δ † ) β ) -1 , we have

From R π 1 ( δ † ) ( H ) = R G ( H ) + R B ( H ) and Lemmas A.18, A.20, A.22, with δ † = Θ(max { 1 /H 1 / 2 , 1 /H 1 / ( β +1) } ) we have

Therefore, by summing regrets from ⌈ T/H ⌉ number of blocks and from V T = O (max { T/H 3 / 2 , T/H ( β +2) / ( β +1) } ) , H = ⌈ T 1 / 2 ⌉ , and the fact that length of time steps in each block is bounded by H , we have

where the second equality comes from Hölder's inequality.

From (34) and (35), we have

Upper Bounding E [ R π 2 ( T )] . We observe that the EXP3 is run for ⌈ T/H ⌉ decision rounds and the number of policies (i.e. π i ( δ ′ ) for δ ′ ∈ B ) is B . Denote the maximum absolute sum of rewards of any block with length H by a random variable Q ′ . We first provide a bound for Q ′ using concentration inequalities. For any block i , we have

Denote by T i the set of time steps in the i -th block. We define the event E 2 ( i ) = {| ̂ µ [ s 1 ,s 2 ] ( a ) -µ [ s 1 ,s 2 ] ( a ) | ≤ √ 14 log( H ) /n [ s 1 ,s 2 ] ( a ) , for all s 1 , s 2 ∈ T i , s 1 ≤ s 2 , a ∈ A ( i ) } and E 2 = ⋂ i ∈ [ ⌈ T/H ⌉ ] E 2 ( i ) . From Lemma A.30, with H = ⌈ √ T ⌉ we have

By assuming that E 2 holds true, we can get a lower bound for µ t ( a π t ) , which may be a negative value from rotting, for getting an upper bound for | ∑ i · H ∧ T t =( i -1) H +1 µ t ( a π t ) | . We can observe that ∑ i · H ∧ T t =( i -1) H +1 µ t ( a π t ) ≤ H . Therefore the remaining part is to get a lower bound for ∑ i · H ∧ T t =( i -1) H +1 µ t ( a π t ) . For the proof simplicity, we consider that when an arm is rotted, then the arm is considered as a different arm after rotting. For instance, when arm a is rotted at time s , then arm a is considered as a different arm a ′ after s . Therefore, each arm can be considered to be stationary. The set of arms is denoted by L . We denote by L + the set of arms having µ t ( a ) ≥ 0 for a ∈ L . We first focus on the arms in L / L + .

Let δ max denote the maximum value in B so that δ max = 1 / 2 . With E 2 and a ∈ L / L + , we assume that ˜ t 2 ( ≥ t 2 ) is the smallest time that there exists t 1 ≤ s ≤ ˜ t 2 with s = ˜ t 2 +1 -2 l -1 for l ∈ Z + such that the following condition is met:

From the definition of ˜ t 2 , we observe that for given ˜ t 2 , the time step s , which satisfies (38), equals to t 1 . Then, we can observe that n [ t 1 , ˜ t 2 ] ( a ) = max {⌈ C 2 log( H ) / (∆ t 1 ( a ) -δ max ) 2 ⌉ , 1 } for some constant C 2 > 0 , which satisfies (38). From n [ t 1 ,t 2 ] ( a ) ≤ n [ t 1 , ˜ t 2 ] ( a ) , we have n [ t 1 ,t 2 ] ( a ) ≤ max { C 3 log( H ) / (∆ t 1 ( a ) -δ max ) 2 , 1 } for some constant C 3 > 0 . Then the regret from arm a , denoted by R ( a ) , is bounded as follows: R ( a ) = ∆ t 1 ( a ) n [ t 1 ,t 2 ] ( a ) ≤ max { C 3 log( H )∆ t 1 ( a ) / (∆ t 1 ( a ) -δ max ) 2 , ∆ t 1 ( a ) } . Since x/ ( x -δ max ) 2 < 1 / (1 -δ max ) 2 = 4 for any x > 1 , we have ∆ t 1 ( a ) / (∆ t 1 ( a ) -δ max ) 2 ≤ 4 . Then we have R ( a ) ≤

max { C 4 log( H ) , ∆ t 1 ( a ) } for some constant C 4 > 0 . Then from |L| ≤ H , we have ∑ a ∈L / {L + } R ( a ) ≤ max { C 4 H log( H ) , H + V H,i } .

Since ∑ a ∈L + R ( a ) ≤ H , we have ∑ a ∈L R ( a ) ≤ H +max { C 4 H log( H ) , H + V H,i } . Therefore from R ( a ) = ∑ t 2 ( a ) t = t 1 ( a ) (1 -µ t ( a )) , we have

which implies that from V H,i ≤ H under Assumption A.12, for some C 5 > 0 , we have

Next we provide a bound for | ∑ i · H ∧ T t =( i -1) H +1 η t | . We define the event E 3 ( i ) = {| ∑ i · H ∧ T t =( i -1) H +1 η t | ≤ 2 √ H log( T ) } and E 3 = ⋂ i ∈ [ ⌈ T/H ⌉ ] E 3 ( i ) . From Lemma A.30, for any i ∈ [ ⌈ T/H ⌉ ] , we have

Then, under E 2 ∩ E 3 , with (37), we have

which implies 1 / 2+ ∑ i · H ∧ T t =( i -1) H r t / ( C 5 H log H +4 √ H log T ) ∈ [0 , 1] or some large enough C > 0 . With the rescaling and translation of rewards in Algorithm 2, from Corollary 3.2. in Auer et al. [2], we have

Remark A.23 . Regarding the utilization of the regret analysis of Corollary 3.2 (EXP3) in Auer et al. [2], we note that the reward for each base can be defined independently of the actual master's action. One might wonder whether the regret analysis for EXP3 can be utilized, considering the fact that the reward from a selected base may depend on the master's action due to the adaptive rotting rates. However, we highlight that the critical aspect of applying EXP3 analysis is whether the rewards from each base are defined independently of the actual action of the master, rather than whether the received (observed) reward from the selected base depends on the master's action. We can construct rewards for each base δ ∈ B at time t when a block starts, denoted as x t ( δ ) , as the reward obtained when the master selects base δ (even though base δ is not actually selected from the algorithm). Then, we can define x t ( δ ) for each δ regardless of the master's actual action. In other words, irrespective of the actually selected base, we define x t ( δ ) for all δ ∈ B as the reward that the master can obtain by selecting δ . In such a case, whatever the selected base by the master is at time t , x t ( δ ) 's remain the same, respectively. This construction is feasible because it's solely for analytical purposes and not necessary for the algorithm's functioning. With this construction of reward for each base, we can utilize EXP3 analysis to obtain a regret bound regarding the master ( R π 2 ( T ) ).

Note that the expected regret from EXP3 is trivially bounded by o ( H 2 ( T/H )) = o ( TH ) and B = O (log( T )) . Then, with (39), we have

Finally, from (33), (36), and (40), with H = T 1 / 2 , we have

which concludes the proof.

A.7.2 Case of δ † V > δ † S

Let π i ( δ ′ ) for δ ′ ∈ B denote the base policy for time steps between ( i -1) H +1 and i · H ∧ T in Algorithm 2 using 1 -δ ′ as a threshold. Denote by a π i ( δ ′ ) t the pulled arm at time step t by policy π i ( δ ′ ) . Then, for δ † ∈ B , which is set later for a near-optimal policy, we have

where and

Note that R π 1 ( T ) accounts for the regret caused by the near-optimal base algorithm π i ( δ † ) 's against the optimal mean reward and R π 2 ( T ) accounts for the regret caused by the master algorithm by selecting a base with δ ∈ B at every block against the base with δ † . In what follows, we provide upper bounds for each regret component. We first provide an upper bound for E [ R π 1 ( T )] by following the proof steps in Theorem 3.3. Then we provide an upper bound for E [ R π 2 ( T )] . Weset δ † to be a smallest value in B which is larger than δ † S = c 1 max { ( S T /T ) 1 / ( β +1) , 1 /H 1 / ( β +1) , ( S T /T ) 1 / 2 , 1 /H 1 / 2 } such that we have δ † = Θ(max { ( S T /T ) 1 / ( β +1) , 1 /H 1 / ( β +1) , ( S T /T ) 1 / 2 , 1 /H 1 / 2 } ) .

π

Upper Bounding E [ R 1 ( T )] . We refer to the period starting from time step ( i -1) H +1 to time step i · H ∧ T as the i -th block . For any i ∈ ⌈ T/H -1 ⌉ , policy π i ( δ † ) runs over H time steps independent to other blocks so that each block has the same expected regret and the last block has a smaller or equal expected regret than other blocks. Therefore, we focus on finding a bound on the regret from the first block equal to ∑ H t =1 1 -µ t ( a π 1 ( δ † ) t ) . We define an arm a as a good arm if ∆ t ( a ) ≤ δ † / 2 , a near-good arm if δ † / 2 < ∆ t ( a ) ≤ 2 δ † , and otherwise, a is a bad arm at time t . In A , let ¯ a 1 , ¯ a 2 , . . . , be a sequence of arms, which have i.i.d. mean rewards following (1). For analysis, we consider abrupt change as sampling a new arm. In other words, if a sudden change occurs to an arm a by pulling the arm a , then the arm is considered to be two different arms; before and after the change. The type of abruptly rotted arms (good, near-good, or bad) after the change is determined by the rotted mean reward. Without loss of generality, we assume that the policy samples arms, which are pulled at least once, in the sequence of ¯ a 1 , ¯ a 2 , . . . , .

Denote by A ( i ) the set of sampled arms, which are pulled at least once, in the i -th block, which satisfies |A ( i ) | ≤ H . We also define A S ( i ) as a set of arms that have been rotted and pulled at least once in the i -th block, which satisfies |A S ( i ) | ≤ S i , where S i is defined as the number of abrupt changes in the i -th block. Let µ [ t 1 ,t 2 ] ( a ) = ∑ t 2 t = t 1 µ t ( a ) /n [ t 1 ,t 2 ] ( a ) . We define the event E 1 = {| ̂ µ [ s 1 ,s 2 ] ( a ) -µ [ s 1 ,s 2 ] ( a ) | ≤ √ 12 log( H ) /n [ s 1 ,s 2 ] ( a ) for all 1 ≤ s 1 ≤ s 2 ≤ H,a ∈ A ( i ) } . From Lemma A.30, as in (6), we have

For the case that E 1 does not hold, the regret is E [ R π ( H ) | E c 1 ] P ( E c 1 ) = O (1) , which is negligible comparing with the regret when E 1 holds true which we show later. Therefore, in the rest of the proof we assume that E 1 holds true.

In the following, we first provide a regret bound over the first block.

For regret analysis, we divide R π 1 ( δ † ) 1 ( H ) into two parts, R G ( H ) and R B ( H ) corresponding to regret of good or near-good arms, and bad arms over time T , respectively, such that R π 1 ( H ) = R G ( H ) + R B ( H ) . We can easily obtain that

from ∆( a ) ≤ 2 δ † for any good or near-good arms a .

Now we analyze R B ( H ) . We divide regret R B ( H ) into two regret from bad arms in A (1) / A S (1) , denoted by R B , 1 ( H ) , and regret from bad arms in A S (1) , denoted by R B , 2 ( H ) such that R B ( H ) = R B , 1 ( H ) + R B , 2 ( H ) . We first analyze R B , 1 ( H ) in the following. We consider arms in A (1) / A S (1) . For the proof, we adopt the episodic approach in Kim et al. [13] for regret analysis. In the following, we introduce some notation. Here we only consider arms in A (1) / A S (1) so that the following notation is defined without considering (rotted) arms in A S (1) . Given a policy sampling arms in the sequence order, let m G be the number of samples of distinct good arms and m B i be the number of consecutive samples of distinct bad arms between the i -1 -st and i -th sample of a good arm among m G good arms. We refer to the period starting from sampling the i -1 -st good arm before sampling the i -th good arm as the i -th episode . Observe that m B 1 , . . . , m B m G 's are i.i.d. random variables with geometric distribution with parameter C (2 δ † ) β for some constant C > 0 , conditional on the value of m G . Therefore, P ( m B i = k ) = (1 -C (2 δ † ) β ) k C (2 δ † ) β , for i = 1 , . . . , m G .

Define ˜ m G H to be the total number of samples of a good arm by the policy π 1 ( δ † ) over the horizon H and ˜ m B i,H to be the number of selections of a bad arm in the i -th episode by the policy π over the horizon H . For i ∈ [ ˜ m G H ] , j ∈ [ ˜ m B i,H ] , let ˜ n G i be the number of pulls of the good arm in the i -th episode and ˜ n B i,j be the number of pulls of the j -th bad arm in the i -th episode by the policy π 1 ( δ † ) over the horizon H . Let ˜ a be the last sampled arm over time horizon H by π 1 ( δ † ) .

With a slight abuse of notation, we use π 1 ( δ † ) for a modified strategy after H . Under a policy π 1 ( δ † ) , let R B be the regret (summation of mean reward gaps) contributed by pulling the j -th bad arm in the i -th episode. Then let R B m G = ∑ i =1 ∑ j ∈ [ m B i ] R B i,j , which is the regret from initially bad arms over the period of m G episodes. For getting R B m G , here we define the policy π 1 ( δ † ) after H such that it pulls H amounts for a good arm and zero for a bad arm. After H we can assume that there are no abrupt changes. For the last arm ˜ a over the horizon H , it pulls the arm up to H amounts if ˜ a is a good arm and ˜ n G ˜ m G H < H . For i ∈ [ m G ] , j ∈ [ m B i ] let n G i and n B i,j be the number of pulling the good arm in i -th episode and j -th bad arm in i -th episode under π , respectively. Here we define n G i 's and n B i,j 's as follows:

i,j m G

If ˜ a is a good arm,

Otherwise,

With a slight abuse of notation, we define S i to be the number of abrupt changes in i -th block. Then, we show that if m G = S 1 , then R B , 1 ( H ) ≤ R B m G .

Lemma A.24. Under E 1 , when m G = S 1 we have

Proof. There are at most S 1 -1 number of abrupt changes over the first block H . We consider two cases; there are S 1 -1 abrupt changes before sampling S 1 -th good arm or not. For the first case, if π 1 ( δ † ) samples the S 1 -th good arm and there are S 1 -1 number of abrupt changes before sampling the good arm, then it continues to pull the good arm for H rounds from E 1 and the definition of π 1 ( δ † ) after H .

Now we consider the second case. If π 1 ( δ † ) samples the S 1 -th good arm before T and there is at least one abrupt change after sampling the arm, then before sampling the S 1 -th good arm, there must exist two consecutive good arms such that there is no abrupt change between sampling the two good arms.

This is a contraction because π 1 ( δ † ) must pull the first good arm up to H under E 1 and S 1 -1 -st abrupt change must occur after H .

Therefore, considering the first case, when m G = S 1 +1 , we have

which implies R B ( H ) ≤ R B m G .

From the above lemma, we set m G = S 1 and analyze R B m G to get a bound for R B , 1 ( H ) in the following lemma.

Lemma A.25. Under E 1 and policy π 1 ( δ † ) , we have

Proof. We can show this theorem by following the proof steps in Lemma A.8.

Now we analyze R B , 2 ( H ) in the following lemma. We denote by V H a cumulative amount of rotting rates in the first block.

Lemma A.26. Under E 1 and policy π , we have

Proof. We can show this theorem by following the proof steps in Lemma A.9.

From R π 1 ( H ) = R G ( H ) + R B ( H ) , (42), and (43), we have

The above regret is for the first block. Therefore, by summing regrets over ⌈ T/H ⌉ number of blocks, we have shown that

Upper bounding E [ R π 2 ( T )] . By following the proof steps in Theorem A.15, we have

Finally, from (41), (44), and (45), with the fact that ∑ S T s =1 E [ ρ t ( s ) ] ≤ V T , H = T 1 / 2 , and δ † = Θ(max { ( S T /T ) 1 / ( β +1) , 1 /H 1 / ( β +1) , ( S T /T ) 1 / 2 , 1 /H 1 / 2 } ) , we have

which concludes the proof.

A.8 Proof of Theorem 4.1: Regret Lower Bound for Slowly Rotting Rewards

We first consider the case when V T = Θ( T ) . Recall that ∆ 1 ( a ) = 1 -µ 1 ( a ) . Then for any randomly sampled a ∈ A , we have E [ µ 1 ( a )] ≥ y P ( µ 1 ( a ) ≥ y ) = y P (∆ 1 ( a ) < 1 -y ) for y ∈ [0 , 1] . Then with y = 1 / 2 , we have E [ µ 1 ( a )] ≥ (1 / 2) P (∆ 1 ( a ) < (1 / 2)) = Θ(1) from constant β > 0 and (1). Then with E [ µ 1 ( a )] ≤ 1 , we have E [ µ 1 ( a )] = Θ(1) . We then think of a policy π ′ that randomly samples a new arm and pulls it once every round. Since E [ µ 1 ( a )] = Θ(1) for any randomly sampled a , we have E [ R π ′ ( T )] = Θ( T ) . Next, we think of any policy π ′′ except π ′ . Then any policy π ′′ must pull an arm a at least twice. Let t ′ and t ′′ be the rounds when the policy pulls arm a . If we consider ρ t ′ = V T then such policy has Ω( V T ) regret bound. Since V T = Θ( T ) , any algorithm has Ω( T ) in the worst case. Therefore we can conclude that any algorithm including π ′ has a regret bound of Ω( T ) in the worst case, which concludes the proof for V T = Θ( T ) .

Now we think of the case where V T = o ( T ) . For the lower bound, we adopt the proof methodology of Theorem 1 in Kim et al. [13] by making necessary adjustments to accommodate V T and β . We note that since higher β implies a reduced chance of sampling a near-optimal arm, the criteria for defining the mean rewards of near-optimal arms becomes less stringent for higher β , which does not appear in the previous work. We first categorize arms as either bad or good according to their initial mean reward values. For the categorization, we utilize two thresholds in the proof as follows. Consider 0 < γ < c < 1 for γ , which will be specified, and a constant c . Then the value of 1 -γ represents a threshold value for identifying good arms, while 1 -c serves as the threshold for identifying bad arms. We refer to arms a satisfying µ 1 ( a ) ≤ 1 -c as 'bad' arms and arms a satisfying µ 1 ( a ) > 1 -γ as 'good' arms. We also consider a sequence of arms in A denoted by ¯ a 1 , ¯ a 2 , . . . . Given a policy π , without loss of generality, we can assume that π selects arms according to the order of ¯ a 1 , ¯ a 2 , . . . . For the rotting rates, we define ϱ = V T / ( T -1) . Then we consider ρ t = ϱ for all t ∈ [ T -1] so that ∑ T -1 t =1 ρ t = V T .

Case of V T = O (1 /T 1 / ( β +1) ) : When V T = O (1 /T 1 / ( β +1) ) , the lower bound of order T β β +1 for the stationary case, from Theorem 3 in Wang et al. [24], is tight enough for the non-stationary case. From Theorem 3 in Wang et al. [24], we have

We note that even though the mean rewards are rotting in our setting, Theorem 3 in Wang et al. [24] remains applicable without requiring any alterations in the proofs providing a tight regret bound for the near-stationary case. For the sake of completeness, we provide the proof of the theorem in the following. Let K 1 denote the number of bad arms a that satisfy µ 1 ( a ) ≤ 1 -c before sampling the first good arm, which satisfies µ 1 ( a ) > 1 -γ , in the sequence of arms ¯ a 1 , ¯ a 2 , . . . . Let µ be the initial mean reward of the best arm among the sampled arms by π over time horizon T . Then for some κ > 0 , we have

By taking expectations on the both sides in (47) and setting κ = Tγ/c , we have

We observe that K 1 follows a geometric distribution with success probability P ( µ 1 ( a ) > 1 -γ ) /p ( µ 1 ( a ) / ∈ (1 -c, 1 -γ ]) = γ ≤ C 1 γ β / (1 + C 2 γ β -C 3 c β ) for some constants C 1 , C 2 , C 3 > 0 from (1), in which the success probability is the probability of sampling a good arm given that the arm is either a good or bad arm. Here we set a constant 0 < c < 1 satisfying 1 -C 3 c β > 0 . Then by setting γ = 1 /T 1 β +1 with κ = T β β +1 /c , for some constant C > 0 we have

where the last equality is obtained from log x ≥ 1 -1 /x for all x > 0 .

Case of V T = ω (1 /T 1 / ( β +1) ) and V T = o ( T ) : When V T = ω (1 /T 1 / ( β +1) ) , however, the lower bound of the stationary case is not tight enough. Here we provide the proof for the lower bound of V 1 / ( β +2) T T ( β +1) / ( β +2) for the case of V T = ω (1 /T 1 / ( β +1) ) . Let K m denote the number of 'bad" arms a that satisfy µ 1 ( a ) ≤ 1 -c before sampling m -th 'good" arm, which satisfies µ 1 ( a ) > 1 -γ , in the sequence of arms ¯ a 1 , ¯ a 2 , . . . . Let N T be the number of sampled good arms a such that µ 1 ( a ) > 1 -γ until T .

We can decompose R π ( T ) into two parts as follows:

R π ( T ) = R π ( T ) ✶ ( N T < m ) + R π ( T ) ✶ ( N T ≥ m ) . (48) We set m = ⌈ (1 / 2) T 1 / ( β +2) V ( β +1) / ( β +2) T ⌉ and γ = ( V T /T ) 1 / ( β +2) with V T = o ( T ) . For the first term in (48), R π ( T ) ✶ ( N T < m ) , we consider the fact that the minimal regret is obtained from the situation where there are m -1 arms whose mean rewards are 1 . In such a case, the optimal policy must sample the best m -1 arms until their mean rewards become below the threshold 1 -γ (step 1) and then samples the best arm at each time for the remaining time steps (step 2). The number of times each arm needs to be pulled for the best m -1 arms until their mean reward falls below 1 -γ is bounded from above by γ/ρ +1 = γ (( T -1) /V T ) + 1 . Therefore, the regret from step 2 is R = Ω(( T -mγ ( T/V T )) γ ) = Ω( T ( β +1) / ( β +2) V 1 / ( β +2) T ) in which the optimal policy pulls arms which mean rewards are below 1 -γ for the remaining time after step 1. Therefore, we have

✶ ✶ ✶ For getting a lower bound of the second term in (48), R π ( T ) ✶ ( N T ≥ m ) , we use the minimum number of sampled arms a that satisfy µ 1 ( a ) ≤ 1 -c. When N T ≥ m and K m ≥ κ , the policy samples at least κ number of distinct arms a satisfying µ 1 ( a ) ≤ 1 -c until T . Therefore, we have

R π ( T ) ✶ ( N T ≥ m ) ≥ cκ ✶ ( N T ≥ m,K m ≥ κ ) . (50) We have γ = Θ( γ β ) from (1) with constant β > 0 . By setting κ = m/γ -m - √ m/γ , with V T = o ( T ) and constant β > 0 , we have

Then from (49), (50), and (51), we have

Next we provide a lower bound for P ( K m ≥ κ ) . Observe that K m follows a negative binomial distribution with m successes and the success probability P ( µ 1 ( a ) > 1 -γ ) / P ( µ 1 ( a ) / ∈ (1 -c, 1 -γ ]) = γ , in which the success probability is the probability of sampling a good arm given that the arm is either a good or bad arm. In the following lemma, we provide a concentration inequality for K m .

Lemma A.27. For any 1 / 2 + γ/m < α < 1 ,

Proof. Let X i for i > 0 be i.i.d. Bernoulli random variables with success probability γ. From Section 2 in Brown [9], we have

From (53) and Lemma A.29, for any 1 / 2 + γ/m < α < 1 we have

in which the first inequality comes from Lemma A.29, which concludes the proof.

From Lemma A.27 with α = 1 -1 / √ m and large enough T , we have

Therefore, from (52) and (54), we have

Finally, from (46) and (55), we conclude that for any policy π , we have

A.9 Proof of Theorem 4.2: Regret Lower Bound for Abruptly Rotting Rewards

First, we deal with the case when S T = 1 or S T = Θ( T ) . When S T = 1 (implying V T = 0 ), from the definition, the problem becomes stationary without rotting instances, which implies E [ R π ( T )] = Ω( √ T ) from Theorem 3 in Wang et al. [24]. When S T = Θ( T ) , we consider that rotting occurs for the first S T -1 rounds with ρ t = 1 for all t ∈ [ S T -1] . Then it is always beneficial to pull new arms every round until S T -1 rounds because the mean rewards of rotted arms are below 0 and those of non-rotted arms lie in [0 , 1] . This means that any ideal policy samples a new arm and pulls it every round until S T -1 . Then for any randomly sampled a ∈ A , we have E [ µ 1 ( a )] ≥ y P ( µ 1 ( a ) ≥ y ) = y P (∆ 1 ( a ) < 1 -y ) for y ∈ [0 , 1] . Then with y = 1 / 2 , we have E [ µ 1 ( a )] ≥ (1 / 2) P (∆ 1 ( a ) < (1 / 2)) = Θ(1) from constant β > 0 and (1). Then with E [ µ 1 ( a )] ≤ 1 , we have E [ µ 1 ( a )] = Θ(1) . Since E [ µ 1 ( a )] = Θ(1) for any randomly sampled a ∈ A , any ideal policy has E [ R π ( T )] ≥ ∑ S T i =1 E [ µ 1 ( a )] = Ω( S T ) = Ω( T ) , which concludes the proof for S T = Θ( T ) .

Now we consider the case of S T = o ( T ) and S T ≥ 2 . We initially provide a regret bound with respect to the cumulative rotting amount of ¯ V T . We first think of a policy π that randomly samples a new arm and pulls it once every round. Then for any randomly sampled a ∈ A , we have E [ µ 1 ( a )] = Θ(1) . Then from constant β > 0 , E [ R π ( T )] = Ω( T ) . Then there always exists ρ t 's satisfying ∑ T -1 t =1 ρ t = T , which implies E [ R π ( T )] = Ω( T ) = Ω( ¯ V T ) .

̸

Now we think of any nontrivial algorithm which must pull an arm a at least twice. Let t ′ and t ′′ be the rounds when the policy pulls arm a ( t ′ < t ′′ ) . If we consider ρ t ′ > 0 and ρ t = 0 for t ∈ [ T -1] / { t ′ } in which ρ t ′ = ∑ T -1 t =1 ρ t and 1 + ∑ T -1 t =1 ρ t ✶ ( ρ t = 0) ≤ S T , then such policy has R π ( T ) = Ω( ∑ T -1 t =1 ρ t ) regret bound because, at time t ′′ , it pulls the arm a rotted by ρ t ′ . Therefore, for any policy π , there always exist a rotting rate adversary satisfying the following expected regret bound of

Next, for the regret bound with respect to S T , we follow the proof steps in Theorem 4.1. However, the regret bound of S T does not depend on the magnitude of rotting rates but on the number of rotting instances. To address this, we need to design a new worst-case in which an adversary makes near-optimal arms rotted to be sub-optimal arms abruptly rather than gradually. We first categorize arms as either bad or good according to their initial mean reward values. For the categorization, we utilize two thresholds in the proof as follows. Consider 0 < γ < c < 1 for γ , which will be specified, and a constant c . Then the value of 1 -γ represents a threshold value for identifying good arms, while 1 -c serves as the threshold for identifying bad arms. We refer to arms a satisfying µ 1 ( a ) ≤ 1 -c as 'bad' arms and arms a satisfying µ 1 ( a ) > 1 -γ as 'good' arms. We also consider a sequence of arms in A denoted by ¯ a 1 , ¯ a 2 , . . . . Given a policy π , without loss of generality, we can assume that π selects arms according to the order of ¯ a 1 , ¯ a 2 , . . . .

Let K m denote the number of bad arms a that satisfy µ 1 ( a ) ≤ 1 -c before sampling m -th good arm, which satisfies µ 1 ( a ) > 1 -γ , in the sequence of arms ¯ a 1 , ¯ a 2 , . . . . Let N T be the number of sampled good arms a such that µ 1 ( a ) > 1 -γ until T .

We can decompose R π ( T ) into two parts as follows:

R π ( T ) = R π ( T ) ✶ ( N T < m ) + R π ( T ) ✶ ( N T ≥ m ) . (57) We set m = S T and γ = ( S T /T ) 1 / ( β +1) with S T = o ( T ) . For getting a lower bound for the first term in (57), R π ( T ) ✶ ( N T < m ) , we consider the fact that the minimal regret is obtained from the situation where there are m -1 arms whose mean rewards are 1 . In such a case, the optimal policy must sample the best m -1 arms until their mean rewards become equal to or below the threshold value of 1 -γ (step 1) and then samples the best arm at each time for the remaining time steps (step 2). In step 1, when the optimal policy pulls an optimal arm, we can think of the case when the mean reward of the arm is abruptly rotted to the value of 1 -γ . This implies that the required number of rounds for step 1 is m -1 . The regret from step 2 is R = Ω(( T -m +1) γ ) = Ω( S 1 / ( β +1) T T β/ ( β +1) ) , in which the optimal policy pulls arms which mean rewards are below or equal to 1 -γ for the remaining time after step 1. Therefore, we have

✶ ✶ ✶ For getting the above, we note that there always exists ρ t 's satisfying ∑ T -1 t =1 ρ t = O ( γm ) = o ( T ) , which implies ∑ T -1 t =1 ρ t ≤ T . Such ρ t 's can be considered for the below. For getting a lower bound of the second term in (57), R π ( T ) ✶ ( N T ≥ m ) , we use the minimum number of sampled arms a that satisfy µ 1 ( a ) ≤ 1 -c. When N T ≥ m and K m ≥ κ , the policy samples at least κ number of distinct arms a satisfying µ 1 ( a ) ≤ 1 -c until T . Therefore, we have

R π ( T ) ✶ ( N T ≥ m ) ≥ cκ ✶ ( N T ≥ m,K m ≥ κ ) . (59) We set γ = P ( µ 1 ( a ) > 1 -γ ) /p ( µ 1 ( a ) / ∈ (1 -c, 1 -γ ]) . Then we have γ = Θ( γ β ) from (1) with constant β > 0 . By setting κ = m/γ -m -m/ ( γ √ m +3) , with S T = o ( T ) and constant β > 0 , we have

Then from (58), (59), and (60), we have

Next we provide a lower bound for P ( K m ≥ κ ) . Observe that K m follows a negative binomial distribution with m successes and the success probability P ( µ 1 ( a ) > 1 -γ ) / P ( µ 1 ( a ) / ∈ (1 -c, 1 -γ ]) = γ , in which the success probability is the probability of sampling a good arm given that the arm is either a good or bad arm. We recall Lemma A.27 for a concentration inequality for K m in the following.

From Lemma A.28 with α = 1 -1 / √ m +3 and large enough T , we have

where the last inequality comes from m/ ( m +3) = ( S T ) / ( S T +3) ≥ 1 / 4 and √ m +3 / ( √ m +3 -1) ≥ 1 . Therefore, from (61) and (62), we have

Overall from (56) and (63), for any π , there exist ρ t 's such that E [ R π ( T )] = Ω(max { S 1 / ( β +1) T T β/ ( β +1) , V T } ) .

A.10 Additional Experiments

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We compare the performance of our Algorithms with benchmarks for smaller or larger β . In Figure 5 (a,b), we can observe that our algorithms outperform the benchmarks for β = 0 . 5 and β = 2 .

A.11 Lemmas for Concentration Inequalities

Lemma A.29 (Theorem 6.2.35 in Tsun [22]) . Let X 1 , . . . , X n be identical independent Bernoulli random variables. Then, for 0 < ν < 1 , we have

Lemma A.30 (Corollary 1.7 in Rigollet and Hütter [18]) . Let X 1 , . . . , X n be independent random variables with σ -sub-Gaussian distributions. Then, for any a = ( a 1 , . . . , a n ) ⊤ ∈ R n and t ≥ 0 , we have

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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studied an extension of multi-armed bandit problem by introducing infinitely many arms and generalized rotting constraints. It provides explicit regret lower bounds and proposes an algorithm with regret upper bound matching the lower bound when $\\beta \\geq 1$. It has been claimed that closing the gap between the lower and upper bounds when $\\beta \\in (0,1) $ remains an open problem.\n\nSTRENGTHS:\n- Extending the multi-armed bandit problem to infinitely many arms with rotting mean rewards is feasible in many real-world applications. Solid theoretical analyses and empirical results are presented to justify the proposed solution.\n\n- The paper in general is well-written and easy to follow.  Since I have not checked the supplementary material step by step, I can not guarantee the correctness of the proofs, but the theoretical results make sense to me.\n\nWEAKNESSES:\n- The paper could be viewed as an extension of [1], and the impact of the paper might not be extremely significant. As mentioned in the paper, when $\\beta \\in (0,1)$, the proposed algorithm can not be proved near optimal at the current stage. The theoretical result could be more impactful if this issue can be solved.\n\n[1] Jung-hun Kim, Milan Vojnovic, and Se-Young Yun. Rotting infinitely many-armed bandits. ICML, pages 11229–11254. PMLR, 2022.\n\n- When the environment parameters $\\beta, V_T,$ and $S_T$ are unknown, a significant amount of additional regret is generated with proposed Algorithm 2. \n\n- The paper could benefit from a more extensive experiment study. For example, the algorithm performances can be compared under different rotting processes (perhaps by varying the rotting rate and adding randomness).\n\nQUESTIONS:\nThe following questions are raised simply for discussion. There is no need to address them in the paper.\n\n- Why the rotting budget $V_T$ and $S_T$ are defined for all selected arms? I would imagine it can fit better with real-world problems to treat the rotting process for each arm independently. Saying if an arm is selected $N$ times, the rotting budget is $f(N)$ where $f$ is a nondecreasing function.\n\n- Corresponding to my second point in Weakness, is it possible to address the unknown parameters without incurring large additional regret? Can the methodology in the following paper [2] be applied in this paper to address the nonstationary rotting rewards? Can $\\beta$ be estimated by selecting multiple new arms?\n\n[2] Chen, Yifang, et al. \"A new algorithm for non-stationary contextual bandits: Efficient, optimal and parameter-free.\" Conference on Learning Theory. PMLR, 2019.\n\nLIMITATIONS:\nThe authors adequately addressed the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the extension of the classic stochastic multi-armed bandit problem to the case where there is a) an infinite set of arms and b) rotting of the arm means. Specifically the rotting behaviour is of the 'rested bandit' variety, where the mean reward of an arm may fall as an immediate consequence of playing it, but the mean of an unplayed arm will not change, in contrast to the restless bandit. The infinite set of arms have i.i.d. mean rewards (no structural assumptions on the set of arms or existence of a reward function over them etc.) and reward observations are sub-Gaussian.\n\nThis paper considers a more general rotting behaviour than Kim et al. (2022, ICML). In Kim et al. (2022) the magnitude of rotting in each round was bounded, in the present paper the more general setting where the total amount of rotting in T rounds is bounded is considered, and the somewhat different setting where the number of rounds in which rotting occurs is considered. These two settings are referred to as the 'slow' and 'abrupt' rotting cases respectively.\n\nAn important parameter in the infinitely-many-armed bandit problem is $\\beta>0$ which controls the probability of an arm being $\\delta$-near optimal. All regret bounds in the paper for mean rewards in [0,1] and consider dependence on the number of rounds $T$ and parameter $\\beta$. $\\beta=1$ corresponds to a uniform distribution on arm means.\n\nThe paper derives a sliding-window-UCB-based approach, which can be tuned to both the slow and abrupt rotting cases when the bounds on rotting behaviour are known, and an adaptive version which aims to learn the rotting behaviour when these parameters are not known. The paper shows tight (in terms of order) regret guarantees on these algorithms, and improved empirical performance over sensible competitors.\n\nSTRENGTHS:\nThe paper studies an interesting extension of prior work to allow for more general rotting constraints. This is likely to be of interest to the multi-armed bandit community and both the methodological and theoretical work present some novelty and careful algorithmic design that merit publication.\n\nI was pleased to see a treatment of the case where key problem parameters were not known, and that largely comparable bounds were achievable in this setting.\n\nGenerally, the paper is written well and concisely, with useful clarifications made around the most important details and a clear relationship to the most closely related prior work.\n\nWEAKNESSES:\nThere are three main weaknesses which I would like to see commented on the rebuttal phase. I think the first is difficult to address fully in a rebuttal phase, but I would like to be reassured that potential issues do not limit the scope of the contribution.\n\n1. All of the theoretical results are ultimately order results only, without identification of the constants in the appendices, and some being somewhat vaguely described as needing to be 'large enough'. The paper would be stronger if these constants (or upper bounds on them) could be identified, to guarantee for which values of $T$ the bounds are meaningful. It would be helpful if the bounds could thus be realised for the cases considered in the experiments and compared to the actual regret of Algorithms 1 and 2.\n\n2. If I am not mistaken the experiments only consider $\\beta=1$? It would be interesting to see if the convergence of the algorithm is replicated for various $\\beta \\neq 1$.\n\n3. The motivation in terms of real applications is not especially strong. It is difficult to imagine a setting where the precise assumptions of the paper are realistic. In a recommendation setting, is it likely that there will be sufficient information to be confident that the click through rate of an item is always non-increasing, yet there is no contextual information available to supplement the model or exploit similar click through rates across similar items? In the clinical setting, is it likely that the rested assumption is realistic (i.e. any loss in efficacy is only determined by the actions of a single decision maker) alongside the immediacy of feedback and time horizons being of a scale such that the algorithms actually exhibit non-linear regret? If it is the case that the benefit of the paper is more fundamental - it presents an understanding of a simpler problem and its insights serve as a foundation to tackling these more complex real-world challenges, then perhaps it would be more helpful to make that clear?\n\nQUESTIONS:\n1. Can you provide clarity on whether your constants in the regret analysis are identifiable, and suggest the values of these for some simple instance of the problem? How would the resultant bounds compare to the empirical performance of Algorithms 1 and 2?\n\n2. What would the experimental results look like for $\\beta \\neq 1$?\n\n3. Is Prop 2.3 missing some kind of additional condition, or clarity over what is meant by worst-case in line 453? It would seem to be that you could construct an example where $\\rho_1=T+1$ and all other $\\rho=0$ and one could achieve $\\sqrt{T}$ regret while satisfying $\\sum{t=1}^{T-1}\\rho_t >T$?\n\n4. Can you put the WUCB definition in an equation display and reference its equation number in the statement of Algorithm 1?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-02"}

comments: SUMMARY:
Authors investigated an adaptive approach for the rotting bandits problem under the infinitely arms assumption.

STRENGTHS:
Authors introduced a new ucb like policy for the mentioned problem, additionally a lower bound analysis has been carried on.

WEAKNESSES:
No clear weaknesses

QUESTIONS:
I wonder how the regret bound behaves with respect to the effective rotting instead of the V_T or S_T. Assuming their values to be large and far from the real rotting, would it be possible to extend this analysis and solution to propose an adaptive result? Additionally, I wonder if also the regret lower bound would be tight in that case.

Similarly, I was wondering how good these results would be for the infinitely many afrmed bandit with no rotting.

LIMITATIONS:
No clear limitations have been found.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-11

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies infinite-armed bandits with rotting rewards. They show a lower bound on regret in terms of the total-variation and number of abrupt changepoints in the change in rewards. They also provide regret upper bounds: (1) a UCB-like algorithm tuned with knowledge of problem parameters gets optimal regret in some regimes and (2) a parameter-free bandit-over-bandit algorithm can attain optimal regret in some regimes when the level of non-stationarity is large.\n\nSTRENGTHS:\n* The infinite-armed bandit problem is well motivated and the problem and results are clearly presented.\n* There are matching upper and lower bounds on regret, at least for the better-understood $\\beta \\geq 1$ setting.\n\nWEAKNESSES:\n* The definition of $V_T,S_T$ are a bit unclear and may not be fully rigorous. They are bounds on the total amount and count of rotting, but the rotting $\\rho_t$ depends on the chosen arm $\\rho_t$ and is thus random (in fact, determined by an adaptive adversary according to Assumption 2.1). But, meanwhile, $V_T,S_T$ seem to be treated as deterministic constants in the whole paper given that they are, for example, used to tune Algorithm 1 to get the best regret bounds. In fact, it would not even make sense for $V_T,S_T$ to _appear_ in the expected regret bounds if they were random. So, the only way the results of this work can rigorously hold is if $V_T,S_T$ bound the respective random quantities $\\sum_{t=1}^{T_1} \\rho_t$ and $1 + \\sum_{t=1}^{T-1} 1\\{\\rho_t\\neq 0\\}$ for all realizations of the randomness, which then means there are some missing assumptions for the results of this paper. For instance, even for a \"nice\" algorithm, $\\sum_{t=1}^{T_1} \\rho_t$ could be very large with some probability in which case $V_T$ is forced to be large as well because of a \"bad realization\". The only other way around this is to make further assumptions about the adversary/design of non-stationarity. In either case, there should be more discussion as the rigor of results presented versus the given assumptions seems unclear to me.\n*  There are no optimal regret upper bounds without parameter knowledge. Also, the bandit-over-bandit result for parameter-free result requires further assumptions on the non-stationarity and so does not apply as generally as the other results of this paper.\n\nQUESTIONS:\nPlease refer to strengths and weaknesses above.\n\nLIMITATIONS:\nNo broader impact concerns.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}