Linear Contextual Bandits with Adversarial Corruptions

We study the linear contextual bandits problem in the presence of adversarial1 corruption , where the interaction between the player and a possibly infinite decision2 set is contaminated by an adversary that can corrupt the reward up to a corruption3 level C measured by the sum of the largest alteration on rewards in each round.4 We present a variance-aware algorithm that is adaptive to the level of adversarial5 contamination C. The key algorithmic design includes ( 1 ) a multi-level partition6 scheme of the observed data , ( 2 ) a cascade of confidence sets that are adaptive to7 the level of the corruption , and ( 3 ) a variance-aware confidence set construction8 that can take advantage of low-variance reward . We further prove that the regret9 of the proposed algorithm is Õ ( C2d √∑T t=1 σ 2 t + C 2 √ dT + CR √ dT ) , where10 d is the dimension of context vectors , T is the number of rounds , R is the range11 of noise and σ2t , t = 1 . . . , T are the variances of instantaneous reward . We also12 prove a gap-dependent regret bound for the proposed algorithm , which is instance-13 dependent and thus leads to better performance on good practical instances . To the14 best of our knowledge , this is the first variance-aware corruption robust algorithm15 for contextual bandits.16 1 Introduction17 Multi-armed bandits algorithms are widely applied in online advertising ( Li et al. , 2010 ) , clinical18 trials ( Villar et al. , 2015 ) , recommendation system ( Deshpande and Montanari , 2012 ) and many other19 real-world tasks . In the model of multi-armed bandits , the algorithm needs to decide which action20 ( or arm ) to take ( or pull ) at each round and receive a reward for the chosen action . In the stochastic21 setting , the reward is subject to a fixed but unknown distribution for each action . In reality , however,22 these rewards can easily be “ corrupted ” by some malicious users . A typical example is click fraud23 ( Lykouris et al. , 2018 ) , where botnets simulate the legitimate users clicking on an ad to fool the24 recommendation systems . This motivates the studies of the bandits algorithms that are robust to25 adversarial corruptions.26 For example , Lykouris et al . ( 2018 ) introduced a bandit model in which an adversary could corrupt27 the stochastic reward generated by an arm pull . They proposed an algorithm and show that the28 regret of this “ middle ground ” scenario degrades smoothly with the amount of corruption injected29 by the adversary . Gupta et al . ( 2019 ) proposed an alternative algorithm which gives a significant30 improvement in regret.31 While the algorithms that are robust to the corruptions have been studied in the setting of multi-armed32 bandits in a number of prior works , they are still understudied in the setting of linear contextual33 bandits . The linear contextual bandits problem can be regarded as an extension of the multi-armed34 bandit problem to linear optimization , in order to tackle an unfixed and possibly infinite set of feasible35 Submitted to 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Do not distribute . actions . There is a large body of literature on efficient algorithms for linear contextual bandits with36 no corruptions ( Abe et al. , 2003 ; Auer , 2002 ; Chu et al. , 2011 ; Dani et al. , 2008 ; Rusmevichientong37 and Tsitsiklis , 2010 ; Abbasi-Yadkori et al. , 2011 ; Li et al. , 2019b ) , to mention a few . The significance38 of this setting lies in the fact that linear regression approaches are widely used in recommendation39 systems and advertising ( Li et al. , 2010 ; Jhalani et al. , 2016 ; Deshpande and Montanari , 2012 ) . Linear40 contextual bandits with adversarial corruptions is an arguably more challenging setting since most of41 the previous corruption-robust algorithms are based on the idea of action elimination ( Lykouris et al.,42 2018 ; Gupta et al. , 2019 ; Bogunovic et al. , 2021 ) , which is not applicable to the contextual bandits43 settings where the decision set is time varying and possibly infinite at each round . In Garcelon et al.44 ( 2020 ) , it is shown that a malicious agent can force a linear contextual bandit algorithm to take any45 desired action T − o ( T ) times over T rounds , while applying adversarial corruptions to rewards with46 a cumulative cost that only grow logarithmically . This poses a big challenge for designing corruption47 robust algorithms for linear contextual bandits.48 In this paper , we make a first attempt to study a linear contextual bandit model where an adversary49 can corrupt the rewards up to a corruption level C , which is defined as the the sum of biggest50 alteration the adversary made on rewards in each round . We propose a linear contextual bandits51 algorithm that is robust to reward corruption , dubbed multi-level optimism-in-the-face-of-uncertainty52 weighted learning ( Multi-level OFUL ) . More specifically , our algorithm consists of the following53 novel techniques : ( 1 ) We design a multi-level partition scheme and adopt the idea of sub-sampling to54 do the robust estimation of the model parameters ; ( 2 ) We maintain a cascade of candidate confidence55 sets corresponding to different corruption level ( which is unknown ) and randomly select a confidence56 set at each round to take the action ; and ( 3 ) We design confidence sets that depend on the variances57 of rewards , which lead to a potentially tighter regret bound.58 Our contributions are summarized as follows:59 • We propose a variance-aware algorithm which is adaptive to the amount of adversarial corruptions60 C. To the best of our knowledge , it is the first algorithm for the setting of linear contextual bandits61 with adversarial corruptions which does not rely on the finite number of actions and other additional62 assumptions.63 • We prove that the regret of our algorithm is in Õ ( C2d √∑T t=1 σ 2 t + C 2 √ dT + CR √ dT ) , where64 d is the dimension of context vectors , T is the number of rounds , R is the range of noise and σ2t , t =65 1 . . . , T are the variances of instantaneous reward . Our regret upper bound has a multiplicative66 dependence on C2 which indicates that our algorithm achieves a sub-linear regret when the67 corruption level satisfies C = o ( T 1/4 ) .68 • We also derive a gap-dependent regret bound Õ ( 1 ∆ · C 2R2d+ 1∆ · d 2C2 maxt∈ [ T ] σ 2 t ) for our69 proposed algorithm , which is instance-dependent and thus leads to a better performance on good70 practical instances.71 Notation . We use lower case letters to denote scalars , and use lower and upper case bold face letters72 to denote vectors and matrices respectively . We denote by [ n ] the set { 1 , . . . , n } . For a vector x ∈ Rd73 and matrix Σ ∈ Rd×d , a positive semi-definite matrix , we denote by ‖x‖2 the vector ’ s Euclidean74 norm and define ‖x‖Σ = √ x > Σx . For two positive sequences { an } and { bn } with n = 1 , 2 , . . . ,75 we write an = O ( bn ) if there exists an absolute constant C > 0 such that an ≤ Cbn holds for all76 n ≥ 1 and write an = Ω ( bn ) if there exists an absolute constant C > 0 such that an ≥ Cbn holds77 for all n ≥ 1 . We use Õ ( · ) to further hide the polylogarithmic factors . We use 1 ( · ) to denote the78 indicator function.79 2 Related Work80 Bandits with Adversarial Attacks : There is a large body of literature on the problems of multi-81 armed bandits with adversarial corruptions . Most research in this area aims to design algorithms that82 achieve desirable regret bound in both stochastic multi-armed bandits and adversarial bandits , known83 as “ the best of both worlds ” guarantees ( Bubeck and Slivkins , 2012 ; Seldin and Slivkins , 2014 ; Auer84 and Chiang , 2016 ; Seldin and Lugosi , 2017 ; Zimmert and Seldin , 2019 ) . These works mainly focus85 on achieving bounds in the worst case and the case where there is no adversary . As a result , these86 algorithms are either not robust to instances that moderate amount of corruptions occur , or suffer87 from restrictive assumptions on adversarial corruptions . Distinctive from the above line of research,88 Lykouris et al . ( 2018 ) focus on a variant of classic multi-armed bandit model in which each pull of an89 arm generates a stochastic reward that may be contaminated by an adversary before it is revealed90 to the player . In their work , the corruption level C is defined as C = ∑ t maxa |rt ( a ) − rtS ( a ) |91 where rtS ( a ) is the stochastic reward of arm a and r t ( a ) is the corrupted reward of arm a at round t.92 They develop algorithms adaptive to the unknown corruption level , which achieves an O ( K1.5C √ T ) 93 regret bound . Gupta et al . ( 2019 ) proposed an improved algorithm that can achieve a regret bound94 with only additive dependence on C.95 On the other hand , many research efforts have also been devoted into designing adversarial attacks96 that cause standard algorithms to fail ( Jun et al. , 2018 ; Liu and Shroff , 2019 ; Gupta et al. , 2019 ; 97 Garcelon et al. , 2020 ) .98 Linear Bandits with Corruptions : Li et al . ( 2019a ) studied stochastic linear bandits with adversarial99 corruptions and achieved Õ ( 1∆ ·d 5/2C+ 1∆2 ·d 6 ) regret bound where d is the dimension of the context100 vectors , ∆ is the gap between the rewards of the best and the second best action in the decision101 set D. The distinction between Li et al . ( 2019a ) and our work is that Li et al . ( 2019a ) considers a102 fixed decision set D throughout all T rounds , while we consider contextual bandits with changing103 decision set observed before each round . Bogunovic et al . ( 2021 ) also studied linear bandits with104 adversarial corruptions and considered the setting under the assumption that context vectors undergo105 small random perturbations , which is previously introduced by Kannan et al . ( 2018 ) . Aside from106 the additional assumption , another major distinction in Bogunovic et al . ( 2021 ) is that the number107 of actions k is finite and the regret bound depends on k in the contextual setting with unknown108 corruption level C. Recently , Lee et al . ( 2021 ) considered corrupted linear bandits with a finite and109 fixed decision set and achieve an instance-independent regret of Õ ( d √ T + C ) . Though both their110 work and the work by Li et al . ( 2019a ) focus on corrupted linear stochastic bandits , Lee et al . ( 2021 ) 111 have a slightly different definition of regret and adopt a strong assumption on corruptions that in112 each round t , the corruptions on rewards are linear in the actions . Neu and Olkhovskaya ( 2020 ) 113 studied linear contextual bandits with a finite decision set ( i.e. , K actions ) and an adversary . Unlike114 our model , they assume that the adversary can add an arbitrary noise to the loss under a limited115 amount and prove an Õ ( ( Kd ) 1 3T 2 3 ) + · √ dT regret bound for their proposed algorithm . Kapoor116 et al . ( 2019 ) considered the corrupted linear contextual bandits setting under a strong assumption on117 corruptions that for any prefix , at most an η fraction of the rounds are corrupted.118 3 Preliminaries119 In this paper , we study linear contextual bandits with adversarial corruptions . We will introduce our120 model and some basic concepts in this section.121 Corrupted linear contextual bandits . We consider the the linear contextual bandits model studied122 in Abbasi-Yadkori et al . ( 2011 ) under the same corruption studied by Lykouris et al . ( 2018 ) . In detail,123 distinctive from the linear contextual bandits Abbasi-Yadkori et al . ( 2011 ) , the interaction between124 the agent and the environment is now contaminated by an adversary . The protocol between the agent125 and the adversary at each round t ∈ [ T ] can be described as follows:126 1 . At the beginning of round t , the environment generates an arbitrary decision set Dt ⊆ Rd where127 each element represents a feasible action that can be selected by the agent.128 2 . The environment generates stochastic reward function r′t ( a ) = 〈a , µ∗〉+ t ( a ) together with an129 upper bound on the standard variance of t ( a ) , i.e. , σt ( a ) for all a ∈ Dt.130 3 . The adversary observes Dt , r′t ( a ) , σt ( a ) for all a ∈ Dt and decides a corrupted reward function131 rt defined over Dt.132 4 . The agent observes Dt and selects at ∈ Dt.133 5 . The adversary observes at and then returns rt ( at ) and σt ( at ) .134 6 . The agent observes rt ( at ) , σt ( at ) .135 Let Ft be the σ-algebra generated by D1 : t , a1 : t−1 , 1 : t−1 , r1 : t−1 and σ1 : t−1.136 At step 2 , µ∗ is a hidden vector unknown to the agent which can be observed by the adversary at the137 beginning . We assume that for all t ≥ 1 and all a ∈ Dt , ‖a‖2 ≤ A , |〈a , µ∗〉| ≤ 1 and ‖µ∗‖2 ≤ B138 almost surely . t ( a ) can be any form of random noise as long as it satisfies139 ∀t ≥ 1 , ∀a ∈ Dt , | t ( a ) | ≤ R , E [ t ( a ) |Ft ] = 0 , E [ 2t ( a ) |Ft ] ≤ σ2t ( a ) . ( 3.1 ) This assumption on t is a variant of that in Zhou et al . ( 2020 ) : We now require the noise to be140 generated for all a ∈ Dt in advance before the adversary decides the corrupted reward function . Our141 assumption on noises is more general than those in ( Li et al. , 2019a ; Bogunovic et al. , 2021 ; Kapoor142 et al. , 2019 ) where they are assumed to be 1-sub-Gaussian or Gaussian.143 At step 3 , we assume that the adversary has observed all the previous information and thus may144 predict which policy the agent will take at the current round . However , since the agent can take a145 randomized policy , the adversary may not know exactly which action the agent will take.146 Corruption level . We define corruption level147 C = 1 R+ 1 T∑ t=1 sup a∈Dt |r′t ( a ) − rt ( a ) | . ( 3.2 ) to indicate the level of adversarial contamination . We say a model is C-corrupted if the corruption148 level is no larger than C.149 Our definition of corruption level is equivalent to the counterpart in Lykouris et al . ( 2018 ) and Gupta150 et al . ( 2019 ) where they define C = ∑T t=1 maxa |r′t ( a ) − rt ( a ) | in our notation of rewards . We151 introduce a factor of 1R+1 since the noise is of range R in our model , while they assume all the152 rewards are in range [ 0 , 1 ] .153 Regret . Since the actions selected by the agent may not be deterministic , we define the regret for this154 model as follows:155 Regret ( T ) = T∑ t=1 〈a∗t , µ∗〉 − E [ T∑ t=1 〈at , µ∗〉 ] . ( 3.3 ) Our definition follows from the definition in Gupta et al . ( 2019 ) where the standard metric in stochastic156 multi-armed bandit models of pseudo-regret is adopted . But note that we need to take the expectation157 on ∑T t=1 r ′ t ( at ) ( the second term in ( 3.3 ) ) , since a randomized policy is applied in each round.158 Gap . Let ∆t be the gap between the rewards of the best and the second best action in the decision set159 Dt as defined in Dani et al . ( 2008 ) which can be formally written as160 ∆t = min a∈Dt , a6∈A∗t ( 〈a∗t , µ∗〉 − 〈a , µ∗〉 ) . ( 3.4 ) where A∗t = argmaxa∈Dt〈a , µ ∗〉 and a∗t is an arbitrary element in A∗t . Let ∆ denotes the smallest161 gap mint∈ [ T ] ∆t.162 4 The Proposed Algorithm163 In this section , we propose a variance-aware algorithm , Multi-level OFUL , in Algorithm 1 , to tackle164 the corrupted linear contextual bandits problem . At the core of our algorithm is an action partition165 scheme to group historical selected actions and use them to select the future actions in different166 groups with different probabilities . Such a scheme is introduced to deal with the unknown corruption167 level . For simplicity , we denote rt ( at ) , σt ( at ) in Section 3 by rt , σt in our algorithm.168 Main difficulty in our setting . We begin with the main difficulty that prevents us from applying169 existing algorithms to our setting . Consider a simpler setting where the agent knows the corruption170 level C in prior , and we have σt = R for all t. Then we can apply OFUL ( Abbasi-Yadkori et al.,171 2011 ) to solve our problem . In detail , in each round we estimate µ∗ by µt , which is the minimizer of172 the following ridge regression problem:173 µt = argmin µ∈Rd λ‖µ‖22 + t−1∑ i=1 [ 〈µ , ai〉 − ri ] 2 . ( 4.1 ) Algorithm 1 Multi-level OFUL 1 : Set the largest level of confidence sets : ` max ← dlog2 2T e. 2 : For ` ∈ [ ` max ] , set Σ1 , ` ← λI , µ1 , ` ← 0 , c1 , ` ← 0 . 3 : Set Σ1 ← λI , µ1 ← 0 , c1 ← 0 . 4 : for t = 1 , · · · , T do 5 : Observe Dt . 6 : for ` = 1 , · · · , ` max do 7 : Set βt , ` and γt , ` as defined in ( 4.5 ) and ( 4.6 ) . 8 : C′t , ` ← { µ|‖µ− µt‖Σt ≤ βt , ` } ∩ { µ|‖µ− µt , ` ‖Σt , ` ≤ γt , ` } . 9 : Ct , ` ← { C′t , ` , C′t , ` 6= ∅ Ct , ` +1 , otherwise . 10 : end for 11 : Set f ( t ) = { ` with probability 2− ` 1 < ` ≤ ` max 1 otherwise . 12 : Select at ← argmaxa∈Dt maxµ∈Ct , f ( t ) 〈µ , a〉 and observe rt , σt . 13 : Set σt = max { ( R+ 1 ) / √ d , σt } . 14 : Σt+1 ← Σt + ata > t /σ2t , ct+1 ← ct + rtat/σ2t , µt+1 ← Σ−1t+1ct+1 . 15 : for ` 6= f ( t ) do 16 : Σt+1 , ` ← Σt , ` , ct+1 , ` ← ct , ` , µt+1 , ` ← µt , ` . 17 : end for 18 : Σt+1 , f ( t ) ← Σt , f ( t ) + ata > t /σ2t , ct+1 , f ( t ) ← ct , f ( t ) + rtat/σ2t . 19 : µt+1 , f ( t ) ← Σ−1t+1 , f ( t ) ct+1 , f ( t ) . 20 : end for By slightly modifying the self-normalized martingale concentration inequality proposed in Abbasi-174 Yadkori et al . ( 2011 ) , we can conclude that µ∗ belongs to the ellipsoid ‖µ− µt‖Σ−1t ≤ βt with high175 probability , where βt = Õ ( R √ d + C √ d ) . Such a confidence bound leads to a final regret which176 has a polynomial dependence on C. However , such a simple approach have two limitations . First,177 the agent does not know C apriori in our setting , thus it is impossible to set βt to be dependent on178 C. Second , vanilla ridge regression estimator does not consider different variances σt in each round,179 thus it only gives a very conservative estimation.180 Action partition scheme . To address the unknown C issue , besides the original estimator µt which181 uses all previous data , Algorithm 1 maintains several additional learners to learn µ∗ at different182 accuracy level simultaneously , and it randomly selects one of the learners with different probabilities183 at each round . Such a “ parallel learning ” idea is inspired by Lykouris et al . ( 2018 ) . In detail,184 we partition the observed data into ` max levels indexed by [ ` max ] and maintain ` max sub-sampled185 estimators µt,1 , · · · , µt , ` max . According to line 11 , the observed data in round t goes into level ` with186 probability 2− ` if 1 < ` ≤ ` max and it goes to level 1 with probability 1− ∑ ` max ` =2 2 − ` = 1/2+2− ` max .187 The intuition is that if 2 ` ≥ C , then the corruption level experienced by level ` 188 Corruptiont , ` = t∑ i=1 1 ( f ( i ) = ` ) R+ 1 · sup a∈Di |ri ( a ) − r′i ( a ) | ( 4.2 ) can be bounded by some quantity that is independent of C. That says , the individual learners whose189 level is greater than logC can learn µ∗ successfully , even with the corruption . For the learners whose190 level is less than logC , we can also control the error by controlling the probability for the agent to191 select them.192 Weighted regression estimator . After introducing the partition scheme , we still need to deal193 with the varying variance ( heteroscedastic ) case . Similar to ( Kirschner and Krause , 2018 ; Zhou194 et al. , 2020 ) , we proposed the following weighted ridge regression estimator , which incorporates the195 variance information of the rewards into estimation:196 µt = argmin µ∈Rd λ‖µ‖22 + t−1∑ i=1 [ 〈µ , ai〉 − ri ] 2/σ2i . ( 4.3 ) Here σt is defined as the upper bound of the true variance σt in line 13 . The closed-form solution to197 ( 4.3 ) is calculated at each round in line 14 . The use of σt , as we will show later , makes our estimator198 more efficient in the heteroscedastic case . Meanwhile , we also apply our weighted regression199 estimator to each individual learner , and their estimator µt , ` can be written as follows:200 µt , ` = argmin µ∈Rd λ‖µ‖22 + t−1∑ i=1 1 ( f ( i ) = ` ) · [ 〈µ , ai〉 − ri ] 2/σ2i . ( 4.4 ) The closed-form solution to ( 4.4 ) is calculated at each round in lines 15–20.201 Final Multi-Level confidence sets . With the estimators µt , µt,1 , · · · , µt , ` max at the beginning of202 round t , we define a cascade of candidate confidence sets as in lines 6–10 , where203 βt , ` = 8 √ d log ( R+ 1 ) 2λ+ tA2 ( R+ 1 ) 2λ log ( 4t2/δ ) + 4 √ d log ( 4t2/δ ) + 2 ` √ d+ √ λB , ( 4.5 ) γt , ` = 8 √ d log ( R+ 1 ) 2λ+ tA2 ( R+ 1 ) 2λ log ( 8t2T/δ ) + 4 √ d log ( 8t2T/δ ) + C ` √ d+ √ λB , ( 4.6 ) with C ` = log ( 2 ` 2/δ ) + 3 . For simplicity , we define204 ` ∗ = max { 2 , dlog2 Ce } ( 4.7 ) as an important threshold in our later proof for regret bound analysis . Later we will prove that Ct , ` 205 contains µ∗ for all ` ≥ ` ∗ , t ≥ 1 with high probability.206 Note that each candidate confidence set can be written as the intersection of two ellipsoids . The207 intuition behind our construction of candidate confidence sets is that we hope that Ct , ` is robust208 enough to handle the 2 ` -corrupted case , i.e. , µ∗ ∈ Ct , ` with high probability . To achieve this , the first209 ellipsoid makes use of the global information and the “ radius ” βt , ` need to contain a factor of 2 ` to210 tolerate a corruption level of 2 ` , and the second ellipsoid makes use of the observed data in level ` 211 since this level only contain a few times of corruptions in 2 ` -corrupted case.212 Action selection . With the candidate confidence sets , we use line 11 to randomly decide one213 confidence set and select an action based on the optimism-in-the-face-of-uncertainty ( OFU ) principle214 in line 12 . Then we update the estimators for the next round t+ 1.215 Remark 4.1 . Our algorithm shares a similar strategy for partitioning the observed data with the216 algorithm in Lykouris et al . ( 2018 ) but note that there is a major difference in that : Lykouris et al.217 ( 2018 ) regard the partition scheme as a “ layer structure ” , i.e. , their algorithm further uses different218 estimators in layers of parallel learners and do action elimination layer by layer in each round . In219 contrast , the sub-sampled estimators in our algorithm are used independently , i.e. , the selected action220 only relies on one of the partitions . As a result , Algorithm 1 does not need to do action elimination,221 thus is capable of handling the cases where the number of actions is huge or even infinite.222 5 Main Results223 In this section we present our main theorem , which establishes the regret bound for Multi-level224 OFUL.225 Theorem 5.1 . Set λ = 1/B2 . Suppose that C = Ω ( 1 ) , R = Ω ( 1 ) , for all t ≥ 1 and all a ∈ Dt , 〈a , µ∗〉 ∈ [ −1 , 1 ] . Then with probability at least 1 − 3δ , the regret of Algorithm 1 is bounded as follows : Regret ( T ) = Õ C2d √√√√ T∑ t=1 σ2t + C 2 √ dT + CR √ dT . Remark 5.2 . When σt , R = Ω ( 1 ) , the regret bound in Theorem 5.1 matches the regret bound of226 OFUL proposed in Zhou et al . ( 2020 ) when the corruption level C is a constant.227 Remark 5.3 . Compared with the Õ ( d √ T + C ) result in Lee et al . ( 2021 ) , our result has a multi-228 plicative quadratic dependence on C , which seems to be worse . However , we want to emphasize that229 we focus on a more challenging contextual bandits setting where the decision sets Dt at each round230 are not identical , which is different from that in Lee et al . ( 2021 ) . Therefore , our result and that in231 Lee et al . ( 2021 ) are not directly comparable.232 Remark 5.4 . Note that this instance-independent regret upper bound also holds in a stronger model233 than the one described in Section 3 , where the adversary can even decide the decision set Dt at each234 round t since our regret bound can hold without any assumption on the decision sets.235 Corollary 5.5 . Under the same conditions as in Theorem 5.1 , if σt given by the environment are all R , the regret of Algorithm 1 is bounded by : Regret ( T ) = Õ ( C2dR √ T ) . We also provide a gap-dependent regret bound.236 Theorem 5.6 . Suppose that C = Ω ( 1 ) , R = Ω ( 1 ) , for all t ≥ 1 and all a ∈ Dt , 〈a , µ∗〉 ∈ [ −1 , 1 ] . Then with probability at least 1− 3δ , the regret of Algorithm 1 is bounded as follows : Regret ( T ) = Õ ( 1 ∆ · C2R2d+ 1 ∆ · d2C2 max t∈ [ T ] σ2t ) . Remark 5.7 . Theorem 5.6 automatically suggests an Õ ( R2d2C2/∆ ) regret bound , by the fact237 σt = O ( R ) . Compared with previous result Õ ( d5/2C/∆ + d6/∆2 ) ( Lee et al. , 2021 ) , our result238 has a better dependence on the dimension d but a worse dependence on the corruption level C. As239 Remark 5.3 suggests , we focus on a more challenging contextual bandits setting , and the worse240 dependence on C might be due to this.241 6 Proof Outline242 First we have the following lemma which is a corruption-tolerant variant of Bernstein inequality for243 self-normalized vector-valued martingales introduced in Zhou et al . ( 2020 ) .244 Lemma 6.1 ( Bernstein inequality for vector-valued martingales with corruptions ) . Let { Gt } ∞t=1 be a filtration , { xt , ηt } t≥1 a stochastic process so that xt ∈ Rd is Gt-measurable and ηt ∈ R is Gt+1-measurable . Fix R , L , σ , λ > 0 , µ∗ ∈ Rd . For t ≥ 1 let ystocht = 〈µ∗ , xt〉 + ηt and suppose that ηt , xt also satisfy |ηt| ≤ R , E [ ηt|Gt ] = 0 , E [ η2t |Gt ] ≤ σ2 , ‖xt‖2 ≤ L. Suppose { yt } is a sequence such that ∑t i=1 |yi − ystochi | = C ( t ) for all t ≥ 1 . Then , for any245 0 < δ < 1 , with probability at least 1− δ we have ∀t > 0,246 ‖µt − µ∗‖Zt ≤ βt + C ( t ) + √ λ‖µ∗‖2 , where for t ≥ 1 , µt = Z−1t bt , Zt = λI + ∑t i=1 xix > i , bt = ∑t i=1 yixi , and βt = 8σ √ d log dλ+ tL2 dλ log ( 4t2/δ ) + 4R log ( 4t2/δ ) . Next , we have that with high probability , all the level ` ≥ ` ∗ only influenced by limited amount of247 corruptions as mentioned in Section 4.248 Lemma 6.2 . Let Corruptiont , ` be defined in ( 4.2 ) . Then we have with probability at least 1− δ , for all ` ≥ ` ∗ , t ≥ 1 : Corruptiont , ` ≤ C ` = log ( 2 ` 2/δ ) + 3 . We denote by Esub the event that the above inequality holds.249 We define the following event to further show that our candidate confidence sets with ` ≥ ` ∗ are250 “ robust ” enough , i.e . Ct , ` contains µ∗ with high probability.251 Definition 6.3 . Let ` ∗ be defined in ( 4.7 ) . We introduce the event E1 as follows.252 E1 : = { ∀ ` ≥ ` ∗ and t ≥ 1 , ‖µ∗ − µt‖Σt ≤ βt , ` and ‖µ∗ − µt , ` ‖Σt , ` ≤ γt , ` } . ( 6.1 ) where βt , ` , γt , ` are defined in ( 4.5 ) and ( 4.6 ) .253 Next lemma suggests that the event E1 happens with high probability.254 Lemma 6.4 . Let E1 be defined in ( 6.1 ) . For any 0 < δ < 1/3 , we have P ( E1 ) ≥ 1− 3δ.255 For simplicity , we define at , ` = argmaxa∈Dt maxµ∈Ct , ` 〈µ , a〉 for each level ` . at can be seen as an256 action vector randomly chosen from at , ` , ` ∈ [ ` max ] . Next two lemmas suggest that under event E1,257 at each round , the gap between the optimal reward and the selected reward can be upper bounded by258 some bonus terms related to at , ` .259 Lemma 6.5 . Suppose E1 occurs . If f ( t ) ≤ ` ∗ , we have 〈a∗t − at , µ∗〉 ≤ 2βt , ` ∗‖at‖Σ−1t +260 2βt , ` ∗‖at , ` ∗‖Σ−1t .261 Lemma 6.6 . On event E1 , if f ( t ) = ` > ` ∗ , we have 〈a∗t − at , µ∗〉 ≤ 2γt , ` ‖at‖Σ−1t , ` .262 Now we provide the proof sketch of Theorem 5.1.263 Proof sketch of Theorem 5.1 . Suppose E1 occurs . The main idea to bound the regret is to decompose264 the total rounds [ T ] into two non-overlapping parts , based on which individual learner is selected at265 that round . In detail , we have266 Regret ( T ) = E [ T∑ t=1 ( 〈a∗t , µ∗〉 − 〈at , µ∗〉 ) ] = E [ T∑ t=1 1 ( f ( t ) ≤ ` ∗ ) ( 〈a∗t , µ∗〉 − 〈at , µ∗〉 ) ] ︸ ︷︷ ︸ I1 + ` max∑ ` = ` ∗+1 E [ T∑ t=1 1 ( f ( t ) = ` ) ( 〈a∗t , µ∗〉 − 〈at , µ∗〉 ) ] ︸ ︷︷ ︸ I2 ( ` ) . ( 6.2 ) Here I1 represents the regret where the the `` low-level '' learner is selected , where the corruption level267 is beyond the learner level.For this case , by Lemma 6.5 , we can directly show that268 I1 ≤ E [ T∑ t=1 1 ( f ( t ) ≤ ` ∗ ) min { 2 , 2βt , ` ∗‖at , ` ∗‖Σ−1t + 2βt , ` ∗‖at‖Σ−1t } ] . ( 6.3 ) We further bound ( 6.3 ) . Let Ft be the σ-algebra generated by as , rs , σs , f ( s ) for s ≤ t − 1.269 Then by the property of our partition scheme ( note that P ( f ( t ) = ` ∗ ) = 2− ` ∗ ) , we can show that270 E [ 1 ( f ( t ) ≤ ` ∗ ) ‖at , ` ∗‖Σ−1t |Ft ] ≤ 2 ` ∗E [ ‖at‖Σ−1t |Ft ] . Therefore , we can further bound I1 by271 I1 ≤ 4 · 2 ` ∗ E [ T∑ t=1 min { 2 , βT , ` ∗‖at‖Σ−1t } ] ︸ ︷︷ ︸ I3 . ( 6.4 ) To further bound I3 , we split [ T ] into 2 parts , I1 = { t ∈ [ T ] |‖at/σt‖Σ−1t > 1 } , I2 = { t ∈272 [ T ] |‖at/σt‖Σ−1t ≤ 1 } to bound I3 . The intuition here is that the cardinality of I1 is bounded , and273 the sum of terms with t ∈ I2 can be bounded using Cauchy-Schwarz inequality.274 ∑ t∈I1 min { 2 , βT , ` ∗‖at‖Σ−1t } ≤ 2|I1| ≤ 2 T∑ t=1 min { 1 , ‖at/σt‖2Σ−1t } ≤ 4d log ( R+ 1 ) 2λ+ TA2 ( R+ 1 ) 2λ , ( 6.5 ) where the first inequality holds since min { 2 , βT , ` ∗‖at‖Σ−1t } ≤ 2 , the second inequality follows275 from the definition of I1 , the third inequality holds by Lemma C.2.276 ∑ t∈I2 min { 2 , βT , ` ∗‖at‖Σ−1t } ≤ βT , ` ∗ √∑ t∈I2 σ2t · √∑ t∈I2 min { 1 , ‖at/σt‖2Σ−1t } ≤ βT , ` ∗ √√√√ ( R+ 1 ) 2T/d+ T∑ t=1 σ2t · √ 2d log ( R+ 1 ) 2λ+ TA2 ( R+ 1 ) 2λ , ( 6.6 ) where the first inequality follows from Cauchy-Schwarz inequality , the second inequality follows277 from the definition of σt and Lemma C.2.278 Substituting ( 6.5 ) and ( 6.6 ) into ( 6.3 ) , we have279 I1 = Õ C2d √√√√ T∑ t=1 σ2t + C 2 √ dT + CR √ dT . ( 6.7 ) Now it remains to bound I2 ( ` ) . By Lemma 6.6 , we have280 I2 ( ` ) ≤ 2E [ T∑ t=1 1 ( f ( t ) = ` ) min { 1 , γt , ` ‖at , ` ‖Σ−1t , ` } ] ︸ ︷︷ ︸ I4 = Õ R√Td+ d √√√√ T∑ t=1 σ2t , ( 6.8 ) where the second equality can be proved by analysis similar to that of ( 6.5 ) and ( 6.6 ) . Finally,281 substituting ( 6.7 ) and ( 6.8 ) into ( 6.2 ) ends our proof.282 283 7 Conclusion and Future Work284 In this paper , we have considered the linear contextual bandits problem in the presence of adversarial285 corruptions . We propose a Multi-level OFUL algorithm , which is provably robust to the adversarial286 attacks . 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Summary: This paper proposes an algorithm that achieves the first corruption-robust result for contextual linear stochastic bandits with infinite arms using the idea of multi-level sampling rates. The regret bound has multiplicative dependence on C^2 and is variance-aware. The algorithm recovers the optimal regret in terms of T and gap when no corruptions are presented.