Learning Neural Contextual Bandits through Perturbed Rewards

√ T ) regret upper bound is still achievable under standard regularity conditions , where T is the number of rounds of interactions and d̃ is the effective dimension of a neural tangent kernel matrix . Extensive comparisons with several benchmark contextual bandit algorithms , including two recent neural contextual bandit models , demonstrate the effectiveness and computational efficiency of our proposed neural bandit algorithm . 1 INTRODUCTION . Contextual bandit is a well-formulated abstraction of many important real-world problems , including content recommendation ( Li et al. , 2010 ; Wu et al. , 2016 ) , online advertising ( Schwartz et al. , 2017 ; Nuara et al. , 2018 ) , and mobile health ( Lei et al. , 2017 ; Tewari & Murphy , 2017 ) . In such problems , an agent iteratively interacts with an environment to maximize its accumulated rewards over time . Its essence is sequential decision-making under uncertainty . Because the reward from the environment for a chosen action ( also referred to as an arm in literature ) under each context is stochastic , a no-regret learning algorithm needs to explore the problem space for improved reward estimation , i.e. , learning the mapping from an arm and its context1 to the expected reward . Linear contextual bandit algorithms ( Abbasi-Yadkori et al. , 2011 ; Li et al. , 2010 ) , which assume the reward mapping is a linear function of the context vector , dominate the community ’ s attention in the study of contextual bandits . Though theoretically sound and practically effective , their linear reward mapping assumption is incompetent to capture possible complex non-linear relations between the context vector and reward . This motivated the extended studies in parametric bandits , such as generalized linear bandits ( Filippi et al. , 2010 ; Faury et al. , 2020 ) and kernelized bandits ( Chowdhury & Gopalan , 2017 ; Krause & Ong , 2011 ) . Recently , to unleash the power of representation learning , deep neural networks ( DNN ) have also been introduced to learn the underlying reward mapping directly . In ( Zahavy & Mannor , 2019 ; Riquelme et al. , 2018 ; Xu et al. , 2020 ) , a deep neural network is applied to provide a feature mapping , and exploration is performed at the last layer . NeuralUCB ( Zhou et al. , 2020 ) and NeuralTS ( Zhang et al. , 2020 ) explore the entire neural network parameter space to obtain nearly optimal regret using the neural tangent kernel technique ( Jacot et al. , 2018 ) . These neural contextual bandit algorithms significantly boosted empirical performance compared to their classical counterparts . Nevertheless , a major practical concern of existing neural contextual bandit algorithms is their added computational cost when performing exploration . Take the recently developed NeuralUCB and NeuralTS for example . Their construction of the high-probability confidence set for model exploration depends on the dimensionality of the network parameters and the learned context vectors ’ representations , which is often very large for DNNs . For instance , a performing neural bandit solution often has the number of parameters in the order of 100 thousands ( if not less ) . It is prohibitively 1When no ambiguity is invoked , we refer to the feature vector for an arm and its context as a context vector . expensive to compute inverse of the induced covariance matrix on such a huge number of parameters , as required by their construction of confidence set . As a result , approximations , e.g. , only using the diagonal of the covariance matrix ( Zhou et al. , 2020 ; Zhang et al. , 2020 ) , are employed to make such algorithms operational in practice . But there is no theoretical guarantee for such diagonal approximations , which directly leads to the gap between the theoretical and empirical performance of the neural bandit algorithms . In this work , to alleviate the computational overhead caused by the expensive exploration , we propose to eliminate explicit model exploration by learning a neural bandit model with perturbed rewards . At each round of model update , we inject pseudo noise generated from a zero-mean Gaussian distribution to the observed reward history . With the induced randomization , sufficient exploration is achieved when simply pulling the arm with the highest estimated reward . This brings in considerable advantage over existing neural bandit algorithms : no additional computational cost is needed to obtain no regret . We rigorously prove that with a high probability the algorithm obtains a Õ ( d̃ √ T ) regret , where d̃ is the effective dimension of a neural tangent kernel matrix and T is the number of rounds of interactions . This result recovers existing regret bounds for the linear setting where the effective dimension equals to the input feature dimension . Besides , our extensive empirical evaluations demonstrate the strong advantage in efficiency and effectiveness of our solution against a rich set of state-of-the-art contextual bandit solutions over both synthetic and real-world datasets . 2 RELATED WORK . Most recently , attempts have been made to incorporate DNNs with contextual bandit algorithms . Several existing work study neural-linear bandits ( Riquelme et al. , 2018 ; Zahavy & Mannor , 2019 ) , where exploration is performed on the last layer of the DNN . Under the neural tangent kernel ( NTK ) framework ( Jacot et al. , 2018 ) , NeuralUCB ( Zhou et al. , 2020 ) constructs confidence sets with DNN-based random feature mappings to perform upper confidence bound based exploration . NeuralTS ( Zhang et al. , 2020 ) samples from the posterior distribution constructed with a similar technique . However , as the exploration is performed in the induced random feature space , the added computational overhead is prohibitively high , which makes such solutions impractical . The authors suggested diagonal approximations of the resulting covariance matrix , which however leave the promised theoretical guarantees of those algorithms up in the air . Reward perturbation based exploration has been studied in a number of classical bandit models ( Kveton et al. , 2019a ; 2020 ; 2019b ) . In a context-free k-armed bandit setting , Kveton et al . ( 2019b ) proposed to estimate each arm ’ s reward over a perturbed history and select the arm with the highest estimated reward at each round . Such an arm selection strategy is proved to be optimistic with a sufficiently high probability . Later this strategy has been extended to linear and generalized linear bandits ( Kveton et al. , 2019a ; 2020 ) . In ( Kveton et al. , 2020 ) , the authors suggested its application to neural network models ; but only some simple empirical evaluations were provided , without any theoretical justifications . Our work for the first time provides a rigorous regret analysis of neural contextual bandits with the perturbation-based exploration : a sublinear regret bound is still achievable in terms of the number of interactions between the agent and environment . 3 NEURAL BANDIT LEARNING WITH PERTURBED REWARDS . We study the problem of contextual bandit with finite K arms , where each arm is associated with a d-dimensional context vector : xi ∈ Rd for i ∈ [ K ] . At each round t ∈ [ T ] , the agent needs to select one of the arms , denoted as at , and receives its reward rat , t , which is generated as rat , t = h ( xat ) + ηt . In particular , h ( x ) represents the unknown underlying reward mapping function satisfying 0 ≤ h ( x ) ≤ 1 for any x , and ηt is an R-sub-Gaussian random variable that satisfies E [ exp ( µηt ) ] ≤ exp [ µ2R2 ] for all µ ≥ 0 . The goal is to minimize pseudo regret over T rounds : RT = E [ ∑T t=1 ( ra∗ − rt , at ) ] , ( 3.1 ) where a∗ is the optimal arm with the maximum expected reward . To deal with the potential non-linearity of h ( x ) and unleash the representation learning power of DNNs , we adopt a fully connected neural network f ( x ; θ ) to approximate h ( x ) : f ( x ; θ ) = Algorithm 1 Neural bandit with perturbed reward ( NPR ) 1 : Input : Number of rounds T , regularization coefficient λ , perturbation parameter ν , network width m , network depth L. 2 : Initialization : θ0 = ( vec ( W1 ) , . . . , vec ( WL ) ] ∈ Rp with Gaussian distribution : for 1 ≤ l ≤ L − 1 , Wl = ( W , 0 ; 0 , W ) with each entry of W sampled independently from N ( 0 , 4/m ) ; WL = ( w > , −w > ) with each entry of w sampled independently from N ( 0 , 2/m ) . 3 : for t = 1 , . . . , T do 4 : if t > K then 5 : Pull arm at and receive reward rt , at , where at = argmaxi∈ [ K ] f ( xi , θt−1 ) . 6 : Generate { γts } s∈ [ t ] ∼ N ( 0 , ν2 ) . 7 : Set θt by the output of gradient descent for solving Eq ( 3.2 ) . 8 : else 9 : Pull arm ak . 10 : end if 11 : end for √ mWLφ ( WL−1φ ( · · ·φ ( W1x ) ) ) , where φ ( x ) = ReLU ( x ) , θ = [ vec ( W1 ) , . . . , vec ( WL ) ] ∈ Rp with p = m + md + m2 ( L − 1 ) , and depth L ≥ 2 . Each hidden layer is assumed to have the same width ( i.e. , m ) for convenience in later analysis ; but this does not affect the conclusion of our theoretical analysis . Existing neural bandit solutions perform explicit exploration in the entire model space ( Zhou et al. , 2020 ; Zhang et al. , 2020 ; Zahavy & Mannor , 2019 ; Riquelme et al. , 2018 ; Xu et al. , 2020 ) , which introduces prohibitive computational cost . And oftentimes the overhead is so high that approximation has to be employed ( Zhou et al. , 2020 ; Zhang et al. , 2020 ) , which unfortunately breaks the theoretical promise of these algorithms . In our proposed model , to eliminate such explicit model exploration in neural bandit , a randomization strategy is introduced in the neural network update . We name the resulting solution as Neural bandit with Perturbed Rewards , or NPR in short . In NPR , at round t , the neural model is learned with the t rewards perturbed with designed perturbations : min θ L ( θ ) = ∑t s=1 ( f ( xas ; θ ) − ( rs , as + γts ) ) 2 /2 +mλ‖θ − θ0‖22/2 ( 3.2 ) where { γts } ts=1 ∼ N ( 0 , ν2 ) are Gaussian random variables that are independently sampled in each round t , and ν is a hyper-parameter that controls the strength of perturbation ( and thus the exploration ) in NPR . We use an l2-regularized square loss for model estimation , where the regularization centers at the randomly initialization θ0 with the trade-off parameter λ . The detailed procedure of NPR is given in Algorithm 1 . The algorithm starts by pulling all candidate arms once . This guarantees that for any arm , NPR is sufficiently optimistic compared to the true reward with respect to its approximation error ( Lemma 4.4 ) . When all the K arms have been pulled once , the algorithm pulls the arm with the highest estimated reward , at = argmaxi f ( xi ; θt−1 ) . Once received the feedback rat , t , the model perturbs the entire reward history so far via a freshly sampled noise sequence { γts } ts=1 , and updates the neural network by { ( xas , ras , s + γts ) } ts=1 using gradient descent . In the regret analysis in Section 4 , we prove that the variance from the added { γts } s∈ [ t ] will lead to the necessary optimism for exploration ( Abbasi-Yadkori et al. , 2011 ) . We adopt gradient descent for analysis convenience , while stochastic gradient descent can also be used to solve the optimization problem with a similar theoretical guarantee based on recent works ( AllenZhu et al. , 2019 ; Zou et al. , 2019 ) . Compared with existing neural contextual bandit algorithms ( Zhou et al. , 2020 ; Zhang et al. , 2020 ; Zahavy & Mannor , 2019 ; Riquelme et al. , 2018 ; Xu et al. , 2020 ) , NPR does not need any added computation for model exploration , besides the regular neural network update . This greatly alleviates the overhead for the computation resources ( both space and time ) and makes NPR sufficiently general to be applied for practical problems . More importantly , our theoretical analysis directly corresponds to its actual behavior when applied , as no approximation is needed .

Summary: The paper studies neural contextual bandits in the realizable setting. It proposes an algorithm that trains the neural network that maps arm contexts to rewards, with perturbed rewards. The paper proves that under NTK analysis the learnt function provides an optimistic estimate of the reward of each arms and therefore sub-linear regret guarantees can be derived. Crucially the dimension term in the regret only depends on the effective dimension of the NTK. Some experiments are performed on real and synthetic datasets where the proposed algorithm performs at par with neuralTS and neuralUCB algorithms while cutting down the running time by a large factor. Thus this can indicate the practicality of the proposed algorithm.