Misspecified Gaussian Process Bandit Optimization

1 Introduction . Bandit optimization has been successfully used in a great number of machine learning and real-world applications , e.g. , in mobile health [ 42 ] , environmental monitoring [ 40 ] , economics [ 27 ] , hyperparameter tuning [ 26 ] , to name a few . To scale to large or continuous domains , modern bandit approaches try to model and exploit the problem structure that is often manifested as correlations in rewards of `` similar '' actions . Hence , the key idea of kernelized bandits is to consider only smooth reward functions of a low norm belonging to a chosen Reproducing Kernel Hilbert Space ( RKHS ) of functions . This permits the application of flexible nonparametric Gaussian process ( GP ) models and Bayesian optimization methods via a well-studied link between RKHS functions and GPs ( see , e.g. , [ 18 ] for a concise review ) . A vast majority of previous works on nonparametric kernelized bandits have focused on designing algorithms and theoretical bounds on the standard notions of regret ( see , e.g. , [ 40 , 9 , 35 ] ) . However , they solely focus on the realizable ( i.e. , well-specified ) case in which one assumes perfect knowledge of the true function class . For example , the analysis of the prominent GP-UCB [ 40 ] algorithm assumes the model to be well-specified and ignores potential misspecification issues . As the realizability assumption may be too restrictive in real applications , we focus on the case where it may only hold approximately . In practice , model misspecifications can arise due to various reasons , such as incorrect choice of kernel , consideration of an overly smooth function class , hyperparameter estimation errors , etc . Hence , an open question is to characterize the impact of model misspecification in the kernelized setting , and to design robust algorithms whose performance degrades optimally with the increasing level of misspecification . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . In this paper , we study the GP bandit problem with model misspecification in which the true unknown function might be ✏-far ( as measured in the max norm ) from a member of the learner ’ s assumed hypothesis class . We propose a novel GP bandit algorithm and regret bounds that depend on the misspecification error , time horizon , and underlying kernel . Specifically , we present an algorithm that is based on the classical uncertainty sampling approach that is frequently used in Bayesian optimization and experimental design . Importantly , our main presented algorithm assumes no knowledge of the misspecification error ✏ and achieves standard regret rates in the realizable case . Related work on GP bandits . GP bandit algorithms have received significant attention in recent years ( e.g. , [ 40 , 11 , 9 , 39 ] ) . While the most popular approaches in the stochastic setting rely on upper confidence bound ( UCB ) and Thompson sampling strategies , a number of works also consider uncertainty sampling procedures ( e.g. , [ 10 , 41 , 6 ] ) . Beyond the standard setting , numerous works have also considered the contextual bandit setting ( e.g. , [ 22 , 44 , 21 ] ) , while the case of unknown kernel hyperparameters and misspecified smoothness has been studied in , e.g. , [ 46 , 3 , 47 ] . [ 46 , 3 ] assume that the reward function is smooth as measured by the known ( or partially known ) kernel , while in our problem , we allow the unknown function to lie outside a given kernel ’ s reproducing space . Related corruption-robust GP bandits in which an adversary can additively perturb the observed rewards are recently considered in [ 4 ] . In Section 2 , we also discuss how the misspecified problem can be seen from this perspective , where the corruption function is fixed and the adversarial budget scales with the time horizon . While the focus of [ 4 ] is on protecting against an adaptive adversary and thus designing randomized algorithms and regret bounds that depend on the adversarial budget , we propose deterministic algorithms and analyze the impact of misspecification . Apart from corruptions , several works have considered other robust aspects in GP bandits , such as designing robust strategies against the shift in uncontrollable covariates [ 5 , 20 , 31 , 38 , 7 ] . While they report robust regret guarantees , they still assume the realizable case only . Our goal of attaining small regret despite the wrong hypothesis class requires very different techniques from these previous works . Related work on misspecified linear bandits . Recently , works on reinforcement learning with misspecified linear features ( e.g. , [ 12 , 45 , 17 ] ) have renewed interest in the related misspecified linear bandits ( e.g. , [ 48 ] , [ 24 ] , [ 30 ] , [ 15 ] ) first introduced in [ 16 ] . In [ 16 ] , the authors show that standard algorithms must suffer ⌦ ( ✏T ) regret under an additive ✏–perturbation of the linear model . Recently , [ 48 ] propose a robust variant of OFUL [ 1 ] that requires knowing the misspecification parameter ✏ . In particular , their algorithm obtains a high-probability Õ ( d p T + ✏ p dT ) regret bound . In [ 24 ] , the authors propose another arm-elimination algorithm based on G-experimental design . Unlike the previous works , their algorithm is agnostic to the misspecification level , and its performance matches the lower bounds . As shown in [ 24 ] , when the number of actions K is large ( K d ) , the “ price ” of ✏-misspecification must grow as ⌦ ( ✏ p dT ) .1 Our main algorithm is inspired by the proposed technique for the finite-arm misspecified linear bandit setting [ 24 ] . It works in the more general kernelized setting , uses a simpler data acquisition rule often used in Bayesian optimization and experimental design , and recovers the same optimal guarantees when instantiated with linear kernel . Several works have recently considered the misspecified contextual linear bandit problem with unknown model misspecification ✏ . [ 14 ] introduce a new family of algorithms that require access to an online oracle for square loss regression and address the case of adversarial contexts . Concurrent work of [ 33 ] solves the case when contexts / action sets are stochastic . Both works ( [ 14 ] and [ 33 ] ) leverage CORRAL-type aggregation [ 2 ] of contextual bandit algorithms and achieve the optimal Õ ( p dT ✏ + d p T ) regret bound . Finally , in [ 32 ] , the authors present a practical master algorithm that plays base algorithms that come with a candidate regret bound that may not hold during all rounds . The master algorithm plays base algorithms in a balanced way and suitably eliminates algorithms whose regret bound is no longer valid . Similarly to the previous works that rely on the CORRAL-type master algorithms , we use the balancing master algorithm of [ 33 ] together with our GP-bandit base algorithm to provide contextual misspecified regret bounds . Around the time of the submission of this work , a related approach that among others also considers misspecified kernel bandits appeared online [ 8 ] . [ 8 , Theorem 3 ] contains the same regret scaling due to misspecification as we obtain in our results . The main difference between the two works is in the proposed algorithms and analysis techniques . Our approach does not require robust estimators and simply uses the standard ones ( i.e. , GP posterior/ kernelized ridge regression mean and variance 1A result by [ 15 ] further shows that this result can be improved to O ( ✏ p KT ) in the small-K regime . estimators ) that can be computed in the closed-form . It can also handle infinite action sets similarly to the classical Bayesian optimization algorithms ; it utilizes a single acquisition function that , in practice , can be maximized via standard off-the-shelf global optimization solvers . We also present a complete treatment of the misspecified problem by showing the failure of standard UCB approaches , algorithms for known and unknown ✏ , an impossibility result , and extensions to the contextual bandit setting . Finally , our main algorithm ( Algorithm 2 ) demonstrates the use of a different acquisition function in comparison to [ 24 , 8 ] that relies on standard and non-robust estimators . Contributions . In this paper , we systematically handle model misspecification in GP bandits . Specifically , this paper makes the following contributions : • We introduce a misspecified kernelized bandit problem , and for known misspecification error ✏ , we present the EC-GP-UCB algorithm with enlarged confidence bounds that achieves cumulative RT = O ( B p TT+ T p T+✏T p T ) regret . Our simple lower bound argument shows that ⌦ ( T ✏ ) regret is unavoidable in the general kernelized setting . • For when ✏ is unknown , we propose another algorithm based on uncertainty sampling and phased exploration that achieves ( up to polylog factors ) the previous regret rates in the misspecified setting , and standard regret guarantees in the realizable case ( when ✏ = 0 ) . • Finally , we consider a misspecified contextual kernelized problem , and show that when action sets are stochastic , our EC-GP-UCB algorithm can be effectively combined with the regret bound balancing strategy from [ 33 ] to achieve previous regret bounds ( up to some additive lower order terms ) . 2 Problem statement . We consider the problem of sequentially maximizing some black-box reward function f⇤ : D ! R over a known compact set of actions D ⇢ Rd . To learn about f⇤ , the learner relies on sequential noisy bandit feedback , i.e. , at every round t , the learner selects xt 2 D and obtains a noisy observation y ⇤ t = f ⇤ ( xt ) + ⌘t , ( 1 ) where we assume independent ( over rounds ) -sub-Gaussian noise ( see Appendix A for definition ) . Let k ( · , · ) denote a positive definite kernel function defined on D ⇥D and Hk ( D ) be its associated Reproducing Kernel Hilbert Space ( RKHS ) of well-behaved functions . Suppose that before making any decision , the learner is provided with a hypothesis class Fk ( D ; B ) = { f 2 Hk ( D ) : kfkk B } , ( 2 ) where every member function has a bounded RKHS norm kfkk = p hf , fi for some B > 0 , that measures the complexity of f with respect to kernel k ( · , · ) . We consider kernel functions such that k ( x , x ) 1 for every x 2 D. Most commonly used kernel functions that satisfy this property are outlined in Appendix A . The standard setting assumes a realizable ( i.e. , well-specified ) scenario in which f⇤ 2 Fk ( D ; B ) , i.e. , the unknown function is a member of the known RKHS with bounded norm . In contrast , in the misspecified setting , we assume that the learner is informed that f⇤ can be uniformly approximated by a member from the given hypothesis class Fk ( D ; B ) , i.e. , min f2Fk ( D ; B ) kf f ⇤ k1 ✏ , ( 3 ) for some ✏ > 0 and max-norm k · k1 . Here , we note that if two functions are close in RKHS norm , then they are also pointwise close but the reverse does not need to be true . Hence , the true function f ⇤ can in principle have a significantly larger RKHS norm than any f from Fk ( D ; B ) , or it might not even belong to the considered RKHS . We also note that in the case of continuous universal kernels [ 28 ] ( i.e. , due to the universal function approximation property of such kernels ) , any continuous function f⇤ on D satisfies the above assumption ( Eq . ( 3 ) ) for any ✏ > 0 and Fk ( D ; B ) with suitably large RKHS norm bound B . This is a difference in comparison to the previously studied misspecified linear setting , and another motivation to study the misspecified kernelized problem . As in the standard setting ( which corresponds to the case when ✏ = 0 ) , we assume that k ( · , · ) and B are known to the learner . The goal of the learner is to minimize the cumulative regret R ⇤ T = TX t=1 max x2D f ⇤ ( x ) f⇤ ( xt ) , ( 4 ) where T is a time horizon . We denote instantaneous regret at round t as r⇤t = f⇤ ( x⇤ ) f⇤ ( xt ) , where x⇤ 2 argmaxx2D f⇤ ( x ) , and note that R⇤T = PT t=1 r ⇤ t . The learner ’ s algorithm knows Fk ( D ; B ) , meaning that it takes D , k ( · , · ) and B as input . Crucially , in our most general considered problem variant , the exact value of misspecification ✏ is assumed to be unknown , and so we seek an algorithm that can adapt to any ✏ > 0 , including the realizable case when ✏ = 0 . Alternatively , one can consider competing with a best-in-class benchmark , i.e. , RT = TX t=1 max x2D f̃ ( x ) f̃ ( xt ) , ( 5 ) where f̃ 2 argmin f2Fk ( D ; B ) kf f ⇤ k1 ( 6 ) and |f̃ ( x ) f⇤ ( x ) | ✏ for every x 2 D. Here , the goal is to minimize cumulative regret in case the true unknown objective is f̃ 2 Fk ( D ; B ) , while noisy observations of f̃ that the learner receives are at most ✏-misspecified , i.e. , y ⇤ t = f̃ ( xt ) +m ( xt ) | { z } f⇤ ( xt ) +⌘t , ( 7 ) where m ( · ) : D ! [ ✏ , ✏ ] is a fixed and unknown function . Since it holds that kf̃ f⇤k1 ✏ , the absolute difference between RT and R⇤T is at most 2✏T , and as will already become clear in Section 3.2 , this difference will have no significant impact on the scalings in our main bound ( since R⇤T = ⌦ ( ✏T ) ) . In our theoretical analysis , we will interchangeably use both regret definitions ( from Eqs . ( 4 ) and ( 5 ) ) .

Summary: This paper considers the problem of regret minimization for Gaussian process bandits. The authors focus on the setting where the model is misspecified and the true reward function is not contained in the underlying RKHS. They provide three algorithms: 1) a ucb based method for the setting where the misspecification error is known, 2) an elimination approach for when it is not known, 3) a contextual bandits version.