Improved Variance-Aware Confidence Sets for Linear Bandits and Linear Mixture MDP

• For linear bandits , we obtain an rOppolypdq b 1 ` řK k “ 1 σ 2 kq data-dependent regret bound , where d is the feature dimension , K is the number of rounds , and σ2k is the unknown variance of the reward at the k-th round . This is the first regret bound that only scales with the variance and the dimension but no explicit polynomial dependency on K. When variances are small , this bound can be significantly smaller than the rΘ ` d ? K ˘ worst-case regret bound . • For linear mixture MDPs , we obtain an rOppolypd , logHq ? Kq regret bound , where d is the number of base models , K is the number of episodes , and H is the planning horizon . This is the first regret bound that only scales logarithmically with H in the reinforcement learning with linear function approximation setting , thus exponentially improving existing results , and resolving an open problem in [ Zhou et al. , 2020a ] . We develop three technical ideas that may be of independent interest : 1 ) applications of the peeling technique to both the input norm and the variance magnitude , 2 ) a recursion-based estimator for the variance , and 3 ) a new convex potential lemma that generalizes the seminal elliptical potential lemma . 1 Introduction . In sequential decision-making problems such as bandits and reinforcement learning ( RL ) , the agent chooses an action based on the current state , with the goal to maximize the total reward . When the state-action space is large , function approximation is often used for generalization . One of the most fundamental and widely used methods is linear function approximation . For ( infinite-actioned ) linear bandits , the minimax-optimal regret bound is rΘpd ? Kq [ Dani et al. , 2008 , Abbasi-Yadkori et al. , 2011 ] , where d is the feature dimension and K is the number of total rounds played by the agent.2 However , oftentimes the worst-case analysis is overly pessimistic , and ˚Equal contribution . 2We follow the reinforcement learning convention to use K to denote the total number of rounds / episodes . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . it is possible to obtain data-dependent bound that is substantially smaller than rOpd ? Kq in benign scenarios . One direction to study is the variance magnitude . As a motivating example , in linear bandits , if there is no noise ( variance is 0 ) , one only needs to pay at most d regret to identify the best action because d samples are sufficient to recover the underlying linear coefficients ( in general position ) . This constant-type regret bound is much smaller than the ? K-type regret bound in the worst case where the variance magnitude is a lower bounded constant . Therefore , a natural question is : Can we design an algorithm that adapts to the variance magnitude , and its regret degrades gracefully from the benign noiseless constant-type bound to the worst-case ? K-type bound ? In RL , exploiting the variance information is also important . For tabular RL , one needs to utilize the variance information , e.g. , Bernstein-type exploration bonus to achieve the minimax optimal regret [ Azar et al. , 2017 , Zanette and Brunskill , 2019 , Zhang et al. , 2020c , a , Menard et al. , 2021 , Dann et al. , 2019 ] . For example , the recently proposed MVP algorithm [ Zhang et al. , 2020a ] , enjoys an rOppolylogpHq ˆ p ? SAK ` S2Aqq regret bound , where S is the number of states , A is the number of actions , H is the planning horizon , and K is the total number of episodes . 34 Notably , this regret bound only scales logarithmically with H . On the other hand , without using the variance information , e.g. , using Hoeffding-type bonus instead of Bernstein-type bonus , algorithms would suffer a regret that scales polynomially with H [ Azar et al. , 2017 ] . Going beyond tabular RL , a recent line of work studied RL with linear function approximation with different assumptions [ Yang and Wang , 2019 , Modi et al. , 2020 , Jin et al. , 2020 , Ayoub et al. , 2020 , Zhou et al. , 2020a , Modi et al. , 2020 ] . Our paper studies the linear mixture Markov Decision Process ( MDP ) setting [ Modi et al. , 2020 , Ayoub et al. , 2020 , Zhou et al. , 2020a ] , where the transition probability can be represented by a linear function of some features or base models . This model-based assumption is motivated by problems in robotics and queuing systems . We refer readers to Ayoub et al . [ 2020 ] for more discussions . For this linear mixture MDP setting , previous works can obtain regret bounds in the form rOppolypd , Hq ? Kq , where d is the number of base models . While these bounds do not scale with SA , they scale polynomially with H , because the algorithms in previous works do not use the variance information . In practice , H is often large , and even a polynomial dependency on H may not be acceptable . Therefore , a natural question is Can we design an algorithm that exploits the variance information to obtain an rOppolypd , logHq ? Kq regret bound for linear mixture MDP ? 1.1 Our Contributions . In this paper , we develop new , variance-aware confidence sets for linear bandits and linear mixture MDP and answer the above two questions affirmatively . Linear Bandits . For linear bandits , we obtain an rOppolypdq b 1 ` řK k “ 1 σ 2 kq regret bound , where σ2k is the unknown variance at the k-th round . To our knowledge , this is the first bound that solely depends on the variance and the feature dimension , and has no explicit polynomial dependency on K. When the variance is very small so that σ2k ! 1 , this bound is substantially smaller than the worst-case rΘpd ? Kq bound . Furthermore , this regret bound naturally interpolates between the worst-case ? K-type bound and the noiseless-case constant-type bound . Linear Mixture MDP . For linear mixture MDP , we obtain the desired rOppolypd , logHq ? Kq regret bound . This is the first regret bound in RL with function approximation that 1 ) does not scale with the size of the state-action space , and 2 ) only scales logarithmically with the planning horizon 3 rOp¨q hides logarithmic factors . Sometimes we write out polylogH explicitly to emphasize the logarithmic dependency on H . 4This bound holds for setting where the transition is homogeneous and the total reward is bounded by 1 . We focus on this setting in this paper . See Section 2 and 3 for more discussions . H . Therefore , we exponentially improve existing results on RL with linear function approximation in term of the H dependency , and resolve an open problem in [ Zhou et al. , 2020a ] . More importantly , our result conveys the positive conceptual message for RL : it is possible to simultaneously overcome the two central challenges in RL , large state-action space and long planning horizon . 1.2 Main Difficulties and Technical Innovations . We first describe limitations of existing works why they can not achieve the desired regret bounds described above . Limitations of Existing Variance-Aware Confidence Sets Faury et al . [ 2020 ] , Zhou et al . [ 2020a ] applied Bernstein-style inequalities to construct a confidence sets of the least square estimator for linear bandits . However , their methods can not be applied directly to obtain the desired data-dependent regret bound . Abeille et al . [ 2021 ] also designed an variance-dependent confidence set for logistic bandits . However in their problem the rewards are Bernoulli and the variance is a function of the mean . We give a simple example to illustrate their limitations . Consider the case where the variance is always σ2 ! 1 . Let px1 , y1q , . . . , pxk´1 , yk´1q be the samples collected before the k-th round . Their confidence set at the k-th round is Θk “ tθ|||θ ´ θ̂k||Λk´1 ď Cpσ ? d ` 1 ` λ1 { 2qu ( See In Equation ( 4.3 ) of Zhou et al . [ 2020a ] and Theorem 1 of Faury et al . [ 2020 ] ) . where Λk´1 “ řk´1 τ “ 1 xτx J τ ` λI is the un-normalized covariance matrix , θ̂k “ Λ´1k´1 řk´1 τ “ 1 yτxτ is the estimated linear coefficients by least squares , λ is a regularization parameter and C is a constant . Consider the case d “ 1 and xk “ a 1 { K for k “ 1 , . . . , K. Their regret bound is roughly K ÿ k “ 1 pσ ? d ` 1 ` λ1 { 2q } xk } Λ´1k ě p1 ` λ 1 { 2q K ÿ i “ 1 } xk } Λ´1k ě p1 ` λ 1 { 2q c K 1 ` λ ě ? K , which is much larger than our bound , O ` ? Kσ2 ` 1 ˘ when σ is very small . For more detailed discussion , please refer to Appendix B . Below we describe our main techniques . Elimination with Peeling . Instead of using least squares and upper-confidence-bound ( UCB ) , we use an elimination approach . More precisely , for the underlying linear coefficients θ˚ P Rd , we build a confidence interval for pθ˚qJ µ for every µ in an -net of the d-dimensional unit ball , and we eliminate θ P Rd if θJµ fails to fall in the confidence interval of pθ˚qJµ for some µ . To build the confidence intervals , we use 1 ) an empirical Bernstein inequality ( cf . Theorem 4 ) and 2 ) the peeling technique to both the input norm and the variance magnitude . As will be clear in the proof ( cf . Section D ) , this peeling step is crucial to obtain a tight regret bound for the example above . The new confidence region provides a tighter estimation for θ˚ , which helps address the drawback in least squares . Generalization of the Elliptical Potential Lemma . Since we use the peeling technique which comes with a clipping operation , we can not use the seminal elliptic potential lemma Dani et al . [ 2008 ] any more . Instead , we propose a more general lemma below , which provides a bound of potential for a general class of convex functions though with a worse dependency on d than the bound in the elliptical potential lemma . We believe this lemma can be applied to other problems as well . Lemma 1 ( Generalized Quadratic Potential Lemma ) . Let fpxq ě 0 be a convex function over R such that fpxqx2 ď fpyq y2 ď 1 and fpxq ě fpyq if x 2 ě y2 ą 0 . Let Bp1q denote the d-dimensional unit ball . Fix ` P p0 , 1s . For any x1 , x2 , . . . , xt P Bp1q and µ1 , µ2 , . . . , µt P Bp1q , we have that t ÿ i “ 1 min # fpxiµiq ři´1 j “ 1 fpxjµiq ` ` 2 , 1 + ď Opd4 logpdt { ` qq . Note that by choosing fpxq “ x2 and µi “ xiΛ ´1 i } xiΛ´1i } with Λi “ ři´1 j “ 1 xjx J j ` ` I , Lemma 1 reduces to the classical elliptic potential lemma [ Dani et al. , 2008 ] . Our proof consists of two major parts . We first establish a symmetric version of Equation ( ? ? ) using rearrangement inequality , and then bound the number of times the energy for some µ ( i.e. , ři j “ 1 fpxjµq ` l2 ) doubles . The full proof is deferred to Appendix C. For linear mixture MDP , we propose another technique to further reduce the dependency on d. Recursion-based Variance Estimation . In linear bandits , generally it is not possible to estimate the variance because the variance at each round can arbitrarily different . On the other hand , for linear mixture MDP , the variance is a quadratic function of the underlying coefficient θ˚ . Furthermore , the higher moments are polynomial functions of θ˚ . Utilizing this rich structure and leveraging the recursion idea in previous analyses on tabular RL [ Lattimore and Hutter , 2012 , Li et al. , 2020 , Zhang et al. , 2020a ] , we explicitly estimate the variance and higher moments to further reduce the regret . See Section 5 for more explanations .

Summary: This paper proposes a new bandit and MDP algorithms for minimizing regret. The bandit algorithm enjoys a variance dependent bound, and the MDP algorithm (linear mixture) enjoys a horizon independent regret bound. The main technical novelty is a novel confidence set construction and a new convex potential lemma. The bandit regret bound has a worse scaling with dimensionality than the standard algorithms but can be significantly smaller depending on the variances.