Title: On the Minimax Regret for Contextual Linear Bandits and Multi-Armed Bandits with Expert Advice
Venue: NeurIPS 2024
Accepted: yes
Keywords: contextual linear bandit

multi-armed bandit with expert advice
Abstract: This paper examines two extensions of multi-armed bandit problems: multi-armed bandits with expert advice and contextual linear bandits.
  For the former problem, multi-armed bandits with expert advice, the previously known best upper and lower bounds have been $O(\sqrt{KT \log \frac{N}{K} })$ and $\Omega( \sqrt{KT \frac{ \log N }{\log K }} )$, respectively.
  Here, $K$, $N$, and $T$ represent the numbers of arms, experts, and rounds, respectively.
  We provide a lower bound of $\Omega( \sqrt{KT \log \frac{N}{K}} )$ for the setup in which the player chooses an expert before observing the advices in each round.
  For the latter problem, contextual linear bandits, we provide an algorithm that achieves $O ( \sqrt{d T \log ( K \min\{ 1, \frac{S}{d} \} )} )$ together with a matching lower bound, where $d$ and $S$ represent the dimensionality of feature vectors and the size of the context space, respectively.

On the Minimax Regret for Contextual Linear Bandits and Multi-Armed Bandits with Expert Advice

Abstract

This paper examines two extensions of multi-armed bandit problems: multi-armed bandits with expert advice and contextual linear bandits. For the former problem, multi-armed bandits with expert advice, the previously known best upper and lower bounds have been O ( √ KT log N K ) and Ω( √ KT log N log K ) , respectively. Here, K , N , and T represent the numbers of arms, experts, and rounds, respectively. We provide a lower bound of Ω( √ KT log N K ) for the setup in which the player chooses an expert before observing the advices in each round. For the latter problem, contextual linear bandits, we provide an algorithm that achieves O ( √ dT log( K min { 1 , S d } )) together with a matching lower bound, where d and S represent the dimensionality of feature vectors and the size of the context space, respectively.

1 Introduction

This paper considers problems of multi-armed bandits with expert advice (MwE) [Auer et al., 2002, Kale, 2014], linear bandits [Dani et al., 2008, Bubeck et al., 2012, Cesa-Bianchi and Lugosi, 2012], and contextual linear bandits (CLB) [Chu et al., 2011, Abbasi-Yadkori et al., 2011, Neu and Olkhovskaya, 2020, Hanna et al., 2023, 2024].

One contribution of this paper is to present the minimax regret for MwE, which addresses the open question posed by Seldin and Lugosi [2016]. For MwE, the well-known EXP4 algorithm [Auer et al., 2002] achieves O ( √ KT log N ) -regret, where K , T and N represent the numbers of arms, rounds and experts. This bound is, however, not always optimal for certain parameter settings of K and N . In fact, as discussed by Seldin and Lugosi [2016], when N = K , the problem is reduced to the standard K -armed bandit problem, for which the minimax regret is Θ( √ KT ) [Audibert and Bubeck, 2009]. This means that the regret bound of EXP4 has a gap of an O ( √ log K ) -factor when N = K . On the side of upper bounds, Kale [2014] addressed this issue by providing an algorithm achieving O (√ KT log + N K ) , 1 where we denote log + x := max { log x, 1 } . This is minimax optimal for the case of N = O ( K ) . However, the minimax optimal bound for arbitrary settings of K and N has been an open question. The best known lower bound Ω (√ KT log N log K ) is shown by Seldin and Lugosi [2016], who conjectured that this lower bound is minimax optimal. This paper provides a solution to this open question by providing a lower bound of Ω (√ KT log + N K ) , which, together with the upper

bound by Kale [2014], implies that the minimax regret is Θ (√ KT log + N K ) . This lower bound is shown for the problem setting in which the player need to choose an expert before observing the advices in each round. As noted in Remark 1 below, though this problem setup is more challenging than the 'classical' setting where the player can observe all expert advice before selecting an arm, almost all known existing algorithms, including those in Table 1, work for this setting.

Contextual linear bandit problems are a generalization of multi-armed bandit problems in which each arm i ∈ [ K ] = { 1 , 2 , . . . , K } is linked to a feature vector φ ( X t , i ) ∈ R d that depends on the context X t at the t -th round, and the expected loss suffered for choosing i is expressed as a linear function of the feature vector: 〈 θ t , φ ( X t , i ) 〉 . In particular, this paper considers the problem setting of the stochastic-context and adversarial-loss model [Neu and Olkhovskaya, 2020, Liu et al., 2024], i.e., problems with stochastic X t and adversarial θ t . We also assume that we are given access to the distribution of contexts, similarly to previous studies, such as those by Neu and Olkhovskaya [2020]. Linear bandit problems can be considered as special cases of contextual linear bandits in which the context space is a singleton.

Upper and lower bounds on regret for (contextual) linear bandits are shown in Table 1. For linear bandit problems, we show a regret upper bound of O (√ dT log + ( K/d ) ) , which is slightly better than the known best bounds of O ( √ dT log K ) by previous studies [Bubeck et al., 2012, Cesa-Bianchi and Lugosi, 2012, Dani et al., 2007]. One notable aspect of this novel bound is that it matches lower bounds for two important special cases of K = 2 d [Dani et al., 2008, Theorem 3], and K = d (equivalent to the standard K -armed bandits). However, specifying the tight minimax bounds for arbitrary values of K remains an open problem.

For contextual linear bandits, this paper presents an upper bound of O ( √ dT log + ( K min { 1 , S/d } )) together with a matching lower bound, where S = |X| is the cardinality of the context space. Note that the regret upper bound of O ( d √ T log T ) by Liu et al. [2024] applies to the more general setting of the adversarial-context and adversarial-loss model. We also note that the problem formulation of contextual linear bandit problems in some existing studies [Neu and Olkhovskaya, 2020, Kuroki et al., 2024, Olkhovskaya et al., 2024] is different from ours. As noted in Liu et al. [2024], however, this problem of different formulations can be reduced to our setting with dimension d |A| , where |A| is the maximum number of actions in their settings. See [Liu et al., 2024] for a more detailed review of previous studies.

To show the regret lower bound for BwE, we follow the approach employed in the study [Chen et al., 2024] on the minimax regret for problems interpolating problems of (full-information) online learning with expert advice and multi-armed bandit. They have shown that, for the generalization of a K -armed bandit problem in which each arm is associated with ν different experts, the minimax regret is Θ ( √ KT log ν ) . Inspired by their construction of the problem instance, we consider a BwE instance in which N ′ = Θ( N/K ) experts give advice for choosing one of two arms and there are independent Θ( K ) copies of such structures. This yields a lower bound of Ω( √ KT log N ′ ) = Ω( √ KT log( N/K )) .

In the proof of regret upper bounds for (contextual) linear bandits, we develop an algorithm based on the follow-the-regularized-leader (FTRL) approach with a Tsallis entropy regularizer with a parameter α ∈ (0 , 1) : ψ α ( w ) = -1 1 -α ∑ i ∈ [ K ] ( w ( i ) α -w ( i )) . This approach can be interpreted as a generalization of the EXP2 algorithm [Bubeck et al., 2012]. In fact, EXP2 can be regarded as the FTRL approach with Shannon entropy regularization, which coincides with the limit of α -Tsallis entropy as α approaches 1 . This paper shows that the regret bounds can be further improved by not fixing α to 1 but by adjusting it appropriately. Similar approaches have been used successfully in various online learning problems including multi-armed bandits [Audibert and Bubeck, 2009], bandits with expert advice [Kale, 2014], graph bandits [Eldowa et al., 2024], and sleeping bandits [Nguyen and Mehta, 2024]. In addition to FTRL with Tsallis entropy regularization, to achieve O (√ KT log + ( KS d ) ) -regret in contextual linear bandits, we combine a novel technique of context-dependent learning rate . More precisely, we introduce learning rate parameters η ( X t ) that change depending on the observed context X t ∈ X . We show that tuning η : X → R > 0 leads to improved regret bounds.

Notation For a natural number n , denote [ n ] = { 1 , 2 , . . . , n } . For two real vectors x = ( x ( i )) i ∈ [ d ] , y = ( y ( i )) i ∈ [ d ] ∈ R d , let 〈 x, y 〉 denote the inner product between x and y , i.e., 〈 x, y 〉 = ∑ i ∈ [ d ] x ( i ) y ( i ) . Let P ( K ) = { w ∈ [0 , 1] K | ∑ i ∈ [ K ] w ( i ) = 1 } denote the set of probability distributions over [ K ] . For any symmetric matrices A and B , denote A glyph[followsequal] B if and only if A -B is positive-semidefinite. Let tr( A ) denote the trace of square matrices A .

2 Multi-armed bandit with expert advice

2.1 Problem setting

In the problem of multi-armed bandits with expert advice (BwE), the player is given the number of arms K and the number of experts N . In each round, each expert j ∈ [ N ] select an advice e t ( j ) ∈ [ K ] , and the player chooses an expert J t ∈ [ N ] . Then, after the player observes the expert advice ( e t ( j )) j ∈ [ N ] , the player pulls the arm I t = e t ( J t ) ∈ [ K ] and gets feedback of the suffered loss glyph[lscript] t ( I t ) ∈ [0 , 1] . The performance of the player is measured by means of regret R T defined as

Remark 1. Weconsider the problem setting in which the player choose the expert J t before observing expert advice ( e t ( j )) j ∈ [ N ] . This setting is more challenging compared to the one where the player can observe all expert advice before selecting an arm, because the available information is more limited. On the other hand, existing algorithms by Auer et al. [2002], Kale [2014] can be applied to this more challenging setting and achieve the regret upper bounds shown in Table 1. Therefore, we have decided to adopt this setting in this study.

Best expert identification The best expert identification (BEI) problem is a variant of BwE problem in which the player aims to identify the expert attaining the minimum value of expected loss from as few feedbacks as possible. We here assume that ( e t , glyph[lscript] t ) follows an identical distribution D independently for all t = 1 , 2 , . . . . Note that, for ( e, glyph[lscript] ) ∼ D , all elements of e and glyph[lscript] may be dependent, and that we impose the independence assumption only between data at different t . For each expert j ∈ [ N ] we define µ j = E ( e,glyph[lscript] ) ∼D [ glyph[lscript] ( e ( j ))] and let j ∗ ∈ [ K ] denote the best expert in terms of the expectation, i.e., j ∗ ∈ arg min j ∈ [ N ] µ j . In the BEI problem, the player repeats selecting

arms and obtaining feedback for an arbitrary number of times T , and then outputs the estimated optimal expert J T ∈ [ K ] .

For a positive number ε , an expert j is called an ε -optimal expert if µ j < µ j ∗ + ε . An algorithm A for BEI is called ( ε, δ ) -probably approximately correct (PAC) if it outputs an ε -optimal expert with probability at least (1 -δ ) . For any algorithm A for BEI, let T ( A , D ) denote the expected value of the number of rounds T until A is terminated when it is applied to the problem instance associated with D .

2.2 Reduction from BEI to BwE

This section shows that BEI problems can be reduced to BwE problems. More precisely, given an algorithm for BwE problems with a regret upper bound, we can construct an ( ε, δ ) -PAC algorithm with a bounded number of queries, as follows:

Lemma 1. Suppose that there exists an algorithm A for BwE such that the regret is bounded as R T ≤ r ( T ) for every T . For an arbitrary ε ∈ (0 , 1) , let T ∗ be such that T ∗ ≥ 2500 · r ( T ∗ ) ε . Then, there exists an ( ε, 0 . 05) -PAC algorithm A ′ for BEI such that T ( A , D ) ≤ T ∗ for any D .

Proof. Consider the following algorithm A ′ for best-expert identification: (i) Run A for T ∗ rounds and let T j be the number of rounds in which the expert j is chosen, i.e., T j = |{ t ∈ [ T ∗ ] | J t = j }| . (ii) Output ˆ J = j with probability T j /T ∗ , i.e., ˆ J is chosen so that Pr[ J = j ] = T j /T ∗ for all j ∈ [ N ] . Let us show that this is an ( ε, 0 . 05) -PAC algorithm.

Let N ε ⊆ [ N ] denote the set of ε -optimal experts and let N c ε = [ N ] \ N ε . Let T ε = ∑ j ∈N ε T j denote the number of choosing ε -optimal experts. We then have R T ∗ ≥ E [ ε · ∑ j ∈N c ε T j ] = ε E [( T ∗ -T ε )] as we suffer per-round regret of at least ε in expectation, in every round when an expert in N c ε is chosen. Hence, from Markov's inequality, we have

where the last inequality follows from the assumption of T ∗ ≥ 2500 · r ( T ∗ ) ε . Hence, from the construction of algorithm A ′ ,

which means that A ′ is an ( ε, 0 . 05) -PAC algorithm for BEI.

2.3 Construction of problem instance

For any p ∈ [0 , 1] , let Ber( p ) represent a Bernoulli distribution of parameter p , i.e., if X ∼ Ber( p ) , then Pr[ X = 1] = p and Pr[ X = 0] = 1 -p .

Without loss of generality, we consider the case that N and K can be expressed as N = N ′ m +1 and K = 2 m + 1 for some positive integers N ′ and m . We denote the set of experts by E = { 0 } ∪ { ( u, v ) } u ∈ [ m ] ,v ∈ [ N ′ ] and the set of arms by A = { 0 } ∪ { ( u, b ) } u ∈ [ m ] ,b ∈{ 0 , 1 } . The expert advice e t is given so that e t (0) = 0 and e t (( u, v )) ∈ { ( u, 0) , ( u, 1) } for all u ∈ [ m ] and v ∈ [ N ′ ] .

Fix u ∗ ∈ [ m ] , v ∗ ∈ [ N ′ ] , and ε ∈ [0 , 1) . We define distributions of ( e, glyph[lscript] ) as follows:

D ( ε ) : When ( e, glyph[lscript] ) follows D ( ε ) , glyph[lscript] (0) follows Ber((1 -ε ) / 2) . For each u ∈ [ m ] , glyph[lscript] (( u, 0)) and glyph[lscript] (( u, 1)) are given by glyph[lscript] (( u, 0)) = b u and glyph[lscript] (( u, 1)) = 1 -b u , where b u follows Ber(1 / 2) . The expert advice is given by e (( u, v )) = ( u, b uv ) , where each b uv follows Ber(1 / 2) for u ∈ [ m ] and v ∈ [ N ′ ] . All elements of glyph[lscript] (0) , ( b u ) u ∈ [ m ] and ( b uv ) u ∈ [ m ] ,v ∈ [ N ′ ] are independent.

D ( ε, u ∗ , v ∗ ) : When ( e, glyph[lscript] ) follows D ( ε, u ∗ , v ∗ ) , glyph[lscript] and e follows the same distribution as D ( ε ) except for e (( u ∗ , v ∗ )) . The value of e (( u ∗ , v ∗ )) is given as e (( u ∗ , v ∗ )) = ( u ∗ , b u ∗ v ∗ ) = ( u ∗ , | glyph[lscript] (( u ∗ , 0)) -b ′ | ) , where b ′ is a random variable that follows Ber(1 / 2 -ε ) independently of the other randomness. In other words, the second element of e (( u ∗ , v ∗ )) is chosen so that glyph[lscript] ( e ( u ∗ , v ∗ )) follows Ber(1 / 2 -ε ) .

The probability distribution of D ( ε, u ∗ , v ∗ ) is constructed so that the marginal distribution of e for ( e, glyph[lscript] ) ∼ D ( ε, u ∗ , v ∗ ) is same as that for D ( ε ) . Indeed, the values of e is determined by ( b uv ) u ∈ [ m ] ,v ∈ [ N ′ ] , and the marginal distribution of b u ∗ ,v ∗ for D ( ε, u ∗ , v ∗ ) is a mixture distribution given as Pr[ glyph[lscript] (( u ∗ , 0)) = 0] · Ber ( 1 2 -ε ) + Pr[ glyph[lscript] (( u ∗ , 0)) = 1] · Ber ( 1 2 + ε ) = 1 2 Ber ( 1 2 -ε ) + 1 2 Ber ( 1 2 + ε ) = Ber ( 1 2 ) , which is equivalent to distributions for D ( ε ) . In addition, if ( e, glyph[lscript] ) ∼ D follows D ( ε, u ∗ , v ∗ ) , the expected loss for choosing expert ( u, v ) is given as

where 1 [ · ] represents the indicator function, i.e., 1 [ E ] = 1 if E is true and 1 [ E ] = 0 otherwise. This means that, for the problem instance of BEI associated with D ( ε, u ∗ , v ∗ ) , the expert ( u ∗ , v ∗ ) is the only ( ε/ 2) -optimal action.

We denote D T ( ε ) = ( D ( ε )) T . Let D T ( ε, u ∗ ) denote the uniform mixture of { ( D ( ε, u ∗ , v ∗ )) T } v ∗ ∈ [ N ′ ] : D T ( ε, u ∗ ) = 1 N ′ ∑ v ∗ ∈ [ N ′ ] ( D ( ε, u ∗ , v ∗ )) T .

2.4 Lower bound for best-expert identification problems

We first consider the special case of m = 1 and provide an instance-specific lower bound for BEI associated with D ( ε ) and {D ( ε, 1 , v ∗ ) } v ∗ ∈ N ′ .

Lemma 2. Suppose m = 1 and let A be an ( ε/ 2 , 0 . 05) -PAC algorithm for BEI. We then have T ( A , D ( ε )) ≥ 1 2 ⌊ ln( N ′ / 4) 4 ε 2 ⌋ =: T ∗ 2 .

glyph[negationslash]

Proof. Let T be the number of rounds the algorithm A proceeds. Let E denote the event that the algorithm A terminate at a round before T ∗ +1 (i.e., T ≤ T ∗ ) and output ˆ J = 0 . Then, from Pinsker's inequality, we have ∣ ∣ Pr [ E | (( e t , glyph[lscript] t )) T ∗ t =1 ∼ D T ∗ ( ε ) ] -Pr [ E | (( e t , glyph[lscript] t )) T ∗ t =1 ∼ D T ∗ ( ε, 1) ]∣ ∣ ≤ D KL ( D T ∗ ( ε, 1) ||D T ∗ ( ε )) ≤ 0 . 25 , where the last inequality follows from Lemma 9 in the supplementary. As A is an ( ε/ 2 , 0 . 05) -PAC algorithm, we have Pr [ E | (( e t , glyph[lscript] t )) T ∗ t =1 ∼ D T ∗ ( ε, 1) ] ≤ 0 . 05 . By applying the union bound, we obtain 1 -Pr [ E | (( e t , glyph[lscript] t )) T ∗ t =1 ∼ D T ∗ ( ε ) ] = Pr [ T > T ∗ | (( e t , glyph[lscript] t )) T ∗ t =1 ∼ D T ∗ ( ε ) ] + Pr [ T ≤ T ∗ , ˆ J = 0 | (( e t , glyph[lscript] t )) T ∗ t =1 ∼ D T ∗ ( ε ) ] ≤ Pr [ T > T ∗ | (( e t , glyph[lscript] t )) T ∗ t =1 ∼ D T ∗ ( ε ) ] + 0 . 05 . Combining these inequalities, we obtain Pr [ T > T ∗ | (( e t , glyph[lscript] t )) T ∗ t =1 ∼ D T ∗ ( ε ) ] ≥ 0 . 95 -Pr [ E | (( e t , glyph[lscript] t )) T ∗ t =1 ∼ D T ∗ ( ε ) ] ≥ 0 . 95 -0 . 25 -Pr [ E | (( e t , glyph[lscript] t )) T ∗ t =1 ∼ D T ∗ ( ε, 1) ] ≥ 0 . 95 -0 . 25 -0 . 05 = 0 . 65 , which implies that T ( A , D ( ε )) ≥ E [ T | (( e t , glyph[lscript] t )) T ∗ t =1 ∼ D T ∗ ( ε ) ] ≥ 0 . 65 T ∗ . This completes the proof.

We can obtain a lower bound for BEI for general m by using Lemma 2.

Theorem 1. Let A be an ( ε/ 2 , 0 . 05) -PAC algorithm for BEI. We then have T ( A , D ( ε )) ≥ m 2 ⌊ ln( N ′ / 4) 4 ε 2 ⌋ = mT ∗ 2 .

2.5 Lower bound for multi-armed bandits with expert advice

We are now ready to provide a lower bound for BwE.

Theorem 2. For any algorithm for BwE and for any sufficiently large T > 0 , there exists a problem instance for which R T ≥ C √ TK log + N K , where C > 0 is a universal constant.

Proof. We fix T and set m = glyph[floorleft] ( K -1) / 2 glyph[floorright] and N ′ = glyph[floorleft] ( N -1) /m glyph[floorright] . Fix ε = Θ (√ m log N ′ T ) so that T < m 2 ⌊ ln( N ′ / 4) 4 ε 2 ⌋ . Assume that R T < C √ TK log + N K := r ( T ) = O ( √ Tm log N ′ ) , where the universal constant C > 0 is sufficiently small so that 2500 · r ( T ) ε ≤ T holds. Then, from Lemma 1, there exists an ( ε, 0 . 05) -PAC BEI algorithm A achieving T ( A , D ) ≤ T for any D . This contradicts Theorem 1, which implies that R T ≥ C √ TK log + N K .

This lower bound, together with the upper bound presented by Kale [2014], means that the minimax regret for BwE is of Θ (√ KT log + N K ) .

3 Contextual linear bandit

3.1 Problem setting

Before the game starts, the player is given a set X of contexts, a finite set [ K ] of arms, a feature mapping φ : X × [ K ] → R d . In each round, the environment chooses loss vector θ t ∈ R d . Then a context X t ∈ X is drawn from a fixed distribution D , and is revealed to the player. The player chooses an action I t ∈ [ K ] and get feedback of glyph[lscript] t ∈ [ -1 , 1] , of which expectation is 〈 θ t , φ ( X t , I t ) 〉 . For any policy π ∗ : X → [ K ] , we define the regret by

We assume that |〈 θ t , φ ( x, i ) 〉| ≤ 1 for all x ∈ X and a ∈ [ K ] . We also pose the following assumption regarding the distribution of the context.

Assumption 1. We assume that the set of contexts is a finite set X = [ S ] and that there exists L ≥ S such that the probability g ( x ) := Pr[ X t = x ] is bounded from below as g ( x ) ≥ 1 /L for all x ∈ [ S ] . Assume that the function g is given.

For any randomized policy p : X → P ( K ) , we denote

Assumption 2. We assume that there exists an exploration policy p 0 : X → P ( K ) such that λ ( p 0 ) < ∞ .

Let λ 0 > 0 denote (an upper bound of) the value of λ ( p 0 ) . We may assume that λ 0 ≤ Ld without loss of generality. In fact, if we set p 0 ( x ) to be a g-optimal design (see, e.g., [Lattimore and Szepesvári, 2020, Section 21.1]) for Z x = { φ ( x, i ) | i ∈ [ K ] } , we then have φ ( x, i ) glyph[latticetop] V ( p ) -1 φ ( x, i ) ≤ 1 g ( x ) φ ( x, i ) glyph[latticetop] V ( p 0 ( x )) -1 φ ( x, i ) ≤ dL holds for all x ∈ X and i ∈ [ K ] .

3.2 Algorithm

The proposed algorithm is based on the framework follow-the-regularized-leader (FTRL) with Tsallis entropy regularization. For α ∈ (0 , 1) , define a regularization function ψ : P ( K ) → R by

Using this regularizer, we define a randomized policy q t : X → P ( K ) on the basis of FTRL, and set an arm-selection policy p t , as follows:

Algorithm 1 Contextual linear bandit algorithm based on FTRL with Tsallis entropy

Input: Feature mapping φ , learning rates ( η ( x )) x ∈X , exploration policy p 0 , parameters γ, α for t = 1 , . . . , T do

Compute p t given by (1) and (2).

Observe X t , choose I t drawn from p t ( X t ) , and get feedback of glyph[lscript] t .

Compute ˆ θ t defined by (3).

end for where ˆ θ t is an unbiased estimator defined below, η ( x ) > 0 is a learning rate parameter that will be specified later, and γ ∈ (0 , 1 / 2) is a parameter that will be chosen depending on η . In each round t , we choose I t following p t ( X t ) and get feedback of glyph[lscript] t . We then compute an unbiased estimator ˆ θ t of θ t defined by

This is in fact an unbiased estimator as

The procedure of our proposed algorithm is summarized in Algorithm 1.

3.3 Regret upper bound

This section provides an upper bound on the regret for Algorithm 1. In the following, we use the symbol β = 1 -α ∈ (0 , 1) for simplicity of notation. We will show a regret upper bound as follows:

Theorem 3. Suppose that 1 / 2 ≤ α < 1 and that γ satisfies

Then, the regret for Algorithm 1 is bounded as follows:

From this theorem, by tuning parameters α and η ( x ) on the basis of T, d, K and g , we obtain the following:

Corollary 1. For sufficiently large T , Algorithm 1 achieves the following:

Remark 2. The proposed algorithm (Algorithm 1) can also work for the infinite context case, in which it enjoys the second regret upper bound in Corollary 1. In fact, this regret upper bound does not include S = |X| or L , and the value of g ( x ) is not required to define η ( x ) in showing this second upper bound. Hence, we can show this bound without the assumption that X is a finite set. In the infinite context case, however, further challenges regarding the computational complexity of the algorithm should be noted. For example, we need to compute V ( p t ) in the algorithm as it appears in the definition of ˆ θ t in (3), which tend to be computationally expensive, depending on the computational model and the setup of distributions.

To prove Theorem 3, we introduce some notations: Denote ˆ glyph[lscript] t ( x ) = (〈 ˆ θ t , φ ( x, i ) 〉) i ∈ [ K ] ∈ R K , which is an unbiased estimator of ( 〈 θ t , φ ( x, i ) 〉 ) i ∈ [ K ] . For any w,w ′ ∈ P ( K ) , let D ( w,w ′ ) denote the Bregman divergence associated with ψ , i.e., D ( w,w ′ ) = ψ ( w ) -ψ ( w ′ ) -〈∇ ψ ( w ′ ) , w -w ′ 〉 . From the standard analysis technique for FTRL (see, e.g., [Lattimore and Szepesvári, 2020, Chapter 28]) and the idea of ghost sample X 0 ∼ D drawn independently from all other randomness (see, e.g., [Neu and Olkhovskaya, 2020]), we obtain the following upper bound:

We note that X 0 is a random variable that does not appear in the decision-making process or algorithms, but appear only in the analysis, and is defined to be independent of X 1 , X 2 , . . . , X T (and therefore is independent of any other variables including q t ). The first component of the right-hand side of this formula can be bounded via the following lemma:

Lemma 4. Suppose that η > 0 , glyph[lscript] ∈ R K and w ∈ P ( K ) satisfy η | glyph[lscript] ( i ) | ≤ 1 -α 4 w ( i ) α -1 for all i ∈ [ K ] . We then have 〈 glyph[lscript],w -w ′ 〉 -1 η D ( w ′ , w ) ≤ 4 η 1 -α ∑ i ∈ [ K ] w ( i ) 2 -α glyph[lscript] ( i ) 2 .

This follows, e.g., directly from the first part of Lemma 9 given by Ito et al. [2024]. To check the sufficient conditions for applying Lemma 4, we use the following lemma:

Lemma 5. It holds for any x ∈ X and i ∈ [ K ] that

Consequently, we have

Combining Lemmas 4 and 5, we obtain the following:

Lemma 6. Suppose that α ≥ 1 / 2 and that γ is given as (5) . Then it holds for any t and x ∈ X that

Further, the expectation of the right-hand side of (8) can be bounded as in the following, which is the key lemma for leading to an improved regret bound of O ( √ dT log + KS d ) .

Lemma 7. We have

Proof. From the definition of ˆ glyph[lscript] t ( x, i ) , for any fixed x ∈ X and i ∈ [ K ] , we have

Denote η 1 = sup x ∈X η ( x ) /g ( x ) β . Then, from (10), we have

where we define u ( x, i ) = g ( x ) q t ( x, i ) and v ( x, i ) = g ( x ) q t ( x, i ) φ ( x, i ) glyph[latticetop] V ( q t ) -1 φ ( x, i ) . We then have (i) u ( x, i ) ≥ 0 for all x ∈ X and i ∈ [ K ] ; ∑ x ′ ∈X ∑ i ′ ∈ [ K ] u ( x ′ , i ′ ) = 1 , (ii) ∑ x ′ ∈X ∑ i ′ ∈ [ K ] v ( x ′ , i ′ ) = d , and (iii) 0 ≤ v ( x, i ) ≤ 1 for all x ∈ X and i ∈ [ K ] . Indeed, the condition (i) is clear from the fact that u ( x, i ) is a probability mass function over X × [ K ] . The condition (ii) follows from

The condition (iii) follows from Lemma 8 and the fact that as g ( x ) q t ( x, i ) φ ( x, i ) φ ( x, a ) glyph[latticetop] glyph[followsequal] V ( q t ) . Let U ⊆ X × [ K ] be the topd subset with respect to the values of u ( x, a ) , i.e., let U be such that | U | = d and u ( x, i ) ≥ u ( x ′ , i ′ ) for any ( x, i ) ∈ U and any ( x ′ , i ′ ) ∈ ( X × [ K ]) \ U . Then, when we consider maximizing ∑ x ∈X ∑ i ∈ [ K ] u ( x, i ) β v ( x, i ) subject to the constraint of (ii) and (iii) on v , the maximum is attained by v ( x, i ) = 1 [( x, i ) ∈ U ] . We hence have ∑ x ∈X ∑ i ∈ [ K ] u ( x, i ) β v ( x, i ) ≤ ∑ ( x,i ) ∈ U u ( x, i ) β ≤ | U | 1 -β ( ∑ ( x,i ) ∈ U u ( x, i ) ) β ≤ d α , where the second inequality follows from Hölder's inequality and the last inequality follows from the condition (i). By combining this with (11), we obtain [LHS of (9)] ≤ η 1 d α 1 -γ . Similarly, denoting η 0 = sup x ∈X η ( x ) , we obtain [LHS of (9)] ≤ η 0 1 -γ E [ ∑ i ∈ [ K ] q t ( X 0 , i ) φ ( X 0 , i ) glyph[latticetop] V ( q t ) -1 φ ( X 0 , i ) ] = η 0 1 -γ E [ tr ( V ( q t ) -1 ∑ i ∈ [ K ] q t ( X 0 , i ) φ ( X 0 , i ) φ ( X 0 , i ) glyph[latticetop] )] = η 0 1 -γ ∑ x ∈X ∑ i ∈ [ K ] v ( x, i ) = η 0 d 1 -γ , which completes the proof.

Now we are ready to provide an upper bound on regret.

We can easily see that Theorem 3 is a direct consequence of Lemmas 3, 6 and 7.

3.4 Regret lower bound

The following theorem implies that the regret upper bound given in Corollary 1 achieved by the Algorithm 1 is tight for S ≤ K ≤ 2 d .

Theorem 4. Suppose any d ′ ≥ 1 , S ≥ 1 and T = Ω( d ′ 2 S ) . Then, for any algorithm for contextual linear bandit problems with K = 2 d ′ , d = d ′ S , and |X| = S there exists a problem instance for which R T = Ω ( d ′ √ ST ) .

This theorem implies a regret lower bound of Ω (√ dT log + ( K min { 1 , S d }) ) . To see this, we first note Theorem 4 implies that, if some d ′ satisfies K ≥ 2 d ′ and d ≥ d ′ S , we can obtain a regret lower bound of R T = Ω( d ′ √ ST ) . Let ( d, K, S ) be an arbitrary given parameter set that satisfies K ≤ 2 d ≤ K S . We then have log 2 K ≤ d ≤ S log 2 K . Define d ′ := glyph[floorleft] log 2 K glyph[floorright] and S ′ := glyph[floorleft] d/ log 2 K glyph[floorright] ≤ S . Then, as we have K ≥ 2 d ′ and d ≥ S ′ log 2 K ≥ S ′ d ′ = Ω( d ) , from Theorem 4, we obtain a regret lower bound of R T = Ω( d ′ √ S ′ T ) = Ω( √ S ′ d ′ Td ′ ) = Ω( √ dTd ′ ) . By combining this with d ′ = Ω(log K ) = Ω ( log + ( K min { 1 , S d } )) , we obtain R T = Ω (√ dT log + ( K min { 1 , S d } ) ) . We hence have the following lower bound:

Corollary 2. For any ( d, K, S ) such that K ≤ 2 d ≤ K S and for any algorithm for contextual linear bandit problems, there exists a problem instance for which R T = Ω (√ dT log + ( K min { 1 , S d }) ) .

glyph[negationslash]

Proof of Theorem 4. Theorem 4 can be shown by using the result of Dani et al. [2008, Theorem 3]. They provide a lower bound of Ω( d √ T ) for (non-contextual) linear bandit problems with K = 2 d . We use S copies of their problem instance of the dimensionality d ′ , to prove Theorem 4. The context X t is drawn from uniform distribution over X = [ S ] , each element of which corresponds to one of the copies of the linear bandit instance. Then, for any x ∈ X , the number of rounds t ≤ T at which X t = x is of Ω( T/S ) with a probability at least 1 / 2 . Hence, the expected cumulative regret suffered for rounds at which X t = x is of Ω( d ′ √ T/S ) for each x ∈ X . By summing this for all x ∈ X , we obtain the lower bound of Ω( Sd ′ √ T/S ) = Ω( d ′ √ ST ) . Note that features mapping φ need to be designed so that φ ( x, i ) and φ ( x ′ , i ′ ) are orthogonal for any x = x ′ and i, i ′ ∈ [ K ] . We can satisfy this condition by setting the dimension of the entire feature space to d = d ′ S .

4 Conclusion

In this study, we investigated the minimax regret in the contexts of the multi-armed bandit with expert advice and contextual linear bandit problems. For the former, we established a regret lower bound of O ( √ KT log N K ) in the setting where the player selects an expert before observing expert advice. This bound matches, up to a constant factor, the upper bound provided by Kale [2014]. Additionally, for the contextual linear bandit problem, we proposed an algorithm that achieves a regret upper bound of O ( √ dT log( K min { 1 , S d } )) . As illustrated in Table 1, this upper bound aligns with the lower bound under certain conditions on the parameters ( K,S,D ) .

Remaining challenges in the problem with expert advice include establishing similar lower bounds when the player can observe expert advice before decision-making in each round. Furthermore, determining the minimax regret in a broader parameter setting for (contextual) linear bandits remains an open problem. Relaxing Assumption 1 on the contextual distribution is also an important direction for enhancing practical applicability.

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J.-Y. Audibert and S. Bubeck. Minimax policies for adversarial and stochastic bandits. In Conference on Learning Theory , pages 217-226, 2009.

Auer, N. Cesa-Bianchi, Y. Freund, and R. E. Schapire. The nonstochastic multiarmed bandit problem. SIAM Journal on Computing , 32(1):48-77, 2002.

Bubeck, N. Cesa-Bianchi, and S. M. Kakade. Towards minimax policies for online linear optimization with bandit feedback. In Conference on Learning Theory , pages 41.1-41.14, 2012.

Cesa-Bianchi and G. Lugosi. Combinatorial bandits. Journal of Computer and System Sciences , 78(5):1404-1422, 2012.

Chen, Y. He, and C. Zhang. On interpolating experts and multi-armed bandits. In International Conference on Machine Learning , pages 6776-6802, 2024.

Chu, L. Li, L. Reyzin, and R. Schapire. Contextual bandits with linear payoff functions. In International Conference on Artificial Intelligence and Statistics , pages 208-214, 2011.

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Eldowa, E. Esposito, T. Cesari, and N. Cesa-Bianchi. On the minimax regret for online learning with feedback graphs. Advances in Neural Information Processing Systems , 36, 2024.

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A Auxiliary lemma

Lemma 8. For any positive-definite matrix A ∈ R d × d and a real vector x ∈ R d , if A glyph[followsequal] xx glyph[latticetop] , it holds that xA -1 x ≤ 1 .

Proof. From the assumption of A glyph[followsequal] xx glyph[latticetop] , we have

B Omitted lemmas and proofs in Section 2

Lemma 9. Suppose m = 1 . It holds for any N ′ , T ∗ ∈ N and ε ∈ [0 , 1 / 2] that

Consequently, if T ∗ ≤ ln( N ′ / 4) 4 ε 2 , then the above value of the KL divergence is at most 1 / 4 .

Proof. For ( e, glyph[lscript] ) , let b 1 ∈ { 0 , 1 } , ( c v ) v ∈ [ N ′ ] ∈ { 0 , 1 } N ′ be binary variables such that glyph[lscript] ((1 , 0)) = b u , glyph[lscript] ((1 , 1)) = 1 -b u , and e ((1 , v )) = | b u -c v | . Then, all the values of ( e, glyph[lscript] ) are determined by glyph[lscript] (0) , b 1 and ( c v ) v ∈ [ N ′ ] . When ( e, glyph[lscript] ) follows D ( ε ) , then ( glyph[lscript] (0) , b 1 , ( c v ) v ∈ [ N ′ ] ) follows E 0 := Ber((1 -ε ) / 2) × Ber(1 / 2) × (Ber(1 / 2)) N ′ . On the other hand, when ( e, glyph[lscript] ) follows D ( ε, 1 , v ∗ ) then ( glyph[lscript] (0) , b 1 , ( c v ) v ∈ [ N ′ ] ) follows E v ∗ := Ber((1 -ε ) / 2) × Ber(1 / 2) × F N ′ ( ε, v ∗ ) , where we define F v ∗ = (Ber(1 / 2)) v ∗ -1 × Ber(1 / 2 + ε ) × (Ber(1 / 2)) N ′ -v ∗ . That is, if ( c v ) v ∈ [ N ′ ] follows F v ∗ , the v ∗ -th element c v ∗ follows Ber(1 / 2) and the other elements follow Ber(1 / 2) independently. Hence, from the data processing inequality, recalling the definition of D T ∗ ( ε ) and D T ∗ ( ε, v ∗ ) given in Section 2.3, we obtain

Let p 0 : { 0 , 1 } T ∗ × N ′ → R and p v : { 0 , 1 } T ∗ × N ′ be the probability mass functions for ( (Ber(1 / 2)) N ′ ) T ∗ and ( F v ∗ ) T ∗ . Then, from the definition of the KL divergence, we have

where we used Jensen's inequality and the fact that ln( x ) is a concave function. The ratio p v ∗ ( c ) p 0 ( c ) of probabilities can be expressed as

glyph[negationslash]

Hence, if v = v ∗ then

where we used the condition that elements of ( c tv ) are independent. Further, if v = v ∗ , we then have

Therefore, we have

which completes the proof.

B.1 Proof of Theorem 1

Proof. For each u ∈ [ m ] , let T u ( A , D ( ε )) denote the expected value of the number of rounds in which the algorithm A observes glyph[lscript] t (( u, 0)) or glyph[lscript] t (( u, 1)) before termination. Then, we will show that T u ( A , D ( ε )) ≥ T ∗ / 2 under the assumption that A is an ( ε/ 2 , 0 . 05) -PAC algorithm.

Fix any u ∈ [ m ] and suppose that A is an ( ε/ 2 , 0 . 05) -PAC BEI algorithm for problems with general m . Then we can construct an ( ε/ 2 , 0 . 05) -PAC BEI algorithm A ′ for problems with m = 1 such that T ( A ′ , D ′ ( ε )) ≤ T u ( A , D ( ε )) , where D ′ ( ε ) and D ( ε ) represent distributions with ( m = 1 , N ′ = N ′ ) and ( m = m,N ′ = N ′ ) , respectively. To show this, consider the following procedure for solving BEI for m = 1 based on A : Run algorithm A . When algorithm A pulls an arm a t other than ( u, 0) or ( u, 1) , then generate e t from D ( ε ) and glyph[lscript] ( a t ) from Ber(1 / 2) independently of e t , and feed them to A . When algorithm A pulls ( u, 0) or ( u, 1) , then query the instance of m = 1 and feed the observed losses to A . Then, this construction provide an ( ε/ 2 , 0 . 05) -PAC BEI algorithm for the instance of m = 1 , and if the instance is associated with D ′ ( ε ) , the number of queries to this distributions has the expected value of T u ( A , D ( ε )) . We hence have T ( A ′ , D ′ ( ε )) ≤ T u ( A , D ( ε )) .

C Omitted proofs in Section 3

C.1 Proof of Lemma 3

Proof. Fix π ∗ : X → [ K ] . From the definitions of p t in (2) and X 0 , and the fact that ˆ θ t is an unbiased estimator of θ t as shown in (4), we have

From [Lattimore and Szepesvári, 2020, Theorem 28.5], for any x ∈ X , ( q t ( x )) t ∈ [ T ] defined by the FTRL procedure (2) satisfies

Combining (14) and (8) with the fact that ψ ( q 1 ( x )) = ψ ( 1 /K ) = -K β -1 α , we obtain the desired inequality.

C.2 Proof of Lemma 5

Proof. From the definition of p t in (2), as we have V ( p t ) glyph[followsequal] γV ( p 0 ) , we have

Further, as we have V ( p t ) glyph[followsequal] (1 -γ ) V ( q t ) glyph[followsequal] (1 -γ ) g ( x ) q t ( x, i ) φ ( x, i ) φ ( x, i ) glyph[latticetop] , from Lemma 8, we have φ ( x, i ) glyph[latticetop] V ( p t ) -1 φ ( x, i ) ≤ 1 (1 -γ ) g ( x ) q t ( x,i ) . This, together with (16), yields (6). From the definition (3) of ˆ θ t , we have

where we used Cauchy-Schwarz inequality and the assumption that | glyph[lscript] t | ≤ 1 . From this and (6), we have (7).

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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of MAB with expert advice and the contextual linear MAB. Specificially, the authors proposed a lower bound of $\\Omega(\\sqrt{KT\\log\\frac{N}{K}})$, improving upon previously known lower bound $\\Omega(\\sqrt{KT\\frac{\\log N}{\\log K}})$ and showing that the previous upper bound $O(\\sqrt{KT\\log\\frac{N}{K}})$ is tight. For the second problem, the authors design an algorithm based on FTRL with Tsallis entropy regularizer and achieve $O(\\sqrt{dT \\log K(\\min(1,S/d))})$ regret bound. A matching lower bound is also proven (under certain condition of $K$) based on an extension of the previous lower bound constructed for linear bandits.\n\nSTRENGTHS:\n- The paper bridges the gap between the lower and the upper bound for MAB with expert advice, solving an open problem in this field.\n- While it is not technically hard to show that a regret minimization algorithm can be transformed to a best expert identification algorithm, the reduction with a specific construction of the expert problem instance leads to a matching lower bound, which is interesting to me.\n- While both the algorithm and the lower bound construction for contextual linear case are kind of standard from a technical perspective, the matching regret bound is good to know.\n\nWEAKNESSES:\nWhile I do not have much concern for the MAB with expert advice problem, for the contextual linear bandit problem, one of my concern is that the algorithm only works for the finite context case, which is very restrictive in real applications. In addition, the implementation of the algorithm requires the knowledge of the context distribution, which is also a limitation of the algorithm. As for the regret bound, while the leading term matches the lower bound, the optimality of the additonal term is unclear. Specifically, in the second bound, $L$ may also be related to $T$ (e.g. $T^{-c}$), making the additional term dominant.\n\nQUESTIONS:\n- Whether Algorithm 1 can be extended to infinite / unknown context case?\n- Whether the additional term in the upper bound for contextual linear bandits is unavoidable?\n\nLIMITATIONS:\nSee \"weakness\" and \"questions\".", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-07"}

comments: SUMMARY:
This paper studies the problem of bandit learning with expert advice. The main contributions are two refined regret bounds: (1) A matching lower regret bound of $\Omega(\sqrt{KT\log(N/K)})$ for multi-armed bandit problem; (2) $\Theta(\sqrt{dT\log(K,\min\{1,S/d\})})$ regret bound for contextual linear bandits.

STRENGTHS:
This authors prove matching regret bounds for bandit problem with expert advice, which is significant enough for an acceptance.

WEAKNESSES:
My concern is about the writing. After reading the paper, I think the high-level intuitions of the construction for lower bounds could be clarified in one or two pages.

I also have a question about the proof of Lemma 3. In equation (15), a lemma in the bandit algorithm book is introduced to derive regret bounds. As far as I can see, the lemma only works for fixed $X_0$. However, in the learning process $p_t$ might depends on the historical contexts $X_0,X_1,X_2,...,X_{t-1}$. I am confused about the usage of the lemma and hope for some explanations.

QUESTIONS:
Please find my question in the comments above.

LIMITATIONS:
Yes.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-12

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this work, the authors tackle the existing gap between upper and lower bounds in bandits with expert advice. An existing lower bound scaled as $\\Omega(\\sqrt{KT \\frac{\\log N}{\\log K}}$, while the state of the art of only provided a $O(\\sqrt{KT \\log (N/K)}$ bound (Kale 2014) \nThe authors close this gap by proposing a novel lower bound that matches the result of Kale 2014 and improves upon the lower bound of Seldin and Lugosi (2016). \n\nThen, the authors also consider the problems of linear bandits and contextual linear bandits. For linear bandits, the authors improve upon the state of the art and provide tighter upper bounds, which match existing lower bounds for various relations of the dimension $d$ and the number of arms $K$.\nThe study of contextual linear bandits in this setting is fairly recent and there, the authors propose novel algorithm that improves upon the state of the art and a novel lower bound that matches their upper bound for certain combinations of dimensions $d$ and number of arms $K$.\n\nSTRENGTHS:\nThis paper provides several significant improvements for variations around the multi-armed bandits. Notably, several of their contributions are lower bounds, one of which closed an open problem that had been open for 8 years.\nThe strategy used to solve this problem is a reduction from the best expert identification problem to the bandits with expert advice setting, and this approach will likely lead to other improvements in different fields. \n\nIn the field of linear bandits, the approach using FTRL with the correct tuning of Tsallis entropy has proved crucial to solving many bandit problems, not just in the adversarial setting but in the more challenging best-of-both-worlds regime. \n\nIn all of these sections, the authors provide detailed proofs that appear correct.\n\nWEAKNESSES:\nIf anything, this paper could benefit from some discussions about the use of FTRL with Tsallis-Inf with tuning of $\\alpha \\in (1/2, 1)$ which is used to solve many variations of the multi-armed bandits problem (for example but not limited to: bandits with feedback graphs (Zimmert and Lattimore (2019), Ito et al. (2022), Dann, Wei and Zimmert (2023) or Decoupling explorations and exploitation in MAB (Rouyer and Seldin (2020),  Jin, Liu and Luo (2023)). \n\nUsing this framework is very beneficial in best-of-both-worlds settings rather than in the purely adversarial or purely stochastic regimes. Have you considered generalizing your results to that setting?\n\nQUESTIONS:\nSee weaknesses.\n\nLIMITATIONS:\nPurely theoretical work. The scope of the results is clearly indicated in the theorems.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-16"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper investigates two significant extensions of multi-armed bandit problems: multi-armed bandits with expert advice (MwE) and contextual linear bandits (CLB). For MwE, the authors close the gap between previously known upper and lower bounds, establishing a matching lower bound of $\\\\Omega\\\\left(\\\\sqrt{KT \\\\log \\\\frac{N}{K}}\\\\right)$, where $K$, $N>K$, and $T$ denote the number of arms, experts, and rounds, respectively. This claim seemingly resolves an open question posed by Seldin and Lugosi (2016), where a $\\\\Omega\\\\left(\\\\sqrt{KT \\\\frac{\\\\log N}{\\\\log K}}\\\\right)$ regret lower bound was shown. For CLB instead, the authors introduce an algorithm that achieves an improved upper bound of $O\\\\left( \\\\sqrt{dT \\\\log\\\\left(K \\\\min\\\\{1, \\\\frac{S}{d}\\\\}\\\\right)\\} \\right)$, where $d$ is the dimensionality of feature vectors, and $S$ is the size of the context space. The authors also provide a matching lower bound, confirming the minimax regret is $\\\\Theta\\\\left( \\\\sqrt{dT \\\\log\\\\left(K \\\\min\\\\{1, \\\\frac{S}{d}\\\\}\\\\right)} \\\\right)$. The results are achieved using the follow-the-regularized-leader (FTRL) approach using the negative Tsallis entropy regularizer for an appropriate tuning of its parameter, and carefully designed context-dependent learning rates.\n\nSTRENGTHS:\nThis work provides relevant contributions to the theoretical understanding of multi-armed bandit problems in terms of the minimax regret.\nEven if the use of the follow-the-regularized-leader (FTRL) approach with negative Tsallis entropy regularization follow from a previous line of work of improved regret bounds in other bandit problems, as made explicit by the authors in the introduction, the regret analysis of the proposed algorithm for CLB requires novel ideas in tuning the learning rate and more care in leveraging the possibility to tune the parameter of the Tsallis entropy.\n\nMore interestingly, the authors manage to provide an improved lower bound for a harder version of bandits with experts advice that matches the regret of the algorithm proposed by Kale (2014).\nThey do so via a proof argument that adapts ideas from the previous work on bandits with feedback graphs by Chen et al. (2023) to MwE, requiring a careful construction of hard problem instances for the latter problem. This construction is nontrivial, interesting, and could potentially help in the design of hard problem instances for proving lower bounds of other related problems.\nFurthermore, introducing logarithmic factors in regret lower bounds is typically challenging and doing so enables a more complete understanding of the problems.\nSimilarly, the lower bound for CLB requires a clever adaptation of previous results.\n\nWEAKNESSES:\nWhile the lower bound for the MwE problem improves on prior results, it requires to consider a **harder** setting than the standard one by restricting the learner.\nSpecifically, they restrict the learner to selecting only experts rather than possibly choosing actions directly, and the learner can observe the expert advice only after making a decision.\nThe presentation is also somewhat misleading, as the authors claim to resolve the open problem left by Seldin and Lugosi (2016) for the “classical” problem of multi-armed bandits with expert advice while it is not exactly the case.\nTo fully address that open question, one needs to consider learners that may select actions irrespectively of the expert advice, and that observe the expert advice before choosing an action.\nNonetheless, the authors themselves make a clear point (rf. Remark 1) that this is indeed a more challenging setting, and my concern mainly revolves about the claims in the abstract and introduction.\nIt should also be clarified that proving the same lower bound for the standard setting remains an open problem that could be interesting to investigate.\n\nSome proofs in the paper feel more like sketches rather than complete proofs. While they provide a good overview of the approach and main ideas, they lack detailed steps and thorough explanations. This makes it challenging for readers to fully verify the results and understand all nuances of the arguments.\nFor instance, the proof of Corollary 1 would benefit from a more detailed explanation of the steps; additionally, an intuitive explanation of the choice of the context-dependent learning rate would help the reader while possibly helping in adapting a similar idea in other related problem settings.\nAnother example is provided by the proof of Theorem 4, which is missing a more formal and explicit computation of the lower bound.\n\nFinally, the results in this current work still do not achieve minimax optimality for the non-contextual linear bandits problem. Indeed, the only known lower bounds that match the upper bound provided in this work hold for the cases $K=d$ and $K=2^d$, respectively. The problem of proving a lower bound for arbitrary values of $K$ is still open and stating this explicitly would make the exposition of the contribution of this paper more transparent and clearer for the reader. A similar question could be made for the contextual version of the problem since the lower bound proved in this paper requires $2^{d/S} \\\\le K \\\\le 2^d$.\n\nQUESTIONS:\n- Could you clarify the doubts that were raised in the weaknesses above?\n- Could you expand a bit on how the argument at lines 278-279 allows us to keep the generality of the values of $S$, $K$, and $d$ as claimed?\n- Do you think the ideas from the different lower bounds for linear bandits could be adapted and combined to prove a regret lower bound for arbitrary values of $K$? Or is there a much larger technical hurdle that needs to be overcome?\n\nMinor comments/typos:\n- Line 99: The argmin is missing $\\\\mu\\_j$\n- Line 104: “denote” instead of “denotes”\n- Line 130: “and the set” instead of “the set”\n- Line 143: “provability” instead of “probability”\n- Line 180: “Lemma” instead of “to Lemma”\n- Line 187: “drawn” instead of ”drown”\n- Line 188: “gets” instead of “get”\n- Line 192: “contexts” instead of “context”\n- In assumption 1, it could be clearer to specify that $L \\\\ge S$ instead of just $L>0$ for it to make sense, otherwise it might not be satisfiable\n- Math display after line 195: $I \\\\sim p(X)$ instead of $I \\\\sim p(x)$\n- Line 196: “that there exists an” instead of “that an”\n- Line 252: the choice of naming the top-$d$ subset of pairs $(x,i)$ as $S$ might cause confusion in the reader, as $S$ is already defined to be the number $S = |\\\\mathcal{X}|$ of contexts\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents new bounds for regret minimization problem in multi-armed bandits with expert advice (MwE) and contextual linear bandits (CLB). For MwE, This paper bridges the gap between existing upper and lower bounds by establishing a new matching minimax optimal lower bound. In the case of CLBs, the authors propose a lower bound and devise an algorithm based on the follow-the-regularized-leader approach, which achieves a matching upper bound.\n\nSTRENGTHS:\n• Novel contribution: The paper introduces new minimax optimal lower bounds for MwE and CLB. Additionally, for CLB, it devises an algorithm that achieves a matching upper bound.\n• Theoretical results: The paper provides comprehensive proofs for each of the newly introduced bounds, offering rigorous theoretical validation of the results.\n\nWEAKNESSES:\n• Motivation: The paper lacks an exploration of practical applications of the proposed work, thus indicating a deficiency in motivation. Incorporating a discussion on potential practical applications would significantly add value of the results presented in the paper.\n• Future Work: The paper does not discuss potential directions for future research.\n• No empirical results verifying the bounds.\n\nI found some typos that the author(s) might want to correct.\n\n• In line 84, ”the player chooses an expert $J_i \\in [K]$ should be replaced by ”the player chooses an expert $J_t \\in [N]$, correct?\n\n• In line 99,”$j^*\\in\\argmin_{j\\in [N]}$ is missing $\\mu_j$\n\n• In line 213: \"This section provide\" to \"This section provides\"\n\nQUESTIONS:\nThe proposed algorithm for contextual linear bandits (CLB) requires an initial exploration policy p0 as one of its input parameters. How is this policy determined at the start of the algorithm? Are there any specific assumptions or conditions on p0 necessary for it to adhere to the presented upper bound?\n\nLIMITATIONS:\nThe authors discuss the assumptions needed for their results to hold.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-31"}