Online Learning with Sublinear Best-Action Queries

Authors: matteo  russo
andrea  celli
riccardo  colini-baldeschi
  meta
federico  fusco
daniel  haimovich
dima  karamshuk
stefano  leonardi
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-14 21:17:56

Abstract: In online learning, a decision maker repeatedly selects one of a set of actions, with the goal of minimizing the overall loss incurred. Following the recent line of research on algorithms endowed with additional predictive features, we revisit this problem by allowing the decision maker to acquire additional information on the actions to be selected. In particular, we study the power of \emph{best-action queries}, which reveal beforehand the identity of the best action at a given time step. In practice, predictive features may be expensive, so we allow the decision maker to issue at most $k$ such queries.

We establish tight bounds on the performance any algorithm can achieve when given access to $k$ best-action queries for different types of feedback models. In particular, we prove that in the full feedback model,  $k$ queries are enough to achieve an optimal regret of $\Theta(\min\{\sqrt T, \frac{T}{k}\})$. This finding highlights the significant multiplicative advantage in the regret rate achievable with even a modest (sublinear) number $k \in \Omega(\sqrt{T})$ of queries.
    
Additionally, we study the challenging setting in which the only available feedback is  obtained during the time steps corresponding to the $k$ best-action queries. There, we provide a tight regret rate of $\Theta(\min\{\frac{T}{\sqrt k},\frac{T^2}{k^2}\})$, which improves over the standard $\Theta(\frac{T}{\sqrt k})$ regret rate for label efficient prediction for $k \in \Omega(T^{2/3})$.

Main paper: 
 introduction: 
 Online learning is a foundational problem in machine learning. In its simplest version, a decision maker repeatedly interacts with a fixed set of n actions over a time horizon T . At each time, the decision maker needs to choose one of a set of actions; subsequently, it receives an action-dependent loss and observes some feedback. These loss functions are generated by an omniscient (but oblivious) adversary and are only revealed on-the-go. The goal of the decision maker is to design a learning algorithm that achieves small regret with respect to the best fixed action in hindsight, i.e., the difference between the decision maker's loss and that of the fixed action. Several online learning algorithms have been developed, characterized by optimal instance-independent regret bound, depending on the feedback model [Cesa-Bianchi and Lugosi, 2006, Slivkins, 2019] .Following the recent literature on algorithms with machine learning-based predictions (see, e.g., the survey by Mitzenmacher and Vassilvitskii [2020] ), we study the case where the learner is allowed to issue a limited number of best-action queries to an oracle that reveals the identity of the best action for that step, so that the learner can choose it. This setting is motivated by scenarios in which obtaining accurate predictions on the optimal choice among numerous actions is possible but comes with significant costs and time constraints. For instance, consider an online platform that continuously moderates posted content (e.g., Meta [Meta, a,b] or Google [Google] ), and the online learning problem it faces: posts are generated one after the other, and the platform's task consists in deciding whether or not to flag the content as harmful. In this application, the platform may do so via (i) content moderation actions that are based on (expert) human reviews (that plays the role of best-action queries), and (ii) automated content moderation decisions, i.e., decisions arising by employing an online learning algorithm. Due to budget or time constraints, access to human reviewing is a scarce resource, and the platform can only employ external reviewers at most k times.While the idea of incorporating hints or queries into online learning models has already been studied [e.g., Bhaskara et al., 2021b Bhaskara et al., , 2023b]] , we are the first to study best-action queries. Bhaskara et al. [2021b] focus on online linear optimization with full feedback, with a query model which outputs vectors that are correlated with the actual losses. In the case of optimization over strongly convex domains, the regret bound improves fromT to log T , even for learners that receive hints for O( √ T ) times. In an alternative model, Bhaskara et al. [2023b] studies comparison queries that allow the decision maker to know in advance, at each time, which among a small number of actions has the best loss. In this model, probing 2 arms is sufficient to achieve time-independent regret bounds for online linear and convex optimization with full feedback, an in the stochastich multi-armed bandit problem. Our model differs from previous ones in two directions: (i) the online learner issues at most k queries (differently from Bhaskara et al. [2023b] ), and (ii) these queries are purely ordinal and the domain is not strongly convex * (so that the logarithmic bound in Bhaskara et al. [2021b] does not apply).Another potential way of addressing how to efficiently using limited moderation is through the abstention learning model, where the learner can "abstain" from making a decision and instead use additional resources to receive the optimal response. This includes models like KWIK and full-information models [Li et al., 2011 , Sayedi et al., 2010 , Zhang and Chaudhuri, 2016 , Cortes et al., 2018 , Neu and Zhivotovskiy, 2020 , Gangrade et al., 2021] .In our model, an online learner repeatedly interacts with n actions over a time horizon T . At the beginning of each time t ∈ [T ] † , the learner chooses one of these actions i t and suffers a loss ℓ t (i t ) generated by an (oblivious) adversary that may depend on both the action and the time; then, it observes some feedback. In this paper, we allow the learning algorithm A to issue a best-action query, for at most k out of T times. When the learner issues a query, an oracle reveals the identity of the best action at that time, i * t , so that the learner can select it. The quality of a learning algorithm is measured via the regret: the difference between its performance and that of the best fixed action in hindsight. The regret of an algorithm A against a sequence of losses ℓ ∈ [0, 1] n×T readsR T (A, ℓ) = t∈[T ] E [ℓ t (i t )] -min i∈[n] t∈[T ] ℓ t (i),where the expectation runs over the (possibly) randomized decisions of the algorithm. We are interested in designing learning algorithms that perform well, i.e., suffer sublinear regret against all possible sequences of losses. For this reason, we denote with R T (A) (without the dependence on ℓ) the worst-case regret of A: R T (A) = sup ℓ R T (A, ℓ). The minimax regret of a learning problem is then the regret achievable by the best algorithm. In our paper we pinpoint the exact minimax regret rates for the problems studied.For the sake of simplicity, we denote with L T (i) the total loss incurred by action i over the whole time horizon:L T (i) = t∈[T ] ℓ t (i). L min T = t∈[T ] ℓ t (i * t )denotes instead the sum of the minimum loss actions at each time. Finally, L T (A k ) = E t∈[T ] ℓ t (i t ) denotes the expected total loss of an algorithm A k issuing at most k best-action queries.Full feedback. We start with the full feedback (a.k.a. prediction with experts) where the learner observes at each time the losses of all the actions after the action is chosen, i.e., all the loss vector ℓ t = (ℓ t (1), ℓ t (2), . . . , ℓ t (n)) after action i t is chosen. We obtain the following results:• We show that by combining the Hedge algorithm with k queries issued uniformly at random (Algorithm 1), we obtain a regret rate of O(min{ √ T , T /k}), see Theorem 2.2.• In Theorem 3.3, we complement this positive result with a tight lower bound: we design a family of instances that forces every algorithm with k queries to suffer the same regret.It is surprising that issuing a sublinear number of queries, say k = T α for α ∈ ( 1 /2, 1), yields a significant improvement on the regret, which becomes T 1-α . Note, the total of the losses incurred by any algorithm in k rounds is at most k, which only affects the overall regret in an additive way; nevertheless, we prove that issuing k queries has an impact on the regret that is multiplicative in k.For instance, T 2 /3 queries are enough to decrease the regret to T 1 /3 , which is well below the Θ( √ T ) minimax regret for prediction with expert without queries [Cesa-Bianchi and Lugosi, 2006] .Label efficient feedback. We then proceed to study a partial feedback model inspired by the label efficient paradigm [Cesa-Bianchi and Lugosi, 2006 ]. In the label-efficient prediction problem, the learner only observes the losses in a few selected times. With label efficient feedback, the learner only observes feedback during the k times where the best-action queries are issued (but only after choosing the action). We obtain the following results:• We modify the (label-efficient) Hedge algorithm [Cesa-Bianchi and Lugosi, 2006] to achieve a regret rate of O(min{T / √ k, T 2 /k 2 }), see Theorem 2.4,• In Theorem 3.4, we show that it is not possible to improve on these rates: there exists instances where all algorithms suffer Ω(T / √ k) regret if k ∈ O(T 2 /3 ) or Ω( T 2 /k 2 ) if k ∈ Ω(T 2 /3 ).We observe that our algorithms improve (for k ∈ Ω(T 2 /3 )) on the regret rate of label efficient prediction with k steps of feedback, which is of order Θ( T / √ k) [Chapter 6.2 in Cesa-Bianchi and Lugosi, 2006] . Also in this case, we observe the surprising multiplicative impact on the regret of the k queries. For instance, k = T 3 /4 queries (which impact on a total loss of the same order but leave unaffected Θ(T ) rounds) are enough to achieve O( √ T ) regret.Stochastic i.i.d. setting. In Section B, we derive (up to polylogarithmic factors) the above results in the stochastic setting, but with different algorithms. Namely, in full feedback, we show how Follow-The-Leader [Auer et al., 2003] achieves regret Θ( T /k) when k ≥ Ω( √ T ), and Θ( √ T ) otherwise. With label efficient feedback, Explore-Then-Commit [Perchet et al., 2015] achieves Θ( T 2 /k 2 ) when k ≥ Ω(T 2 /3 ), and Θ( T / √ k) otherwise. ‡Issuing a best-action query has the effect of inducing a possibly negative instantaneous regret. In fact, the benchmark that the algorithm is comparing against at time t is the loss of the best fixed action over the whole time horizon, which may naturally be larger, at time t, than the best action i * t in that specific time. Overall, the magnitude of the total "negative" regret that is generated by k queries is at most O(k), which is typically sublinear in T , and affects linearly on the overall regret. We prove that this sublinear number of queries has a (surprising) multiplicative effect on the overall regret bound.Our analysis reveals that any algorithm employing a uniform querying strategy can be characterized by a total loss decomposed into two terms: a k /T fraction of the best dynamic loss, representing the sum of the smallest losses at each time, and the total loss of the same algorithm operating without queries, scaled by a factor of 1 -k /T . Additionally, within our proof, we conduct a more refined analysis of the Hedge algorithm in the no-querying setting. Specifically, we can express the regret as a function of the difference between the best dynamic loss and the best-fixed cumulative loss in hindsight. This formulation demonstrates a noteworthy enhancement in regret rates compared to the conventional bound of O( √ T ) when the discrepancy between the best dynamic loss and the best-fixed cumulative loss in hindsight is relatively small. Concerning lower bounds, their construction is more challenging than their "no queries" counterpart. In fact, we need to design hard instances against any query-augmented learning algorithm, and provide tight bounds in both k and T. The lower bounds we construct are stochastic, this implies that the minimax regret we find are tight, even in the stochastic model, where the losses are drawn i.i.d. from a fixed but unknown distribution. To exemplify this, let us consider the following natural instance, which provides a simple proof of the Ω( √ T ) regret lower bound in the adversarial setting (without queries). The instance is composed of two arms, whose rewards are i.i.d. Bernoulli distribution with probability 1 /2. Any learning algorithm achieves T /2 regret, while the best-fixed arm in hindsight is expected to achieve an extra Θ( √ T ) term. § Now, if the learner is given the power to issue (order of) √ T queries, then its regret naturally drops from (order of) √ T to constant.2 Hedge with k Best-Action QueriesIn this section, we propose two algorithms that address online learning with full and label efficient feedback. They are built by combining the Hedge algorithm [Chapter 2 in Cesa-Bianchi and Lugosi, 2006] to uniform queries. We start by presenting some known properties of Hedge (in full feedback) that are crucial key for the main results of this section.Lemma 2.1. Consider the Hedge algorithm Hedge η ( l) run on loss sequence l ∈ [0, U ] n×T with learning rate η < 1 /U. Then, for all action i ∈ [n], it holds that,LT (Hedge η ( l)) ≤ 1 1 -U η • LT (i) + log n η ,where LT (Hedge η ( l)) = t∈[T ],i∈[n] p t (i) • lt (i) is the expected cumulative loss of Hedge η ( l) and LT (i) is the cumulative loss of action i.Proof. We have that, by definition,W T +1 ≥ w T +1 (i)w t (i) • exp -η lt (i) = t∈[T ] exp -η lt (i) = exp -η LT (i) .We also know thatW t+1 ≤ W t •   1 -η i∈[n] p t (i) • lt (i) + η 2 i∈[n] p t (i) • l2 t (i)   .Thus,W T +1 ≤ n • t∈[T ]   1 -η i∈[n] p t (i) • lt (i) + η 2 i∈[n] p t (i) • l2 t (i)   ,which combined with the earlier bound and taking logarithms of both sides, gives-η LT (i) ≤ log n + t∈[T ] log   1 -η i∈[n] p t (i) • lt (i) + η 2 i∈[n] p t (i) • l2 t (i)  § This is a simple corollary of the expected distance of a random walk.≤ log n -η t∈[T ],i∈[n] p t (i) • lt (i) + η 2 t∈[T ],i∈[n] p t (i) • l2 t (i) ≤ log n -(η -U η 2 ) t∈[T ],i∈[n] p t (i) • lt (i).The second inequality above follows from log(1 + z) ≤ z for z ∈ R, and the third by observing thatl2 t (i) ≤ U lt (i), since lt (i) ∈ [0, U ].The lemma follows by rearranging the terms above.2.1 Full Feedback: An O( T log n k ) Regret Bound Theorem 2.2. Consider the problem of online learning with full feedback and k best-action queries, then there exists an algorithm A k that guaranteesR T (A k ) ≤ min T log n, T log n k .We prove that Algorithm 1 exhibits the desired regret guarantees. In particular, as we illustrate next, this algorithm performs uniform querying, i.e., they choose a uniformly random subset Q ⊆ [T ] of size k where to issue best-action queries. In the case of uniform queries, a useful simplification can be made. Observation 2.3. Let A 0 be an algorithm with no querying power, and consider A k , obtained from A 0 by equipping it with k uniformly random queries across the time horizon T or with an independent query on each step time step with probability k /T . Similarly, let i 0 t and i t be the actions selected by A 0 and A k at time t. Then, for all adversarial sequences ℓ ∈ [0, 1] n×T of action losses,E [ℓ t (i t )] = 1 - k T • E ℓ t (i 0 t ) + k T • E [ℓ t (i * t )] ,for all t ∈ [T ], and thusEQUATIONAlgorithm 1 Hedge with Best-Action Queries 1: Input: Sequence of losses ℓ t (i) and query budget k ∈ [T ] 2: Sample k out of T time steps uniformly at random and denote this random set by Q3: Set η = log n T when k ≤ √ T , otherwise η = k T 4: Initialize w 1 (i) = 1 for all i ∈ [n] 5: for t ∈ [T ] do 6: if t ∈ Q then 7: Observe i * t = arg min i∈[n] ℓ t (i) 8:Select action i * t 9:else 10:Let W t = i∈[n] w t (i) 11:Select action i with probability p t (i) = wt(i) Wt 12:Observe ℓ t (i) for all i ∈ [n]13:Updatew t+1 (i) = w t (i) • exp (-η(ℓ t (i) -ℓ t (i * t ))) for all i ∈ [n]Proof of Theorem 2.2. Let us first note that Algorithm 1 without queries is an instantiation Hedge η ( l) applied to losses lt (i) = ℓ t (i) -ℓ t (i * t ) ∈ [0, 1]. Let this algorithm be denoted as A 0 . Then, applying Lemma 2.1 and expanding the terms, we obtainL T (A 0 ) ≤ L T (i) 1 -η + log n η(1 -η) - ηL min T 1 -η .Let A k be Algorithm 1 with k best-action queries. By Observation 2.3, specifically (1), it holds thatL T (A k ) ≤ (1 -k /T )L T (i) 1 -η + (1 -k /T ) log n η(1 -η) - η(1 -k /T )L min T 1 -η + k T • L min T ⇐⇒ R T (A k , i) ≤ (1 -k /T ) log n η(1 -η) + T η -k T (1 -η) (L T (i) -L min T ) ≤ min T log n, T log n k ,where the last inequality holds by setting η = max log n /T , k /T .Before proceeding further, let us provide some technical intuition of why Theorem 2.2 should hold.The additive term k T • L min T (given in Equation ( 1)) alters the choice of the learning rate, which is allowed to be more aggressive, thus impacting the regret in a multiplicative way. To be more specific, in the usual Hedge performance analysis, we do not care about the negative term -ηL min T 1-η . This negative term together with the additive k T • L min T term given by best-action queries allows us to set the optimal η to be larger than the usual (order of) 1 / √ T . In other words, the additive impact of the k T • L min T term permits a multiplicative gain in regret as the learning rate η is modified and increased. 
label efficient: 
 Feedback: An O( T 2 log n k 2 ) Regret BoundWe extend Algorithm 1 to a setting where feedback is given only during querying time steps, with the only difference that the update rule is performed just after the querying time steps and nowhere else across the time horizon. We prove the following theorem:Theorem 2.4. For all adversarial sequences ℓ ∈ [0, 1] n×T of action losses and for all k ≥ T log T 2-1, in the label efficient query model, there exists an algorithm A k that guaranteesR T (A k ) ≤ 2 • min T 2 log n k , T 2 log n k 2 .Algorithm description. We first describe the algorithm A k we use: Let X t ∼ Ber k/T be a Bernoulli random variable, for some k ≤ k to be specified later. A k issues a best action query if, at time step t, X t = 1 and unless the query budget is exhausted. Once the query budget is exhausted, the algorithm stops querying. Otherwise, it performs the usual update rule on losseslt (i) = T k • (ℓ t (i) -ℓ t (i * t )) • I {X t = 1}.The algorithm then simply selects action I t = i * t if X t = 1 and action I t = i with probability p t (i) if X t = 0. Moreover, we denote by X ≤t = (X 1 , . . . , X t ), I ≤t = (I 1 , . . . , I t ).For the sake of the analysis, we introduce another algorithm A ′ k , which is the same as algorithm A k with the only (but crucial) difference that it issues a query if and only if X t = 1, regardless of whether or not query budget is exhausted. We thus bound the regret of A k in terms of the regret of A ′ k . Lemma 2.5. For all adversarial sequences ℓ ∈ [0, 1] n×T of action losses, in the label efficient query model, algorithmA ′ k guarantees R T (A ′ k ) ≤ min T 2 log n k , T 2 log n k2 .Proof. For algorithm A ′ k , we have that its counterpart without queries, A ′ 0 , is an instantiation Hedge η ( l), with l = l. Thus, by Lemma 2.1, we obtainEQUATIONas long as η < k/T . We now recognize that, sinceE lt (i) | X ≤t-1 , I ≤t-1 = ℓ t (i) -ℓ t (i * t ), then i∈[n] E p t (i) • lt (i) X ≤t-1 , I ≤t-1 = i∈[n] p t (i) • (ℓ t (i) -ℓ t (i * t )).Therefore, by the tower property of expectation applied around (2), we haveL T (A ′ 0 ) = i∈[n] E p t (i) • lt (i) = i∈[n] p t (i) • (ℓ t (i) -ℓ t (i * t )) ≤ 1 1 -T k η • L T (i) -L min T + log n η ≤ k k -T η • L T (i) + log n η - T η k -T η L min T ,where the last inequality holds since T η ≤ k. By Observation 2.3, we getL T (A ′ k ) ≤ 1 - k T • k k -T η • L T (i) + log n η - (T -k)η k -T η L min T + k T L min T ,which means that the regret is upper bounded byR T (A ′ k , i) ≤ k log n η( k -T η) + T 2 η -k k T ( k -T η) (L T (i) -L min T ) ≤ min T 2 log n k , T 2 log n k2 ,where the last inequality holds by setting η = max 1T k log n 2 , k k √ 2T 2, and then by noticing, in the latter case, that k-k k √ 2T ≥ √ 2k.With this lemma, we are ready to prove Theorem 2.4, where we bound the regret of algorithm A k .Proof of Theorem 2.4. We consider event E = {|Q| ≤ k}, so that, slightly abusing notation, we write the regret of algorithm A k asR T (A k ) = E [R T (A k ) | E] • P [E] + E R T (A k ) | Ē • P Ē ≤ E [R T (A k ) | E] + T • P Ē .To upper bound the second summand above, we haveEQUATIONby Hoeffding's inequality applied on the binomial random variable |Q| with expectation k, and aslong as k ≥ k -T log T 2 + 1.For what concerns the first summand, we recognize that under event E, A k and A ′ k are exactly the same algorithm. Thus, it holds thatE [R T (A k , i) | E] = E [R T (A ′ k , i) | E]. Moreover, if E R T (A k ) | Ē ≥ 0, and since P [E] ≥ 1 -1 /T , then E [R T (A ′ k , i) | E] ≤ T T -1 • R T (A ′ k , i) ≤ T T -1 • min T 2 log n k , T 2 log n k2 , by Lemma 2.5. If, instead, E R T (A k ) | Ē < 0, we also know that, by an identical derivation to (3), E R T (A k ) | Ē • P Ē ≥ -1. Therefore, E [R T (A ′ k , i) | E] ≤ T T -1 • (R T (A ′ k , i) + 1) ≤ T T -1 • min T 2 log n k , T 2 log n k2 + 1 ,again by Lemma 2.5. Overall, we obtainR T (A k ) ≤ 2 • min T 2 log n k , T 2 log n k 2 . 
lower bounds: 
 In this section, we construct two of randomized instances of the learning problem, which induce a tight lower bound on the minimax regret rates for both feedback models. We define the random variables Z t as the feedback observed by the algorithm at the end of time t. In the full feedback model, Z t = ℓ t , while in the label efficient setting, Z t = ℓ t only if a query is issued at time t (and Z t is an empty n-dimensional vector otherwise). Furthermore, we denote with Z ≤t the array containing the feedback Z 1 , Z 2 , . . . , Z t until time t.We start by describing the two stochastic instances, which are characterized by two distributions over two losses. As a notational convention, we denote the losses with ℓ t , and introduce two probability measures P + , P -(and their corresponding expectations E + , E -). Let ε, q ∈ [0, 1] be two parameters we set later (with ε ≤ q), we have that the losses of the n = 2 actions are distributed as follows:(ℓ t (1), ℓ t (2)) =       (1, 1) w.p. 1 2 under both P + and P - (0, 0) w.p. 1 2 -2q under both P + and P - (0, 1) w.p. q + ε under P + and w.p. q -ε under P - (1, 0) w.p. q -ε under P + and w.p. q + ε under P - .We now introduce and prove a general Lemma on the expected regret R ± T (A k ) suffered by any deterministic algorithm A k which issues at most k queries, against the i.i.d. sequence of valuations drawn according to P ± . Since we want a result that holds for both feedback models, we introduce the random set F which contains the times where A k actually observes the losses; note, N F = |F | is equal to T in full feedback and to the number queries issued in the partial information model. Lemma 3.1. For any determinstic algorithm A k which issues at most k best-action queries, we have:R + T (A k ) + R - T (A k ) ≥ exp - 5ε 2 q E + [N F ] • (T -k)ε 2 -2(q -ε)k.Proof. The best action i * under P + , respectively P -is the first, respectively the second, one, with an expected loss ofEQUATIONOn the other hand, the best realized action i * t yields an expected loss ofEQUATIONMoreover, if the algorithm chooses a suboptimal action i t ̸ = i * , its expected instantaneous regret is:EQUATIONLet now N + , respectively N -, be the random variable that counts the number of times that A k selects action 1, respectively 2, in times that are not in F (i.e., where the choice of 1 is not due to a query). Combining (4),( 5), and (6), we have the following:R ± T (A k ) ≥ E ± t / ∈F (ℓ t (i t ) -ℓ t (i * )) + t∈F (ℓ t (i * t ) -ℓ t (i * )) = 2ε • E ± [N ∓ ] -(q -ε)E ± [N F ] .Since N F ≤ k, and N + + N -≤ T -k, we have the following bound on the regret:R + T (A k ) ≥ P + N + ≤ T -k 2 • (T -k)ε -(q -ε)k R - T (A k ) ≥ P -N + > T -k 2 • (T -k)ε -(q -ε)k.Summing the above two expressions, we obtainEQUATIONAt this point, we apply the Bretagnolle-Huber Inequality [Theorem 14.2 in Lattimore and Szepesvári, 2020] to bound the first term on the right-hand side of the above inequality:P + N + ≤ T -k 2 + P -N + > T -k 2 ≥ 1 2 • exp -D KL P + Z ≤T , P - Z ≤T, where P ± Z≤T is the push-forward measure on all the possible sequences of feedback observed by A k under P ± . The lemma is concluded by combining the above inequality with the following claim, and plugging it into (7).Claim 3.2. It holds thatD KL P + Z ≤T , P - Z ≤T ≤ 5ε 2 q E + [N F ] .Proof. Let us observe that, once we fix the feedback history until time t -1, i.e., fix a realization of the feedback Z ≤t-1 , we have thatD KL P + Zt|Z ≤t-1 , P - Zt|Z ≤t-1 = I {t ∈ F } • D KL P + Zt|t∈F , P - Zt|t∈F ,where P + Zt|Z ≤t-1 (respectively P - Zt|Z ≤t-1 ) is the push-forward measure over {0, 1} 2 when losses are drawn according to P + (respectively P -), conditioning on the previous observations. The equality above holds because (i) algorithm A k observes feedback if and only t ∈ F (by definition), and (ii) A k is deterministic and whether or not t ∈ F may only depend on the past. We now upper bound the KL-divergence term above:D KL P + Zt|Z ≤t-1 , P - Zt|Z ≤t-1 = (q + ε) • log 1 + 2ε q -ε + (q -ε) • log 1 - 2ε q + ε ≤ (q + ε) • 2ε q -ε -(q -ε) • 2ε q + ε = 4ε 2 q q 2 -ε 2 ≤ 5ε 2 q ,where the first inequality follows from log(1 + z) ≤ z for all z ∈ R, and the last holds as long as we choose ε < q √ 5 . To complete our derivation, we express the overall KL divergence exploiting the tower property of conditional expectation:D KL P + Z ≤T , P - Z ≤T = t∈[T ] E + D KL P + Zt|Z ≤t-1 , P - Zt|Z ≤t-1 ≤ 5ε 2 q E + [N F ] ,where expectation is taken over all possible feedback realizations Z ≤t-1 , and the last inequality follows by earlier derivations.This concludes the proof of the lemma.We now show how to use the above lemma to derive the lower bounds. We start with full feedback. Theorem 3.3. In the full feedback query model, for all k ∈ [T ], we have the following lower bounds:• For any algorithm A k that has access to k < c 0 √ T queries, it holds that R T (A k ) ≥ c 0 √ T 4 .• For any algorithm A k that has access to k ≥ c 0√ T queries, it holds that R T (A k ) ≥ c 1 T k .Where c 0 = 1 /(e 8 √ 5) and c 1 = 1 /(320e 2 ) are universal constants.Proof. In full feedback, the algorithm always observes the losses, so that the feedback variable Z t = ℓ t and N F = T . We prove the Theorem via Yao's minimax Theorem: we prove that any deterministic algorithm A k fails against the random instance composed as follows: with probability 1 /2 the losses are drawn i.i.d. according to P + , otherwise, they are drawn i.i.d. according to P -. We can then apply Lemma 3.1 and obtain that the expected regret of A k against such mixture is equal toE [R T (A k )] = 1 2 (R + T (A k ) + R - T (A k )) ≥ exp - 5ε 2 q T • (T -k)ε 2 -2(q -ε)k. (8) Now, if k < c 0 √ T , we set ε = 2 √ 5T and q = 1 4 in (8) to get E [R T (A k )] = 1 2 (R + T (A k ) + R - T (A k )) ≥ √ T 2e 8 √ 5 - √ T 4e 8 √ 5 ≥ √ T 4e 8 √ 5 , Otherwise, if k ≥ c 0 √T , then consider the following choice of the parameters: ε = 1 40ek + 4e-1 40eT and q = 5ε 2 T = T 320e 2 k 2 + 4e-1 160e 2 k + (4e-1) 2 320e 2 T . Plugging these parameters in (8), we get:E [R T (A k )] = 1 2 (R + T (A k ) + R - T (A k )) ≥ T ε 4e -5ε 2 kT + 1 - 1 4e • εk = T 320e 2 k + 4e -1 160e 2 + (4e -2) 2 k 320e 2 T ≥ T 320e 2 k .As similar analysis can be carried over for the label efficient setting. Theorem 3.4. In the label efficient feedback model, for all k ∈ [T ], we have the following lower bounds:• For any algorithm A k that has access to k < c 0T √ k queries, it holds that R T (A k ) ≥ c 0 T 4 √ k .• For any algorithm A k that has access to k ≥ c 0T √ k queries, it holds that R T (A k ) ≥ c 1 T 2 k 2 .Where c 0 = 1 /(e 8 √ 5) and c 1 = 1 /(320e 2 ) are universal constants.Proof. In the label efficient feedback model, Z t is meaningful only for times in F . We then prove the result by Yao's minimax principle, by showing that any determinisitic algorithm A k which issues at most k queries suffers the desired regret against the instance that uniformly chooses between P + and P -. We can apply Lemma 3.1 (nothing that N F ≤ k) to get:E [R T (A k )] = 1 2 (R + T (A k ) + R - T (A k )) ≥ exp - 5ε 2 q k • (T -k)ε 4 -(q -ε)k.Once again, we have two cases. If k ≥ T e 8 √ 5k , we choose ε = 2 √ 5k and q = 1 4 to getE [R T (A k )] = 1 2 (R + T (A k ) + R - T (A k )) ≥ T 2e 8 √ 5k - T 4e 8 √ 5k ≥ T 4e 8 √ 5k . Otherwise, if k ≥ T e 8 √5k , and we choose ε= T 40ek 2 + 4e-1 40ek and q = 5ε 2 T = T 2 320e 2 k 3 + (4e-1)T 160e 2 k 2 + (4e-1) 2 320e 2 k , to obtain E [R T (A k )] = 1 2 (R + T (A k ) + R - T (A k )) ≥ T ε 4e -5ε 2 k 2 + 1 - 1 4e • εk = T 2 320e 2 k 2 + T (4e -1) 160e 2 k + (4e -1) 2 320e 2 ≥ T 2 320e 2 k 2 .Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [Yes] .Justification: Please refer to sections 2 and 3.Guidelines:• The answer NA means that the paper does not include theoretical results.• All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced. • All assumptions should be clearly stated or referenced in the statement of any theorems.• The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. • Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. • Theorems and Lemmas that the proof relies upon should be properly referenced. 
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 Appendix: A.1 Proof of Lemma 2.1 Lemma 2.1. Consider the Hedge algorithm Hedge η ( l) run on loss sequence l ∈ [0, U ] n×T with learning rate η < 1 /U. Then, for all action i ∈ [n], it holds that,LT (Hedge η ( l)) ≤ 1 1 -U η • LT (i) + log n η ,where LT (Hedgeη ( l)) = t∈[T ],i∈[n] p t (i) • lt (i)is the expected cumulative loss of Hedge η ( l) and LT (i) is the cumulative loss of action i.Proof. We have that, by definition,W T +1 ≥ w T +1 (i)w t (i) • exp -η lt (i) = t∈[T ] exp -η lt (i) = exp -η LT (i) .We also know thatW t+1 ≤ W t •   1 -η i∈[n] p t (i) • lt (i) + η 2 i∈[n] p t (i) • l2 t (i)   .Thus,W T +1 ≤ n • t∈[T ]   1 -η i∈[n] p t (i) • lt (i) + η 2 i∈[n] p t (i) • l2 t (i)   ,which combined with the earlier bound and taking logarithms of both sides, gives-η LT (i) ≤ log n + t∈[T ] log   1 -η i∈[n] p t (i) • lt (i) + η 2 i∈[n] p t (i) • l2 t (i)   ≤ log n -η t∈[T ],i∈[n] p t (i) • lt (i) + η 2 t∈[T ],i∈[n] p t (i) • l2 t (i) ≤ log n -(η -U η 2 ) t∈[T ],i∈[n] p t (i) • lt (i).The second inequality above follows from log(1 + z) ≤ z for z ∈ R, and the third by observing that l2 t (i) ≤ U lt (i), since lt (i) ∈ [0, U ]. The lemma follows by rearranging the terms above.B Best-Action Queries in the Stochastic i.i.d. SettingIn this section, we show how the results provided in Section 2 can be obtained in the stochastic i.i.d. setting (defined next) using the Follow-The-Leader algorithm, albeit a suboptimal dependence in n.In the stochastic i.i.d. setting, each action is associated with a fixed but unknown distribution D i supported in [0, 1]. Action i's loss at time step t, ℓ t (i), is drawn independently from distribution D i . We denote with µ(i) the expected loss of action i, and with i * = arg min i∈[n] µ(i) be the action of lowest expected loss. We measure the performance of a learning algorithm A, by considering its regret with respect to the best action distribution:R T (A) = t∈[T ] E [ℓ t (i t ) -ℓ t (i * )] .We denote by ∆ i = µ(i) -µ(i * ) be the gap between the expected loss of the best action and that of action i, and byΨ i = E [|ℓ(i) -ℓ(i * )|]the expected absolute value of such gap (note, we omit the dependence on time as the losses are drawn i.i.d. across time). Whenever the learner issues a query, then the identity of the action i * t with best realized loss is revealed, so that the learner gets, in expectation, E [ℓ t (i * t )] = E min i∈[n] ℓ t (i) .The following facts and simple results are particularly useful for analyzing algorithms in the stochastic generation model. P m τ =1 Y τ -µ ≥ γ ≤ exp - 3γ 2 6σ 2 + 2cγ,where σ 2 = m τ =1 E Y 2 τ -E [Y τ ] 2 .Moreover, when Y τ 's are also identically distributed, we haveσ 2 = m(E Y 2 τ -E [Y τ ] 2 ) ≤ mE [|Y τ |] =: mΨ. Thus, P m τ =1 Y τ -µ ≥ γ ≤ exp - 3γ 2 6mΨ + 2cγ.Proposition B.2. Let T, n ∈ N be the time horizon and number of experts respectively, with T ≥ n-1. For all distributions D i of actions losses, and any algorithm A, its regret isEQUATIONwhereΨ := E [|min i̸ =i * ℓ(i) -ℓ(i * )|] and ∆ := E [min i̸ =i * ℓ(i) -ℓ(i * )].Proof. We first observe that, by the i.i.d. assumption, we can always make the algorithm query in the first k time steps, without losing generality. Therefore, as Q = {t | t ≤ k}, the algorithm suffers E min i∈[n] ℓ t (i) in the first k time steps, which, by independence, givest≤k E min i∈[n] ℓ t (i) = kE min i∈[n]ℓ(i) .We also have the following useful observation that allows us to rewrite the maximum in a convenient form. Namely,EQUATION)The claim follows by the definitions of Ψ and ∆.Thus, in the stochastic case, the goal of a regret-minimizing algorithm is to minimize the first summand above since the second is characteristic of the instance at hand. Observation B.3. Let T, n ∈ N be the time horizon and number of experts respectively, withT ≥ n -1. For all distributions D i of actions losses, Ψ i -∆ i ≤ Ψ -∆. Proof. Since E min j∈[n] ℓ(j) ≤ E [min{ℓ(i * ), ℓ(i)}] for all i ∈ [n], we have ∆ -Ψ = 2 • E min j∈[n] ℓ(j) -ℓ(i * ) ≤ 2 • E [min{ℓ(i), ℓ(i * )} -ℓ(i * )] = ∆ i -Ψ i ,where the last equality also follows from Equation (10) applied to actions i, i * only.B.2 Label Efficient Feedback:Explore-Then-Commit with Ω(T 2 /3 ) Best-Action QueriesWe consider the label efficient feedback case first, where feedback is given only during querying time steps. The techniques used in this section are key for the full feedback case, and easier to illustrate.We provide a simple variation of the Explore-Then-Commit (ETC) algorithm Perchet et al. [2015] that, in the first k ≥ Ω(T 2 /3 ) time steps, is given free access to the identity of the best action before committing to the action to select. We have the following theorem: R T (A k ) ≤ min 3T log 2nT 2k , 2nT 2 ln T k 2 .We split the proof of this theorem in two lemmas that immediately imply it, one for k < T 2 /3 and one otherwise.Lemma B.5. If k < T 2 /3 , then R T (A k ) ≤ 3T log 2nT 2k .Proof. For all actions i, we define the clean event asE i = {|µ k (i) -µ(i)| < ε},where ε = log 2nT 2k . Similarly, let E be the intersection of all the E i 's for i ∈ [n]. By Hoeffding's inequality [Equation (5.8) in Chapter 5 of Lattimore and Szepesvári, 2020], we have the following:P Ē = P ∪ i∈[n] Ēi ≤ i∈[n] P Ēi ≤ 2n • exp -2kε 2 ≤ 1 T .Consider now the expected instantaneous regret suffered by ETC at time t + 1. We have two cases: Either the clean event holds, so that the instantaneous regret is at most 2ε t , or it does not, in which case the instantaneous regret is at most 1. By the law of total probability, we have the following:E [ℓ t+1 (i t ) -ℓ t+1 (i)] ≤ P Ēt + E [ℓ t+1 (i t ) -ℓ t+1 (i)|E t ] ≤ 1 T + 2ε t ≤ 3ε t .Summing the expected instantaneous regrets for all t yields the desired bound ofR T (A k ) ≤ 3T log 2nT 2k ,as we use the lower bound Ψ -∆ ≥ 0 to say that in the first k times the regret the algorithm suffers is at most 0.Lemma B.6. If T 2 /3 ≤ k ≤ 4T /9, then R T (A k ) ≤ 2nT 2 ln T k 2 .Proof. We start by showing the statement for 2 actions, and then generalize to n.The case of 2 actions. Let us recall that, by Proposition B.2, we haveR T (A k ) = |{t ≥ k + 1 | i t = 2}| • ∆ - k 2 • (Ψ -∆),where we have assumed that µ 1 ≤ µ 2 , without loss of generality. We suffer positive regret in the last T -k time steps when we select action 2 over 1, which happens if and only if μk (1) ≤ μk (2). Hence, the first summand of the above regret expression simply becomes∆(T -k) • P [μ k (2) ≤ μk (1)].Note thatP [μ k (2) ≤ μk (1)] = P k t=1 ℓ t (1) -ℓ t (2) ≥ 0 , so if we define g t = ℓ t (1) -ℓ t (2), whose expected value is E [g t ] = -∆ and P [|g t | ≤ 1] = 1, then, by Fact B.8, P k t=1 g t ≥ 0 = P k t=1 g t + k∆ ≥ k∆ ≤ exp - 3k 2 ∆ 2 6kΨ + 2k∆ ≤ exp - k∆ 2 2Ψ + ∆ .We distinguish two cases, namely ∆ 2 2Ψ+∆ ≤ ln T k and ∆ 2 2Ψ+∆ > ln T k . In the latter case, we get thatR T (A k ) = ∆(T -k) • P [μ k (2) ≤ μk (1)] - k 2 • (Ψ -∆) ≤ ∆T • exp - k∆ 2 2Ψ + ∆ - k 2 • (Ψ -∆) ≤ ∆T • exp (-ln T ) = ∆ ≤ 1.Otherwise, we obtain that Ψ ≥ ∆ 2 • k∆ ln T -1 , and soR T (A k ) = ∆(T -k) • P [μ k (2) ≤ μk (1)] - k 2 • (Ψ -∆) ≤ ∆T - k∆ 4 • k∆ ln T -3 ≤ T 2 k 2 + 3T 4k - 27 16 • ln T ≤ 2T 2 ln T k 2 .The first inequality holds since the maximizing∆ = 4T -3k 2k 2 • ln T , which is consistent with 0 ≤ ∆ ≤ Ψ ≤ 1 since T 2 /3 ≤ k ≤ 4T /9.Generalizing to n actions. We again assume that the first is the best action, and apply Proposition B.2, to obtainR T (A k ) = T t=k+1 j̸ =1 E I μk (j) ≥ max i̸ =j μk (i) (ℓ t (j) -ℓ t (1)) - k 2 • (Ψ -∆) = (T -k) • j̸ =1 ∆ j • P μk (j) ≥ max i̸ =j μk (i) - k 2 • (Ψ -∆) ≤ T • j̸ =1 ∆ j • P [μ k (j) ≥ μk (1)] - k 2 • (Ψ -∆),which holds by the independence of realizations across time steps. Let us define g t (j) = ℓ t (1)-ℓ t (j), so that,Ψ j = E [|g t (j)|], E [g t (j)] = -∆ j and P [|g t (j)| ≤ 1] = 1.Then, by Fact B.8, we obtainP [μ k (j) ≥ μk (1)] = P k t=1 g t (j) ≥ 0 ≤ exp - k∆ 2 j 2Ψ j + ∆ j .Recall that, by Observation B.3, Ψ -∆ ≥ Ψ j -∆ j . Therefore, let us sum and subtract to the regret upper bound the sum k 2n • j̸ =1 (Ψ j -∆ j ), and getR T (A k ) ≤ j̸ =1 T ∆ j • exp - k∆ 2 j 2Ψ j + ∆ j - k 2n • (Ψ j -∆ j ) + k 2 •   1 n • j̸ =1 (Ψ j -∆ j ) -(Ψ -∆)   ≤ j̸ =1 T ∆ j • exp - k∆ 2 j 2Ψ j + ∆ j - k 2n • (Ψ j -∆ j ) .For each j ̸ = 1, an identical derivation to the one in the case of two actions would yield the same regret rate, with k/n in place of k. Thus, overall, we obtainR T (A k ) ≤ 2nT 2 ln T k 2 ,which concludes the proof.B.3 Full Feedback: Follow-The-Leader with Ω √ T Best-Action QueriesWe provide an algorithm that, equipped with best-action queries for k time steps and with full feedback, achieves regret O( T /k) for k ≥ Ω( √ T ), and O( √ T ) otherwise. This means that, with the addition of feedback, fewer queries are needed to switch to a much smaller regret rate.The idea is to use the Follow-The-Leader paradigm Auer et al. [2003] . That is, we select the best action in the first k time steps since queries give us its identity for free, and successively, we select the action maximizing the empirical average so far, μt-1 (i) = 1 t-1 • t-1 τ =1 ℓ τ (i).Algorithm 3 Follow-The-Leader with Best-Action QueriesInput: Sequence of losses ℓ t (i) ∼ D i and k ∈ [T ] query budget for t ∈ [T ] do if t ≤ k then Observe i * t = arg min i∈[n] ℓ t (i) Select action i * t Observe ℓ t (i) for all i ∈ [n] else Select action j = arg min i∈[n] μt-1 (i)We have the following guarantee: Theorem B.7. Let T, n ∈ N be the time horizon and number of actions respectively, with T ≥ n -1. For all distributions D i of actions losses, Algorithm 3 with query access k ≥ 2 √ T guaranteesR T (A k ) ≤ min 3 2T log 2nT , 5nT k .For our purposes, only a weaker version of Fact B.1 is needed:Fact B.8. Let Y 1 , . . . , Y m be i.i.d. random variables such that E [Y τ ] = -∆ Y , E [|Y τ |] = Ψ Y , and P [|Y τ | ≤ 1] = 1, for all τ ∈ [m]. Then, it holds that P m τ =1 Y τ ≥ 0 ≤ exp - m∆ 2 Y 2Ψ Y + ∆ Y .We split the proof of this theorem in two lemmas that immediately imply it, one for k < 2 √ T and one otherwise. Next, we subsume the notation of Theorem B.7. Lemma B.9.If k < 2 √ T , then R T (A k ) ≤ 3 √ 2T log 2nT .Proof. For all times t and actions i, we define the clean event asE i,t = {|µ t (i) -µ(i)| < ε t }, where ε t = log 2nT 2t. Similarly, let E t be the intersection of all the E i,t 's for i ∈ [n] . By Hoeffding's inequality [Equation (5.8) in Chapter 5 of Lattimore and Szepesvári, 2020], we have the following:P Ēt = P ∪ i∈[n] Ēi,t ≤ i∈[n] P Ēi,t ≤ 2n • exp -2tε 2 t ≤ 1 T .Consider now the expected instantaneous regret suffered by FTL at time t + 1. We have two cases: Either the clean event holds, so that the instantaneous regret is at most 2ε t , or it does not, in which case the instantaneous regret is at most 1. By the law of total probability, we have the following:E [ℓ t+1 (i t ) -ℓ t+1 (i)] ≤ P Ēt + E [ℓ t+1 (i t ) -ℓ t+1 (i)|E t ] ≤ 1 T + 2ε t ≤ 3ε t .Summing the expected instantaneous regrets for all t yields the desired bound ofR T (A k ) ≤ 3 2T log 2nT ,as we use the lower bound Ψ -∆ ≥ 0 to say that in the first k times the regret the algorithm suffers is at most 0.Lemma B.10. If k ≥ 2 √ T , then R T (A k ) ≤ 5nT k .Proof. We start by showing the statement for 2 actions, and then generalize to n.The case of 2 actions. We again assume that µ 1 ≤ µ 2 without loss of generality. FTL's regret, thus, readsEQUATIONwhere the second equality follows from the independence of step t from all the preceding steps, and the inequality by Fact B.8. We distinguish three cases, namely∆ 2 2Ψ+∆ ≤ 1 T , 1 T < ∆ 2 2Ψ+∆ ≤ ln T k , and ∆ 2 2Ψ+∆ > ln T k .In the first case, we have that Ψ ≥ ∆ 2 • (T ∆ -1) and so,R T (A k ) ≤ ∆T - k∆ 4 • (T ∆ -3) ≤ T k + 3 2 + 27k 64T ≤ 2T k ,where the second inequality follows since the ∆ maximizing the expression is∆ = 2 k + 3 4T . This implies that Ψ ≥ T 2k 2 -3 32T + 1 2k , which is consistent with 0 ≤ ∆ ≤ Ψ ≤ 1 since k ≥ 2 √ T .In the last case, ∆ 2 2Ψ+∆ > ln T k , we getR T (A k ) = ∆(T -k) • P [μ t-1 (2) ≤ μt-1 (1)] - k 2 • (Ψ -∆) ≤ ∆T • exp - t∆ 2 2Ψ + ∆ - k 2 • (Ψ -∆) ≤ ∆T • exp (-ln T ) = ∆ ≤ 1.We are left to show the case where 1 T < ∆ 2 2Ψ+∆ ≤ ln T k . To this end, let us observe that, for all T ≥ 2, it holds thatT t=m+1 exp - t r ≤ r.for all 1 ≤ m, r ≤ T . This is the case becauseT t=m+1 exp - t r = e -k/r -e -T /r e 1/r -1 ≤ re -k/r ≤ r, since 1 + z ≤ e z for all z ∈ R. Therefore, R T (A k ) ≤ 2∆ • 2Ψ + ∆ ∆ 2 - k 2 (Ψ -∆) = Ψ • 4 ∆ - k 2 + 2 + k∆ 2 .We distinguish two subcases: On the one hand, assume that∆ > 8 k , then 4 ∆ -k 2 < 0. Since Ψ ≥ k∆ 2 2 ln T -∆ 2 , we have R T (A k ) ≤ k∆ 2 2 ln T - ∆ 2 • 4 ∆ - k 2 + 2 + k∆ 2 = 2k∆ ln T - k∆ 2 4 ln T + 3k 4 ≤ 3 ln T 8 + 4 + 1 ln T 8 - 4 k - 1 k • ln T 16 -1 ≤ ln T 2 ,since the expression is maximized for ∆ = 8+ln T 2k > 8 k , as long as T > e 8 . This implies that Ψ ≥ 8 k ln T -ln T 8k , which is consistent with 0 ≤ ∆ ≤ Ψ ≤ 1. On the other hand, ∆ ≤ 8 k , and thus4 ∆ -k 2 ≥ 0. We use that Ψ ≤ T ∆ 2 -∆, and getR T (A k ) ≤ T ∆ 2 -∆ 2 • 4 ∆ - k 2 + 2 + k∆ 2 = 2T ∆ - kT ∆ 2 4 + 3k∆ 4 ≤ 4T k + 9k 16T ≤ 5T k , since the expression is maximized for ∆ = 4 k + 3 2T < 8 k . This implies that Ψ ≤ 8T k 2 + 4 k + 3 8T , which is consistent with 0 ≤ ∆ ≤ Ψ ≤ 1 since k ≥ 2 √ T .Generalizing to n actions. The generalization is very similar to the one in the proof of B.4. Indeed, the regret equalsR T (A k ) = T t=k+1 j̸ =1 E I μt-1 (j) ≤ min i̸ =j μt-1 (i) (ℓ t (j) -ℓ t (1)) - k 2 • (Ψ -∆) = T t=k+1 j̸ =1 ∆ j • P μt-1 (j) ≤ min i̸ =j μt-1 (i) - k 2 • (Ψ -∆) ≤ T t=k+1 j̸ =1 ∆ j • P [μ t-1 (j) ≤ μt-1 (1)] - k 2 • (Ψ -∆) ≤ j̸ =1 ∆ j • T t=k+1 exp - t∆ 2 j 2Ψ j + ∆ j - k 2 • (Ψ -∆),where the last inequality follows by Fact B.8. Identically to the proof of Theorem B.4, we sum and subtract the term k 2n • j̸ =1 (Ψ j -∆ j ), and getR T (A k ) ≤ j̸ =1 ∆ j • T t=k+1 exp - t∆ 2 j 2Ψ j + ∆ j - k 2n • (Ψ j -∆ j ) + k 2 •   1 n • j̸ =1 (Ψ j -∆ j ) -(Ψ -∆)   ≤ j̸ =1 ∆ j • T t=k+1 exp - t∆ 2 j 2Ψ j + ∆ j - k 2n • (Ψ j -∆ j ) .For each j ̸ = 1, an identical derivation to the one in the case of two actions would yield the same regret rate, with k/n in place of k. Thus, overall, we obtainR T (A k ) ≤ 5nT k ,which concludes the proof.Correlated hints. A first model that is close to ours has been introduced by Dekel et al. [2017] , which studies Online Linear Optimization when the learner has access to vectors correlated with the actual losses. Surprisingly, this type of hint guarantees an exponential improvement in the regret bound, which is logarithmic in the time horizon when the optimization domain is strongly convex. Subsequently, Bhaskara et al. [2020] generalize such results in the presence of imperfect hints, while Bhaskara et al. [2023b] prove that Ω( √ T ) hints are enough to achieve the logarithmic regret bound. While these works share significant similarities with ours, we stress that their results are not applicable to the standard prediction with experts model, given the non-strong convexity of the probability simplex over the actions.Queries/Ordinal hints. Bhaskara et al. [2023b] studies online learning algorithms augmented with ordinal queries of the following type: a query takes in input a small subset of the actions and receives in output the identity of the best one. With this twist, it is possible to bring the regret down to O(1) by observing only 2 experts losses in advance at each time. Although our best-action query is stronger (as it compares all the actions), our learner is constrained in the number of times it can issue such queries.Algorithms with predictions. Other works that explore the interplay between hints and feedback are [Bhaskara et al., 2021a , Shi et al., 2022 , Cheng et al., 2023 , Bhaskara and Munagala, 2023 , Bhaskara et al., 2023a] . More broadly, our work follows the literature on enhancing performance through external information (predictions). This has been extensively applied to a variety of online problems to model partial information about the input sequence that can be fruitfully exploited if accurate for improving the performance of the algorithms. Examples include sorting [Bai and Coester, 2023] , frequency estimation [Aamand et al., 2023] , various online problems [Purohit et al., 2018, Gollapudi and Panigrahi, 2019] such as metrical task systems [Antoniadis et al., 2023b,c] , graph coloring [Antoniadis et al., 2023a] , caching [Lykouris and Vassilvitskii, 2021] , scheduling [Lattanzi et al., 2020 , Jiang et al., 2022] , and several others.• While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.• The answer NA means that the paper does not include experiments.• If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. • If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. • Depending on the contribution, reproducibility can be accomplished in various ways.For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. • While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. • The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors).• It should be clear whether the error bar is the standard deviation or the standard error of the mean. Justification: We study a basic theoretical question that has been studied before.Guidelines:• The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.• If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. • The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).* The model with k actions can be captured by a linear function where each entry corresponds to the loss of a specific discrete action, and the continuous action space is the probability simplex over the n discrete actions (which is not strongly convex).† We adopt the notational convention that [x] stands for the set of the first x natural numbers.‡ We use Θ(x) to hide polylogarithmic terms in x. 


Decision: All the referees are unanimous in the fact that this paper contributes a new model for online learning where best-action feedback can be elicited at a cost. The model is well-motivated, the analysis of the proposed algorithm shows surprisingly strong results, and the paper is overall very well written. I am glad to accept it for presentation at Neurips.

Reviews: review: paper_summary: paper_summary: This paper considers the standard prediction with expert advice setting of online learning, but with the twist that the learner may issue, up to $k$ times, a "best-action query" before making a prediction, in which case the identity of an expert incurring the smallest loss in the round is revealed to the learner (and of course they can choose that expert on that round). Two settings are investigated: the standard experts setup wherein the losses of all experts are revealed after each round, and the 'label-efficient prediction' (LEP) setting, where the losses are not revealed unless the learner issues a query for the same (which the authors identify with the best-action query, i.e., all losses are only revealed when the learner makes a best-action query).

The paper shows that these $k \ll T$ best-action queries have an impressive effect, improving the regret in the experts setting to $O(\min(\sqrt{T}, T/k)$, and in the LEP setting to $O(\min(T/\sqrt{k}, T^2/k^2)$. Surprisingly, the method achieving this is just hedge, but run with losses of the form $\ell_{t,i} - \min_i \ell_{t,i}$ instead of just $\ell_{t,i}$ (and the appropriate modification a la prior work on LEP for this setting). The approach to showing this uses the standard analysis of the hedge algorithm, and uses the additive advantage of the best-action queries to allow the method to set an aggressively large learning rate (when $k$ is large enough) without suffering a strong penalty for the same, leading to improved regret bounds. The paper concludes by showing matching lower bounds for this setup, in a minimax sense.
strength_weakness: strength_weakness: I don't see major weaknesses in the paper.  I think the only change that is necessary is that the discussion of the related work in section C should be moved to the main paper, since this provides valuable context to the investigation presented within (but this should be easily accommodated with the extra page, if the paper is accepted).

Perhaps another point of improvement lies in deepening the discussion of scenarios where the model being studied has pertinence. The moderation setup certainly is interesting and natural, but are there other situations where the authors see the relevance of best-action queries?
questions: questions: Suggestion: One approach to modeling problems like using limited moderation effectively that I have seen is through the abstention model of learning, wherein the learner may "abstain" on a query, and receive a "best-response" by utilising extra resources. I think work on online abstention may thus be a useful point of contact with the literature that the paper misses, and might be worth including a discussion on. This includes both the KWIK model, and the full-information model. Useful points of contact with this literature are below.  

Li, Littman, Walsh, and Strehl, Machine Learning 2011, Sayedi, Zadimoghaddam, and Blum NIPS10; Zhang and Chowdhuri, COLT 16; Cortes et al., ICML 18; Neu and Zhivotovskiy, COLT 20; Gangrade et al., Neurips 21
limitations: limitations: This is fine
paper_summary: paper_summary: This paper considers an online learning with actions model where the learner is allowed to make $k$ "best action queries". Such a query at step $t \in [T]$ will return $i_t^\ast \in \arg\min_{i \in [n]} \ell_t(i)$; an action that minimizes the loss at step $t$. The loss values are bounded (in $[0, 1]$ w.l.o.g) and may be generated by an oblivious adversary. The best action query is performed _before_ the learner makes the decision (arm choice). The regret of the learming algorithm is defined in the usual way.

They first consider the full feedback setting, where _after_ making the arm choice, the learner observes the entire loss vector $(\ell_t(1),\ldots,\ell_t(n))$. In this setting, without best action queries, the Hedge algorithm gives $\tilde{O}(\sqrt{T})$ regret for bounded losses (and this is well-known to be optimal).

(i) In the full feedback setting (augmented with best action queries), the authors show that the Hedge algorithm with $k$ uniformly random best action queries gives an (expected)  regret of $\tilde{O}\left(min\\{\sqrt{T},T/k\\}\right)$. This beats the standard regret bound when $k$ is $\omega(\sqrt{T})$.

(ii) They also show a matching lower bound (up to $\log n$ factors) that any full-feedback algorithm with $k$ best action queries (not necessarily random) will suffer $\Omega(\sqrt{T},T/k)$ (expected) regret.

They then consider a label-efficient feedback model w.r.t the best action queries. Here the restriction in feedback is entirely in terms of observing the loss values (there is no outcome space) and linked to the best action queries themselves. The learner can make $k$ best action queries; if the learner chooses to make a query at time step $t$, it receives the full feedback after it makes the choice. Otherwise, it receives no feedback.

(i) In the label-efficient feedback setting (augmented with best action queries), the authors show that the label-efficient Hedge algorithm (does not update the probabilities when there is no feedback) with random best action queries ($\leq k$ of them, but not quite uniform random unlike in full feedback) achieves $O\left(\min(T/\sqrt{k},T^2/k^2)\right)$ regret. This beats the known regret bound for label-efficient Hedge (from Cesa-Bianchi and Lugosi 2006) when $k$ is $\omega(T^{2/3})$. 

(ii) They also show a matching lower bound for regret (up to $\log n$ factors) of any algorithm with $k$ query label efficient feedback.

In the appendix, they show nearly identical bounds for full feedback and label-efficient feedback in the stochastic losses setting. Here the algorithms are Follow the leader and Explore then commit, and the best action queries are simply made in the first $k$ rounds.
strength_weakness: strength_weakness: * Even though the best action query model yields theoretically interesting results, it is a bit too strong compared to some of the existing work. Even if only sublinearly many queries are made, each query is supposed to give the _correct best action (among all)_ for that time step before making the decision, and the number of queries needed is still polynomial in $T$ (at least $T^{1/2}$) to get any interesting result w.r.t the regret (in both full feedback and label efficient feedback). When the authors link best action queries to human expert advice in the introduction section, the assumption on the queries becomes somewhat impracticable.
questions: questions: * In the equations after line 178, is the last equality an inequality ($\leq$) using $T\eta \leq \hat{k}$?
limitations: limitations: * There is no negative social impact.

* The major practical limitation is the strong requirements of the query model. It would be much more feasible if (i) the queries returned an "approximate" best action among all, (ii) the queries returned the exact best action among a subset of actions or (iii) interesting results were obtainable with logarithmically many queries.
paper_summary: paper_summary: The paper’s title concisely summarizes the topic. They study both the “full feedback” setting, where the learner observes the entire loss vector (what the loss would have been for each possible action) on each time step, and the “label-efficient feedback” setting, where the learner only observes the loss vector on time steps when a query is issued.

The results are as follows:
1. An algorithm for the full feedback setting which achieves regret $O(\min(\sqrt{T}, T/k))$ with k best action queries. Interestingly, the queries are issued uniformly at random.
2. An algorithm for the label-efficient feedback setting which achieves regret $O(\min(T/\sqrt{k}, T^2/k^2))$. The queries are also issued mostly at random here (specifically, uniformly at random until the budget is exhausted).
3. Lower bounds for both of the above results which are asymptotically tight.
strength_weakness: strength_weakness: I found the paper to be quite technically dense. Although the description of the results in the introduction was clear, the authors don’t provide much intuition for their results. I had a hard time identifying where (on a technical level) the multiplicative power of k comes from. Conveying technical intuition along with technical results is very beneficial for other researchers trying to make use of the insights from this paper. I suspect the impact of the current version of paper is limited by the presentation. 

For transparency, I will mention that I am not an expert in this sub-area, and it is possible that experts would not have the same complaint. However, to the extent that the intended audience of this paper is researchers in related but not identical fields, I think this weakness is significant.

See also the “limitations” section below.
questions: questions: Can you provide technical intuition for where the multiplicative power of the best-action comes from?

Below are minor comments that the authors are free to use or discard. I’m including them in this section because I don’t know where else to put them, but I don’t expect responses.
-  Line 71: “In our paper we pinpoint the exact minimax regret rates for the problems studied.” It might be worth saying “asymptotically tight” instead of “exact” (although it’s pretty clear from context).
- Lines 84 - 86 are quite compelling.
- Line 93: Maybe mention that the learner still observes the entire loss vector, not just the best action.
- Lines 132 - 133: “combining the Hedge algorithm [Chapter 2 in Cesa-Bianchi and Lugosi, 133 2006] to uniform queries” → “with uniform queries”?
- Line 200: “and Z_t is an empty n-dimensional vector” → “and Z_t is an empty n-dimensional vector otherwise”?
limitations: limitations: In Question 2 of the checklist, the authors state:
> The paper has no significant limitation. Being theoretical in nature, there is a set of model assumptions

I found this answer to be disappointing. Theoretical papers absolutely have limitations. Are the model assumptions realistic? If not, to what extent do the results crucially hinge on those assumptions? Do the algorithms have inefficiencies that would impede or preclude practical usage? Are the technical insights too complex to be easily understandable by other researchers? These are just example limitations; I’m not claiming that any of these particular examples apply to this paper.
paper_summary: paper_summary: In this paper, the authors studies the classical problem of online learning in the additional context of algorithms with predictions, or learning-augmented algorithms. In online learning, a learning algorithm must repeatedly select from a set of options, each associated with a loss function, and the goal is to minimize the regret, i.e., the difference between the (expected) loss of the algorithm and that of the best fixed option in hindsight. The loss function may be adversarially generated based on time and prior actions, which makes obtaining good regret traditionally hard. By utilizing predictions to give the learning algorithm some additional information about the unknown loss function, we may be able to obtain better regrets and break impossibility bounds.

The authors incorporate 'best-action queries', which are queries that informs the algorithm the best option at the current time step. Prior work considers different form of predictions or queries, such as a vector correlated with the actual loss function, or the best option amongst a small subset of all the options. This work differs from prior works both in the form of the advice, and that the learning algorithm is only allowed to query at $k$ out of $T$ time steps.

WIthin this advice model, the authors show that in either the full feedback model, in which the full loss function is revealed to the algorithm after every time step, or the label efficient feedback model, in which the algorithm is only given the full loss function if it issues a query in the current round, the query budget $k$ multiplicatively affects (decreases) regret. This is a surprising result since the regret per round is upper bounded by $1$, so $k$ rounds can only increment the regret additively instead of multiplicatively. The algorithm the authors propose is a very simple and intuitive modification of the classical Hedge, or multiplicative weights, algorithm, except that in $k$ out of $T$ rounds, randomly sampled according to some procedure, the algorithm queries the best-action and follows it instead of a weighted selection amongst all options.
strength_weakness: strength_weakness: I find the author's presentation somewhat rushed and unsatisfactory, and makes significant assumptions on the reader's background knowledge. Coming from a learning-augmented algorithm background, I find many online learning aspects of the paper not sufficiently defined or explained in a self-contained way, despite having some familiarity with the field. Some minor points of complaint are listed in the Questions section below.

More importantly, a majority of the main corpus of the paper consists of technical proofs following one another, without much introduction or intuition in between to give the readers a high-level guideline of what is going on in the paper. It is very hard, without full familiarity of relevant literature, to follow the author's proofs, and understand what part a claim or statement plays in the grand scheme of proving the main theorems. As a result, while I believe the author's claims are overall correct, I am unable to confidently confirm the technical soundness of the paper.

Overall I still believe that this is an exciting paper; but its presentation can be improved significantly.
questions: questions: The authors discuss the potential of using noisy, possibly erroneous predictions or queries in the conclusion. From my understanding, the learning-augmented algorithms community put a lot of emphasis on utilizing machine-learned oracles that can possibly be accurate but have no rigorous guarantees to facilitate and *augment* classical algorithms to retain both *consistency* (good performance when predictions are accurate) and *robustness* (good performance even when predictions are arbitrarily bad). As a result, I would be very interested in seeing discussions and extensions in this noisy prediction regime in the future.

As illustrated above, I believe that a lot of intuition, such as high-level description of the proof strategy, and the usage and purpose of each component lemma or claim in colloquial terms, would be very helpful to the overall presentation in the main corpus, and can help guide readers to follow the flow of logic.

Some minor details I would like to point out:
- Line 35-36: "... the platform's task **consists in** deciding whether..." should be "consists of"?
- Line 49: "...with full feedback, **an** in the **stochastich** multi-armed...".
- The phrases "loss" and "feedback" are both used a lot, perhaps interchangeably, which is somewhat confusing. This is especially important in the label-efficient feedback section, in which it is not immediately clear what is "partial" about the "partial feedback" given to the algorithm. Is it just that the algorithm receives the loss function only when it makes a query, or is the received loss function itself partial?
- Line 102-103: The entire sentence within the parenthesis does not make sense to me, grammatically.
- Line 120: The introduction contains multiple mentions of the classical Hedge algorithm, but never explains, in any level of detail, what is the Hedge algorithm. From their pseudocode of Algorithm 1, which is described as a modification of Hedge, I can extrapolate that Hedge is the classical multiplicative weight algorithm, but the authors did not even make this clear in the introduction.
- Line 142: "We have the following theorem." Which theorem is this sentence exactly referencing to? Theorem 2.2 is the natural candidate but its statement predates this sentence.
- Line 162: I find it somewhat slightly strange that of the two algorithms presented in the paper, one is given a pseudocode block, while the other is only given a description.
- Line 196: "...we construct two **of** randomized instances...".
- Line 199-200: "(and $Z_t$ is an empty $n$-dimensional vector)". Is the scope of this sentence if a query is not issued at time $t$?
limitations: limitations: The authors sufficiently discuss limitations and future directions in the conclusion section. There are no ethical concerns or limitations in their work.

Comments: comment: Thanks for the response. I am satisfied and am maintaining my positive rating.
Thank you for your response. I maintain my rating. Overall I agree with the other reviewers that the paper asks a very timely and interesting question about whether we can utilize reliable queries (and hopefully in the future, learning-augmentation with unreliable predictions) to facilitate online learning, and I very much hope to see the paper structured better with more intuition and self-contained proofs, and suggestions from me and other reviewers incorporated, in the camera-ready version, if accepted.