Online Learning with Sublinear Best-Action Queries
Authors: Matteo Russo
Andrea Celli
Riccardo Colini Baldeschi
Federico Fusco
Daniel Haimovich
Dima Karamshuk
Stefano Leonardi
Niek Tax
Venue: NeurIPS 2024 poster (2024)
Keywords: Online learning
Learning Theory
Low-regret algorithms
Primary area: online_learning

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 $k \in \Omega(T^{2/3})$.

Online Learning with Sublinear Best-Action Queries
Matteo Russo Sapienza University of Rome, Italy mrusso@diag. uniromal... it
Andrea Celli Bocconi University, Italy andrea. wellii@umiboccontt
Riccardo Colini-Baldeschi Meta, Central Applied Science, UK rickuzōmeta.com
Federico Fusco Sapienza University Rome, Italy Tuscof@diag uniromal. it
Daniel Haimovich
Dima Karamshuk
Meta, Central Applied Science, UK danielha@meta. com
Meta, Central Applied Science, UK karamshuk@meta.ccom com
Stefano Leonardi
Niek Tax
Sapienza University of Rome, Italy Soonaddi@digg. uniromal..t.
Meta, Central Applied Science, UK ni diek@meta.com com
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 acquire additional information on the actions to be selected. In particular, we study the power of best-action queries, which reveal beforehand the identity of the best action at given time step. In practice, predictive features may be expensive, so we allow the decision maker the performance
issue most &s such queries. We establish tight bounds any algorithm can achieve when given access to & best-action queries for different types of feedback models. In particular, we prove that in the full feedback model, k queries are enough achieve an optimal regret of O (min VT, T/K}). This finding highlights the significant multiplicative advantage in the regret rate achievable with even modest (sublinear) number the challenging setting in which the only available feedback is obtained during the time steps corresponding to the & best-action queries. There, we provide tight regret rate of regret rate for label efficient prediction for l E S(TT/8).
S(VT)) of queries. Additionally, we study
T*)+3),, which improves over the standard (T/VVR
1 Introduction
Online learning is foundational problem in machine learning. In its simplest version, a decision maker repeatedly interacts with fixed set of actions over time horizon T. At each time, the decision maker needs to choose one of a set of actions; subsequently. it receives an action eetion-dependent loss and observes some feedback. These loss functions are generated by an omniscient (but oblivious) adversary and are only revealed on-the-go. go. The goal of the decision maker is to design a learning algorithm that achieves small regrer with respect 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
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

---

have been developed, characterized by optimal instance-indepente regret bound, depending on the feedback model
and Lugosi, 2006, Slivkins. 2019].
Following the recent literature on algorithms with machine learning- 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 '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 scarce resource, and the platform can only employ external reviewers at most & times.
While the idea of incorporating hints or queries into online learning models has already been studied [e.g., Bhaskara et al., 2021b, 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 from VT to log T, even for learners that receive hints for O(vT) 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 small number of actions has the best loss. this model, probing 2 arms is sufficient to achieve indeenndeenndett 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 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 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 al., 2010, Zhang and Chaudhuri, 2016, Cortes et al., 2018, Neu and Zhivotovskiy, 2020, Gangrade et al., 2021].
1.1 Our Model
In our model, an online learner repeatedly interacts with n actions over time horizon T. At the beginning of each time / E [T]], the learner chooses one of these actions in and suffers a loss (ix) 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 & out of T times. When the learner issues query, an oracle reveals the identity of the best action at that time, i7. 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 E [0, 1 reads
R (A,
Σ E (6(() CEIT
min Σ (()), sEin)
NE[T
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 Rr(AA) (without the dependence on e) the worst-case regret of A: (AA) = sup, (r, E). The minimax regret of learning problem then the regret achievable by the best algorithm. In our paper we pinpoint the exact minimax regret rates for the problems studied.
"The model with & actions can be captured by linear function where each entry corresponds to the loss of specific discrete action, and continuous action space is the probability simplex over the discrete actions (which is not strongly convex).
We adopt the notational convention that [zc stands for the set of the first natural numbers.

---

For the sake of simplicity, we denote with Lr(() the total loss incurred by action over the whole time horizon: Lr(i)= Everyy (+)). - ExErry (<())) denotes instead the sum of the minimum loss actions at each time. Finally, Lr (Ats) E Eie(r f((ia) denotes the expected total loss of an algorithm Ag issuing at most & best-action queries.
1.2 Our Results
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 (&, (1), &1), (2)...... (n)) after action i, 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(V? 2/=]), 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 & queries to suffer the same regret.
It is surprising that issuing sublinear number of queries, say k = T for E (1/2, 1), yields a significant improvement on the regret, which becomes 7'-a, Note, the total of the losses incurred by any algorithm in & rounds is at most k, which only affects the overall regret in an additive way; nevertheless, we prove that issuing queries has an impact the regret that is multiplicative in k. For instance, T queries are enough to decrease the regret T which is well below the O ((VT) minimax regret for prediction with expert without queries
and Lugosi, 2006].
Label efficient feedback. We then proceed 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 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-Biandee Bianchi and Lugosi, 2006] to achieve a regret rate of Omin (T/VE, T²/ T²/x²) see Theorem 2.4,
In Theorem 3.4, show that it is not possible to improve these rates: there exists instances where all algorithms suffer Q(T/vT) regret if l E O(T%/s) or (2(1/1)) if k 12(T4/3).
We observe that our algorithms improve (for k 13(7 on the regret rate of label efficient prediction with is steps of feedback, which of order O(T/VR) [Chapter 6.2 in Cesa-Bi Besa-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 queries (which impact on a total loss of the same order but leave unaffected (TT) rounds) are enough to achieve OVVT) 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 ((T/k) when & 2 Q S((TT), and é(vT) otherwise. With label efficient feedback, Saplore-Then-Comee [Perchet et al., 2015] achieves @(T"/1² when k 2 227 and (T/ otherwise.*
1.3 Technical Challenges
Issuing a best-action query has the effect of inducing possibly negative instantaneous regret. In fact, the benchmark that the algorithm is comparing against at time is the loss of the best fixed action over the whole time horizon, which may naturally be larger, time t, than the best action iž in that specific time. Overall, the magnitude of the total "negative" regret that is generated by A queries is at most Okk), which is typically sublinear in T. and affects linearly on the overall regret. We prove that this sublinear number of queries has (surprising) multiplicative effect on the overall regret bound.
We use (x)) hide polylogarithmic terms in
3

---

Our analysis reveals that any algorithm employing uniform querying strategy can be characterized by a total loss decomposed into two terms: A/r 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 - 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 differenne 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(VT) 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-au -aggmeted learning algorithm, and provide tight bounds in both & 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 2((VT) regret lower bound in the adversarial setting (without queries). The instance 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 extra ((VT) term. Now, if the learner is given the power to issue (order of) VT queries, then its regret naturally drops from (order of) vT to constant.
2 Hedge with k Best-Action Queries
In 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-Bianch, 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,,(e) run on loss sequence i E |0, UJxXT with learning rate rj < 1/u. Then, for all action E [n]] holds that,
1
(Ir()++
+ log m
LreHedge, (2) < Un
EEETTEE where For (Hedge, - (o) (i) is the expected cumulative loss of Hedge, (2) and (i) is the cumulative loss of action i.
Proof. We have that, by definition.
Wr+1 2 wr- exp (-n2((0)=
II exp (-nū(() exp (-ntr((i) <E|T]
We also know that
W411 S W.
1 11 Σ pr(1) (e(1) + 17² Σ pr(1) (r()). 22(0))
kE(M)
xE(m)
Thus,
WT+1 n
II
1- I ((() + +1 Σ pr(1)) (()))
Σr((i)
xE[T]
=E(m]
RE(m)
which combined with the earlier bound and taking logarithms of both sides (since they are both positive), gives
Li( & log n +
Σ log
1 1-n Σ p(()) 4(1) + 17 Σ (o))) 2(0
2E[T]
sE(n)
1E|m)
*This is simple corollary of the expected distance of random walk.

---

< logn 17 Σ pr(i) Ee(i) +1²
pc(i)- (E[T]
Σ
(E[T),iE|m]
log n. (1 Un²)
Σ P((t) Ee(().
The second inequality above follows from log(1 2) S for R, and the third by observing that
(1))) S (1), since (i) E [0, U.. The lemma follows by rearranging the terms above.
2.1 Full Feedback: An O Flog
Regret Bound
Theorem 2.2. Consider the problem of online learning with full feedback and k best-action queries, then there exists algorithm An that guarantees
Tlog n
Rr((As) I min TTogg
We prove that Algorithm I exhibits the desired regret guarantees. In particular, illustrate next, this algorithm performs uniform querying, i.e., they choose a uniformly random subset Q S [T] of size l where to issue best-action queries. In the case of uniform queries, and when feedback and action taken by the algorithm are not correlated, a useful simplification can be made.
Observation 2.3. Let Ag an algorithm with no querying power, with full or efeeciet Consider Av. obtained from Ao by equipping it with k uniformly random queries across the time horizon or with an independent query on each step time step with probability k/T. Similarly, let ig and is be the actions selected by Ag and As at time L. Then, for all adversarial sequences E [0, 1|nXT of action losses,
E B[G(1))= = (1-0)-(1-77 4)
k + E(G()) [&( +
for all (T), and thus
LT(A)) (1 k T
LT(A)) k Lmin
(1)
Algorithm Hedge with Best-Action Queries
1: Input: Sequence of losses (i) and query budget & E [7)
2: Sample out of time steps uniformly at random and denote this random set by Q
3: Set 1)= V lagn when I VT, otherwise = T
&
4: Initialize w, (1)
for all E [n]
5: for i. (T) do
6: if then Observe I = arg min (e(i) Select action iž
7:
8:
9:
else
Let W, -i (i)
Select action with probability (i) wi(()
Wi
Observe (1()) for all E [n]
13:
Update (1) = wx(t) exp (-n(L(()) - le(1t))) for all i e [n]]
Proof of Theurem 2.2. Let us first note that Algorithm without queries is an instantiation Hedge,, (2)
applied to losses (i) (z(i) ((tt) E [0, Let this algorithm be denoted Ag. Then, applying Lemma 2.1 and expanding the terms, we obtain
Lr(i) 1-
log T.piin
(y(A)) <
+
7 - 1)
1 1-n
5

---

Let A& be Algorithm with & best-action queries. By Observation 2.3, specifically (I), it holds that
(T(AA) (1 Li))
(1 (1-+/T) k/T) logn
7711 (l-+/T)
k 1 1-1
k/T)
1-7 +
LTT
17(1 - 11)
T
t/T) log
Tnj
T Tlog
- RT(A
T(1 11) (Ir(()) - I.qiin S min
V
Flogn n,
711 - 7)
k
{ (Vion/TT, H/T).
where the last inequality holds by setting 1) max
Before proceeding further, let us provide some technical intuition of why Theorem 2.2 should hold. The additive term $ Lipin (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,
-744 This in the usual Hedge performance analysis, we do not care about the negative term 1-
negative term together with the additive $ Lytin term given by best-action queries allows to set the optimal yj to be larger than the usual (order of) 3/vT. In other words, the additive impact of the L term permits multiplicatiie gain in regret as the learning rate 7) is modified and increased. T
2.2 Label Efficient Feedback: An O - log Regret Bound
We extend Algorithm 1 to setting where feedback is given only during querying time steps, with the only difference that the update rule 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 & [0, 1| of action losses and for all k 2
T T 2
in the label efficient query model, there exists an algorithm As, that guarantees
2 log n T log n
Rr((st) 2-mmin min
(TV
222
Algorithm deseription. We first describe the algorithm A₃ we use: Let X, Ber (f/z) be Bernoulli random variable, for some ie < k to be specified later. Ak issues a best action query if, time step t, X, 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 losses (i) = E- (1) - (e(t)) I (X, 1}. The algorithm then simply selects action I, = it if X, = and action 1, i with probability pi( if X, 0. Moreover, denote by X X.), X.),
(Xi....,
Lg (z)).
For the sake of the analysis, introduce another algorithm Al. which is the same algorithm A with the only (but crucial) difference that it issues a query and only if X2 = regardless of whether or not query budget is exhausted. We thus bound the regret of A in terms of the regret of Ale-
Lemma 2.5. For all adversarial sequences e 1]xx of action losses, in the label efficient query model, algorithm Ale guarantees
2 log n T log n
(A)) S min T
822
Proof. For algorithm A'., we have that its counterpart without queries, Ab. is an instantiation Hedge, (e), with e é. Thus, by Lemma 2.1, we obtain
L (x()) tE[T)
Σ (i( (a(1) 1 tn
<
1
(ix(()= log n
17

---

as long as 1) < 8/2. We now recognize that, since
[((i)) XXs- ((() (((t)) then
Σ
p(()) p
E
((0)) (
Σ
pe(i) (l((1) (((()))
ic(m)
(c(m)
Therefore, by the tower property of expectation applied around (2), we have
(T(A)) Σ E
p((1)
2(00]
Σ
pr(i) (Le(1) - 2(()))
vE(M)
iE(m]
1
[T((i)
min
log
k (LT(() + log
k T LT
T'n min
T 1 E
1
k- Tn
"
where the last inequality holds since Tn S By Observation 2.3, we get
LT(A)) - [
1: i
(Lr(i)
+ log n + 7
(T - k)m
LT
T Lp
T
TT)
k. T"
which means that the regret is upper bounded by
(r S klogn
T kk 1(kk Mk--Th Tn) T(k Tn) (Ir(())
2 log n
T log
+
T.pin
S min T
22
where the last inequality holds by setting 7 max
logg a
ki /2T²
and then by noticing, in
T
the latter case, that le
v2k.
V2T
With this lemma, we are ready to prove Theorem 2.4, where we bound the regret of algorithm A
Proof of Theorem 2.4. We consider event & = { s k}. so that, slightly abusing notation, we write the regret of algorithm A& as
(x{) = EE (Ax) E] IP [8] + E [RT (Ax)
El IP E S
E (TT
+
To upper bound the second summand above, have
2(k +1 R))
1 = 1, <
P E < T exp
(3)
T
T
by Hoeffding's 's inequality applied on the binomial random variable [Q] with expectation k and as long A
T log
L.
For what concerns the first summand, we recognize that under event E, As and A'e are exactly the same algorithm. Thus, it holds that E [Rr(AR, i) E] E [R i) E). Moreover, if E Rr (As)
E 2 0, and since P [E] Z 1 -//r, then
T
T T-1
2 log
T logn
(RT El S T-1
RI (A,,
min T
122
by Lemma 2.5. If, instead, E (Ax)) El O, also know that, by identical derivation to (3),
Rr (At) E P [E] 1. Therefore,
2 2lo n
T log n
E i) El < T-1 (0((4)))+)))
min T
+1 111
-
k
1.2
again by Lemma 2.5. Overall, obtain
2 llog n
T log n
(r()))) S min T
2
7

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3 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, as the feedback observed by the algorithm the end of time t. In the full feedback model, Ze 4, while in the label efficient setting, Z l only if a query is issued at time t (and Z an empty n-dimensional vector otherwise). Furthermore, we denote with Zgo the array containing the feedback Z1, ..,..., Ze until time t.
We start by describing the stochastic instances, which characterized by two distributions over two losses. As a notational convention, we denote the losses with l,, and introduce two probability measures p+, IP- (and their corresponding expectations E+, -) Let e, E [0, 1] be parameters we set later (with E S Q), we have that the losses of the n = 2 actions are distributed as follows:
(1, 1)
{ under both IP+ and P (0,0) w.p. 2 2q under both P+ and PP-
w.p.
(4(1),4(2)
(0, w.p. 9 e under P+ and w.p. under IP-
(1,0) w.p. q under P+ and w.p. q + under IP-
We now introduce and prove general Lemma the expected regret (AA)) suffered by any deterministic algorithm As which issues at most l queries, against the i.i.d. sequence of valuations drawn according I Since we want result that holds for both feedback models, we introduce the random set F which contains the times where A& actually observes the losses; note, NE = |F] is equal T in full feedback and the number queries issued in the partial information model.
Lemma 3.1. For any determinstic algorithm A; which issues at most k best-action queries, we have:
(*x))) + R2 exp (- E (Nr)) (>n))
5E²
(T k)e
2(g -e)k
Proof. The best action i' under P+ respectively PP- is the first, respectively the second, one, with an expected loss of
E [+(())] q-
On the other hand, the best realized action य yields an expected loss of
(e,(7)) = E ( (2))) (q - E).
(5)
Moreover, the algorithm chooses a suboptimal action i, # i", its expected instantaneous regret is: - E
+ -
[(((*)] - = E [L] E [<() 2e.
(6)
Let now N4, respectively N_, be the random variable that counts the number of times that A selects action respectively 2, in times that are not in F (i.e., where the choice of 1 is not due a query). Combining (5), and (6), we have the following:
(*x)))
2 = Σ (4(())) -(4(())))((11)
(400)) REF
LIF
28 E
[N+] (g [N)=)(n)))
Since NF S. k. and N4 + N_ TT
have the following bound on the regret:
N < T- k
(+{))) > p+
(T - <ke -(q e)
2
T - k
(Ax) F
N+
(T - k)s (q - =e)k.
2
Summing the above two expressions, we obtain
-
T T-k)) ) (T -T- k)e
(AAs) + (Ak) 2
pt
N
<
+
P
N >
2(g - -e)k.
2
2
(7)
8

---

At this point, we apply the Bretagnolle-Hon Huber Inequality [Theorem 14.2 in Lattimore and Szepesvári, 2020] to bound the first term on the right-hand side of the above inequality:
T-K p+ N+
T T- 2
exp (-DKI. Serr:Pzg )) (ZT))
2
where P Z is the push-forward measure on all the possible sequences of feedback observed by As under !== The lemma concluded by combining the above inequality with the following claim, and plugging it into (7).
Claim 3.2. It holds that DKL ( : Zsr
P
5e E [Np].
Proof. Let observe that, once we fix the feedback history until time t 1, i.e., fix a realization of the feedback Z we have that
DEL 2e Zzi P 2017g-11) - DEL (PZitemiPP P PZ(
where P-
2,125-1 (respectively 2 ) is the push-forward measure over {0, 1}2 when losses are drawn according to p+ (respectively IP conditioning the previous observations. The equality above holds because (i) algorithm A, observes feedback if and only t E F (by definition), and (ii) A& is deterministic and whether not t F may only depend on the past. We upper bound the KL-divergence term above:
2E
2
DxL 2.12g-11 P (212e( (e) .log
1
+ (q - e).
loo
q +
2e
2
4E2
5Ea
q (q - E)
+
q² E2
9
where the first inequality follows from log( + 2) < 2 for all R. and the last holds as long we
choose & < v3. To complete our derivation, we express the overall KL divergence exploiting the
q
tower property conditional expectation:
5E2
DKL (Pt P
Σ EETT]
E
DKL (PZ P (21Zer))
E [N[N,
Zer
where expectation taken over all possible feedback realizations Zςt-1, and the last inequality follows by earlier derivations.
This concludes the proof of the lemma.
We now show how use the above lemma 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 Aş that has access to < oo Tn queries, it holds that Rr (As) 2 Co -
/T
For any algorithm Ay that has access to 2 covT queries, holds that Ri (As) 2 C1
Where co = =1/(e" V5) and c 1/(32003) are universal constants.
Proof. In full feedback, the algorithm always observes the losses, that the feedback variable Z, E and NF = T. We prove the Theorem via Yao's minimax Theorem: we prove that any deterministic algorithm A₁ fails against the random instance composed as follows: with probability 1/2 the losses are drawn i.i.d. according to It+, otherwise, they are drawn i.i.d. according to IP_, We can then apply Lemma 3.1 and obtain that the expected regret of Aş against such mixture is equal to
5E2 (--r)
(T- k)
(Ae)) = (A + (Ae) 2 exp
2(g =)k.
(8)

---

Now, i < coVT, we set
2 v5T
and g = 1 in (8) to get
E [[RT(A:)]= [ -
VT
VT
VT
2e V5
4
455v5"
Otherwise, if Co T, then consider the following choice of the parameters:
4e-1
T0ck
10e7
and q = =5s27= 3206" K
T
4p-1
4e=1)² "206"T"
Plugging these parameters in (8), we get:
IGUE
E [ 1 (A)) + (Ax)) 2 kT
Ts
(1 4e
+
ek
4e
T
4
(4e 2)
+
320c
160e
320c2T
320c
A similar analysis can be carried over for the label efficient setting.
Theorem 3.4. In the label efficient feedback model, for all l E [T), we have the following lower bounds:
T For any algorithm As that has access to l < co V queries, it holds that RT(AAt) 2 -00 Avk"
T
T For any algorithm As that has access to 2 cu VT.
queries, it holds that Rr(As)) 2 01 T 12
Where co 1/(e8 V5) and e₁ 1/(32003) are universal constants.
Proof. In the label efficient feedback model, Ze is meaningful only for times in F. We then prove the result by Yao's minimax principle, by showing that any determinisitic algorithm As which issues 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 NE I k) to get:
E [RT(A:) = (Ax) + (Ax)) 2 exp (- (-be2k)
5
TT
K)E
k
(q E)k.
4
Once again, we have two cases. ls < co Vk* we choose
E
v5k
2
and
= to get
1 1 (R (Ax) + (Ax)) 2 E [R =
T
T
T
2e 5k
4e V5k
4e /5k
Otherwise, if k 2 co र.. and we choose E (de-1)²
and q
5
T
(4e 1y2
Tock
Wek
320e+
160
320c to obtain
TE 22 + 1
1
E [RT(As)] = (At( + R
2 4e
eek
4e
T(4e - 1)
(4e 1)²
T2
+
320
160c2k
320e2
220c212
Conclusions
Our work introduces best-action queries in the context of online learning. We provide tight minimax regret in both the full feedback model and in the label efficient one. We establish that leveraging sublinear number of best action queries is enough improve significantly the regret rates achievable without best-action queries. Promising avenues for future research involve integrating best-action queries with diverse feedback forms, extending beyond full feedback, such as bandit feedback, partial monitoring, and feedback graphs (where, in particular, Lemma 2.1 does not hold). Moreover, our work only studies the case where queries are perfect, i.e., the queried oracle gives the correct identity of the best action at that time step with probability 1. Imagining a noisy oracle that gives the correct identity of the best action only with probability 1/n + also an interesting future direction this work leaves open.
10

---

Acknowledgments
Andrea Celli is partially supported by MUR PRIN 2022 project 2022R45NBB funded by the NexIGencrationEU program. Federico Fusco, Stefano Leonardi and Matteo Russo are partially sup- ported by the ERC Advanced Grant 788893 AMDROMA "Algorithmic and Mechanism Design Re- search in Online Markets", by the FAIR (Future Artificial Intelligence Research) project PE0000013, funded by the NextGenerationEU program within the PNRR-PE-AN scheme (M4C2, investment 1.3, line on Artificial Intelligence),, and by the PNRR MUR project 1 3-SoBigData.it Stefano Leonardi and Matteo Russo are also partially supported by the MUR PRIN grant 2022EKNE5K (Learning in Markets and Society).
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12

---

A Omitted Proofs
A.1 Proof of Lemma 2.1
Lemma 2.1. Consider the Hedge algorithm Hedge,, (2) run on loss sequence & & [0, Ujxx with learning rate Tj < 1/u. Then, for all action E [n]. it holds that,
1
(Ir(i)) log
(THHedg (t))
( S Un
where Lr (Hedge, = EEETT, (o( (i) is the expected cumulative loss of Hedge, (e) and (i) is the cumulative loss of action i.
Proof. We have that, by definition,
WT+1 2 wr-1(()we(()) exp (-nl((1))
II exp (-n(((t) exp (-ntx((i) (E(T]
We also know that
W41 S W,
1- Σ
pe(1) 2,(1) + Σ Pr(1) ((()..2(111
(E(m)
AE/m]
Thus,
WT+1 n. II
1 1 Ept)) (.()) 400) + n² Σ p.(1) (2(1)
:E(T]
=E(m]
<E(m)
which combined with the earlier bound and taking logarithms of both sides, gives
-n) < log +
Σ log
- 1 Σ (r( 401 +1²
Σ (i( (0(
LE|T
VEIn
iE|ml
< log -
Σ
Pr(1) E,(1) + 1
Σ
Pe(1) (1)1
EE iE(T).iE(n)
(E(T]iE[M)
< log (1 Un)
Σ
Po(1) (1).
The second inequality above follows from log(1 2) $ for R, and the third by observing that (2))) < (i), since (i) E [0,U] U]. The lemma follows by rearranging the terms above.
B Best-Action Queries in the Stochastic i.i.d. Setting
In this section, we show how the results provided in Section can be obtained in the stochastic i.i.d. setting (defined next) using the Seilo--The-eaaeeee algorithm, albeit suboptimal dependence in n.
In the stochastic i.i.d. setting, each action associated with fixed but unknown distribution D; supported in [0. 1]. Action i's loss at time step (i), is drawn independently from distribution D;- We denote with (i) the expected loss of action i, and with i* = arg minje(ai μ(1) be the action of lowest expected loss. We measure the performance of learning algorithm A, by considering its regret with respect to the best action distribution:
(T
Σ (4(1)) (())) LE[T]
We denote by A, = (ui)= u((i) the gap between the expected loss of the best action and that of action i, and by V, E [i())) E(i" }] the expected absolute value of such gap (note, 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 é with best realized loss is revealed, so that the learner gets, in expectation, EE [it)] E [minte[m] (())).
13

---

B.1
Useful Facts and Observations
The following facts and simple results are particularly useful for analyzing algorithms the stochastic generation model.
Fact B.1 (Bernstein's s Inequality). Let YI,..,,Yn Ym be independent random variables such that
E [Y=] and IP [| S c| 1, for >> 0 and all E [m]]. Then, for y >> O, we have
y, 1 2 3 exp (- 66² + 20y
3y2
E [Y=]² Moreover, when Y,'s 's are also identically distributed, we have x² = (E [ Y+ 2) S mF [|*=|] =: m. Thus,
where 0² -Er= [
Σ Y, - n 2 s exp
m
3-2
G.rV + 207
Proposition B.2. Let T, n. E NN be the time horizon and number of experts respectively, with 2 n n-1. For all distributions Dj of actions losses, and any algorithm A, its regret is
T
k
Rr(AA)) = Σ
A.
(V
A),
(9)
2
1=k+11
where v := -E [[min E(i) {(i')] and A == E mmnn f(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 S k), the algorithm suffers [minkeja] (i) in the first time steps, which, by independence, gives
Σ I
min (1) = kE min (())
FEI
(E(n)
We also have the following useful observation that allows us to rewrite the maximum a convenient form. Namely.
((it)
min ((i) vifi"
min E(1) - E(i*)
min E(i) EE
(10)
2
The claim follows by the definitions of Y and A.
Thus, in the stochastic case, the goal of a regret-minimizing algorithm to minimize the first summand above since the second is characteristic of the instance at hand.
Observation B.3, Let T.n E N be the time horizon and number of experts respectively, with T T2n 1. For all distributions D; of actions losses, V A. v A.
Proof. Since E [minje(m] {()) < E [min [min((((")) (((i'), for all i E [n)]. we have
- V = 2E
min (()) {(i') jE(n
< 2 (()) ((-)) A, Vi,
where the last equality also follows from Equation (10) applied to actions i, 1 only.
B.2 Label Efficient Feedback:Explore-TTees Comn-ommit with 1(T Best-Action Queries
We consider the label efficient feedback case first, where feedback is given only during querying time steps. The techniques used in this section key for the full feedback case, and easier to illustrate.
We provide a simple variation of the Explore Then-Comen that, in the first & 2 2(T%/s 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:
(ETC) algorithm Perchet et al. [2015]
14

---

Algorithm 2 Exxpore-Thenn-Comee with Best-Action Queries
Input: Sequence of losses (2) D; and & [2'] query budget
for t E [T] do
if t then Observe 8 arg min (()
Select action 1 Observe (1)) for all i E [n]
else
Select action arg min.e(m] (1)
Theorem B.4. Let T. N be the time horizon and number of actions respectively, with T 2 1. For all distributions D; of actions losses, Algorithm with query access k E [1T)9] guarantees
log 2nT
2nT In
Rr(As) S min
3T
2k
k-2
We split the proof of this theorem in two lemmas that immediately imply it, one for lt < T and one otherwise.
Lemma B.S. Ifk < T then R2 (Ac) S 37
log 2nr
Proof. For all actions i, we define the clean event
& = i m(i) <e),
where
Similarly, let & be the intersection of all the E for i E [n]. By Hoeffding's inequality [Equation (5.8) in Chapter of Lattimore and Szepesvári, 2020], have the following:
log 2x 24
E
[ S
Σ [ iE(n]
S 2n exp (-2ks S 4
Consider now the expected instantaneous regret suffered by ETC at time / + 1. We have two cases: Either the clean event holds, that the instantaneous regret most 25,, does not, which case the instantancous regret is at most 1. By the law of total probability, we have the following:
E (fc+ ( [
[Le+1
l Ei) S
2s, S 3
Summing the expected instantaneous regrets for all yields the desired bound of
log 2nT
Rr(AA) S 37
2k
as we use the lower bound V A 2 0 to say that in the first times the regret the algorithm suffers most O.
Lemma B.6. 1fT < k S 4T/9, then (rx(As) < In
Proof. We start by showing the statement for actions, and then generalize to n.
The case of actions. Let recall that, by Proposition B.2, have
(AA))) [t 2 k 23 A
(U A))
where we have assumed that μ4 S μ2, without loss of generality. We suffer positive regret in the last T ls time steps when we select action 2 over 1, which happens and only if jux (1) S uu (2). Hence, the first summand of the above regret expression simply becomes A(T k) (2x S jAx (1)).
Note that
(2) S (a) =
Σ 4(11 - (2)))
15

---

so if we define 9r = (1) - (2)., whose expected value is E [ by Fact B.8,
- and P 119.
S 1] = 1, then,
St 2 0 =
9t + KA 2 AA
3 A
S exp
6k + 2K
A
S exp
2V + A
We distinguish two cases, namely 24
T and
A
T
In the latter case, we get that
2V+A
k
Rr(As) = A(T k))
(2) S P4
(1)
( A)
2
RA2
I AT exp
( - A)
2V + A
2
< AT exp (-In7) = 1
Otherwise, we obtain that 1 2 ( kA - 1), and so
(n
k
RT(AA)) = A(T k) P Jine (2) - (x(1
(V A)
2
KA
kA
I AT
3
4
I
T
3T
27
I
122
4k
16)
2T In
T
12
The first inequality holds since the maximizing A
4T-3k 2x
In which is consistent with 0 < A <
T < 1 since 8/ < k < 47/9.
Generalizing to n actions. tion B.2, to obtain
We again assume that the first is the best action, and apply Proposi-
T
(Ac) = Σ
E
I
(x( 2 max (4:6) - E
2 (v A)
(=k+1 1f1
=(T -T- Σ D5 P [+ا(ي) 2 max (l()))
k
-2 - A)
ifj
j
< T Σ A ((x(t) 2 (n(11) 2 (vV (V- A),
##1
which holds by the independence of realizations across time steps. Let define (r( (,(1) so that, 1 = E (T)). [g.()) = A5 and IP [1g)())
< 1 = 1. Then, by Fact B.8, we obtain
k
A
P ( 2 Ru (1)) = P
()) 2 0
S exp
2v , + +
Recall that, by Observation B.3, V A 2 Vr A Therefore, let us sum and subtract to the regret upper bound the sum 2n Eşボr (V) A,), and get
AA k (V, 1,))
(T(As) - Σ
TA exp
2V, A,
2n
j#1
16

---

k 2
1
Σ
( A,) (v (V- 1
n
jfl
R 24, A
h (V; (V: -1))
TA exp
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 obtain
2nT In T
Rr(As)) S
which concludes the proof.
B.3 Full Feedback: Follow- Follow-The-Leader with 9 (VT) Best-Action Queries
We provide algorithm that, equipped with best-action queries for l time steps and with full feedback, achieves regret OTT/s) for k 2 (((T)), and O ((VT) 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-Leadee paradigm Auer al. [2003]. That is, we select the best action first & time steps since queries give us its identity for free, and successively, we select the action maximizing the empirical average so far, (1) T-1 Σt= (1).
Algorithm 3 Follow-The Tee----Th--eeddee with Best-Action Queries
Input: Sequence of losses (1) ~ D; and & [T] query budget
for / E [T] do
ift t then Observe i; arg min ((()) Select action i Observe (t) for all i [n]
else
Select action arg min 1t)
We have the following guarantee:
Theorem B.7. Let T. E be the time horizon and number of actions respectively, with T 2 n 1. For all distributions D; of actions losses, Algorithm with query access l 2 2/T guarantees
5n
(Ax) S min 3//22 2T log 2nT
k
For our purposes, only a weaker version of Fact B. is needed:
Fact B.8. Let YI,..., Ym be i..dd random variables such that E [Y,]
-y, E [| My, and
P P| < 1] = for all T E [m]. Then, holds that
mA 20 + Ay
P
Y
2
0
S
exp
We split the proof of this theorem in two lemmas that immediately imply it, one for & < 2VT and one otherwise. Next, we subsume the notation of Theorem B.7.
Lemma B.9. Ifk 2vT, then (r()s)) < /7 log 2nT.
Proof. For all times and actions é, we define the clean event
E (1), (i)l < ex).
17

---

where E,
log 2nT Similarly, let & be the intersection of all the E 's for i E [n]. By Hoeffding's
20
inequality [Equation (5.8) in Chapter 5 of Lattimore and Szepesvári, 2020], have the following:
P & = P [
Σ TP [Ea *E(m)
S 2n exp (-2te S T
Consider now the expected instantaneous regret suffered by FTL at time t + 1. We have two cases: Either the clean event holds, that the instantaneous regret most 25, or it does not, in which case the instantancous regret is at most 1. By the law of total probability, we have the following:
E (()) - (1)) = P [&]] + E (
f(+1 (f) - +
25 S 35g.
Summing the expected instantaneous regrets for all & yields the desired bound of
RT S 3v 2T log 2n',
as we use the lower bound L at most 0.
2 0 to say that in the first times the regret the algorithm suffers
Lemma B.10. If 2 TVT, then (Ax) I
5 K
Proof. We start by showing the statement for 2 actions, and then generalize to n.
The case of 2 actions. reads
We again assume that μ4 S #2 without loss generality. FTL's regret, thus,
T
RT = E
(l(2) - (( - (o(1) S Fu 1
2 ( -)
[=+++
k
A
P pīt-1 (2) S pr-1 2
(v A)
t=k+1
112
k
<A
exp
(v A) 2
(11)
1+1 2
A
where the second equality follows from the independence of step t from all the preceding steps, and
A the inequality by Fact B.8. We distinguish three cases, namely 20+4 S $ T
A
< lu and
20+4
k
20+4 > I In the first case, we have that v N zz (TA 1) and so,
k
A
(Ax) < AT
(TA 3)
4
T
3
27k
k
64T
27
k
where the second inequality follows since the A maximizing the expression is A A= - implies that v
3 This
27
Z which is consistent with A < v since k: > 2.T.
24
32T
In the last case, 20+
T K
we
get
k ( A)
(Ax) A(T-A) .P (Pt-1(22 & 1c-1(1)]
2
1
$ AT exp
2 (v - A)
2 + A
< AT. .expeepp T) =
< 1.
18

---

We are left to show the case where + it holds that
<
20+
A²
S
TT
k.
To this end, let us observe that, for all T 2 2.
Σ t=m+1
exp
ST.
for all 1 S m,r S T. This is the case because
Σ
c
kk/r k/r
e//r-1 S re -k/r ST,
TT/r
exp
1=++1
since 1 + 2 c² for all 2 E IR. Therefore,
2
A
Rr((A)) S 2A-
2
V
-)= = v
+ 2 +
A
2
2
We distinguish two subcases: On the one hand, assume that A
then
A/4 5/4 < 0. Since
y
2hm 2. have
kAa
kΛ²
A
4
Ka 2
(A((:) S
+
+
+
2n
2
2
2kA
h 2
3k
IT
4hT
4
3hn
4
1
In T
+
4+
8
16 -1)
8
InT
T
2
since the expression is maximized for 8tin 2k
>
8
as long as T > e.. This implies that
E.
V 2
8 ln which is consistent with 0 <
A
<
V
<
1. On the other hand, A s k. and thus
knT 8k
4
2 0. We use that V < TA--A and get
A
2
TA2
A
4 k
k
(Ax)s) S
+
2
2
2
2
KT
2
3k
2TA
+
47'
9k
k
16T
5T
k
since the expression is maximized for A
1/2 +
3
k. This implies that v <
8T A2 +
3
which
2T
8T
consistent with < A < < since k 2 T.
Generalizing to n actions. the regret equals
The generalization is very similar to the one in the proof of B.4, Indeed,
(T()))=
Σ
E ()-
< min n.- ii- (0) (((6)) - 6(11))
k ( A
i4)
2
b+++1
##1
h:
Σ += 1+1
Σ
A P ()-
min pi- (i)
(T A)
Σ
Σ
A P p (1) S (1)]
k
( - A)
19

---

A 2V, + A
k - -v A),
$
<
cxp
j#1
t=++1
where the last inequality follows by Fact B.8. Identically to the proof of Theorem B.4, we sum and subtract the term
&
Σ j (4, A3), and get
2n
T
t
(Ac)
A
Σ
exp
(V, A3)
2V, + A
2n
1#1
1=
k
1
+
V
A,) - - -A)
n1
j+1
Σ
A
Σ t=++1
exp
! 2 + A
k
(v) A,)
2n
j71
For each j f 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 obtain
5nT
(T((s)) S
k
which concludes the proof.
C Further Related Work
Correlated hints. A first model that close to ours has been introduced by Dekel et al. [2017], which studies Online Linear Optimization when the learner has access 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 ((VT) 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.
Oueries/Ordinal hints. Bhaskara et al. [2023b] studies online learning algorithms augmented with ordinal queries the following type: a query takes in input 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 experts losses in advance 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 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 variety of online problems to model partial information about the input sequence that can be fruitfully exploited if accurate for improving the performance 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.
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Answer: [NA].
Justification: The paper has no significant limitation. Being theoretical in nature, there is set of model assumptions (see Question 3). The model is realistic the number of queries we can issue is limited. Moreover, if we supposed that the external queries come from a human (which is true in many applications), we could assume that these reviews are noiseless (see, e.g., httpht/t////rrassprrccccyyc wew.:eeeeeeeeeeeeccoo.......ee.e wwwp:/t.ttaeeeeeeeeeeeemmmeeeeee /neeg//gb// httpp//ipppiiicooocccccaaaa////a meta. htlp///accoo/oo/gg/oo//oooolclla httpppppt.....oooooooooooooollss
This, of course, no longer holds when queries are generated by external learning algorithm, where some degree of noise inevitable. Our current model does not account for potentially faulty queries. Indeed, our algorithms "blindly trust" the query, and the bounds presented in our analysis rely on the assumption that the query is perfect.
com cwtps//oocccc////cccccoccccccc/e
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Reviews:
[{"review_id": "fmsgRbisoE", "reviewer": "Reviewer_Sy4T", "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).\n\nThe 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.", "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.  \n\nLi, 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": "This is fine", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "I think that this paper is both pertinent and timely: the investigation of how augmented information can improve regret in classical online learning settings has been an interesting direction of research in the recent years, and the paper makes a significant contribution to this body of work. I particularly find the improvement in the LEP setting to be remarkable. The paper is further well written and easy to understand, with existing arguments being used in an interesting new way.", "weaknesses": "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).\n\nPerhaps 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?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the questions and suggestions, which we will implement in the final version of the paper.\n\n**Weakness.** Perhaps another point of improvement ... best-action queries?\n\n**Answer.** One other application of our model is in fraud detection. In this context, an online learning algorithm aims to identify incoming points as either potential frauds or benign content. This is critically important for platforms like Booking.com, where customers defrauded by fictitious hotels or BnBs must be reimbursed. Proactively identifying potential fraud saves the platform substantial amounts of money. Specifically, Tax, Jan de Vries, de Jong, Dosoula, van den Akker, Smith, Thuong, and Bernardi (MLHat, 2021) provide evidence of a machine learning system designed to detect potentially fraudulent listings on Booking.com. This system is supplemented by expert human oversight, which can be called upon a limited number of times. These instances of expert involvement can be interpreted as best-action queries.\n\n**Suggestion.** One approach to modeling problems ... Gangrade et al., Neurips 21.\n\n**Answer.** We thank the reviewer for providing this point of contact. We will make sure to expand upon this connection in the final version of the paper related work.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rbZANaapj0", "reviewer": "Reviewer_aUFF", "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.\n\nThey 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).\n\n(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 $\\omega(\\sqrt{T})$.\n\n(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.\n\nThey 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.\n\n(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 $\\omega(T^{2/3})$. \n\n(ii) They also show a matching lower bound for regret (up to $\\log n$ factors) of any algorithm with $k$ query label efficient feedback.\n\nIn 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.", "questions": "* In the equations after line 178, is the last equality an inequality ($\\leq$) using $T\\eta \\leq \\hat{k}$?", "limitations": "* There is no negative social impact.\n\n* 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.", "rating": 5, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 2, "strengths": "* The model (best action queries) is theoretically interesting and the results are quite nice in the sense that you can get polynomial improvements in the regret by incorporating _sublinearly-many random_ best action queries with the standard (full feedback and label efficient feedback) Hedge algorithms.\n* The paper is well-written. Almost all the proofs are cleanly presented in the main matter itself. They also survey and compare some of the existing work on augmenting online prediction algorithms with additional information.", "weaknesses": "* 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the questions and suggestions, which we will implement in the final version of the paper.\n\n**Question.** In the equations after line 178, is the last equality an inequality using $T\\eta \\leq \\hat k$?\n\n**Answer.** Yes, thank you for pointing this out.\n\n**Limitation.** 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.\n\n**Answer.** We thank the reviewer for this suggestion. In several applications (see, e.g., https://transparency.meta.com/en-gb/policies/improving/content-actioned-metric/, https://transparency.meta.com/en-gb/policies/improving/prioritizing-content-review/, https://support.google.com/adspolicy/answer/13584894), these external queries come from human reviewers, for whom we can assume that query responses are perfect.  We agree that in some applications, i.e., when the queries are generated by external learning algorithms, some degree of noise may be inevitable. We believe that addressing also these applications is a compelling and interesting direction of research, and we believe that our work, which provides a nearly complete understanding of the perfect prediction case, may work as a natural starting point along such direction.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YoUa0ye8cK", "reviewer": "Reviewer_VoCp", "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.\n\nThe results are as follows:\n1. 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.\n2. 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).\n3. Lower bounds for both of the above results which are asymptotically tight.", "questions": "Can you provide technical intuition for where the multiplicative power of the best-action comes from?\n\nBelow 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.\n-  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).\n- Lines 84 - 86 are quite compelling.\n- Line 93: Maybe mention that the learner still observes the entire loss vector, not just the best action.\n- Lines 132 - 133: “combining the Hedge algorithm [Chapter 2 in Cesa-Bianchi and Lugosi, 133 2006] to uniform queries” → “with uniform queries”?\n- Line 200: “and Z_t is an empty n-dimensional vector” → “and Z_t is an empty n-dimensional vector otherwise”?", "limitations": "In Question 2 of the checklist, the authors state:\n> The paper has no significant limitation. Being theoretical in nature, there is a set of model assumptions\n\nI 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.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "I find the model of sublinear best-action queries to be quite natural, and the authors also do a good job of motivating it. One example given in the paper which I found compelling was an algorithm for flagging harmful content which can escalate to a human review a limited number of times.\n\nI also found the results to be impressive. As the authors state,\n\n> 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.\n\nThis seems like a potentially powerful insight to me.", "weaknesses": "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. \n\nFor 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.\n\nSee also the “limitations” section below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the questions and suggestions, which we will implement in the final version of the paper.\n\n**Question 1.** Can you provide technical intuition for where the multiplicative power of the best-action comes from?\n\n**Answer.** We are happy that the reviewer shares our enthusiasm for the (somewhat) surprising effectiveness of our query model. We provide here two high-level observations that may help a better understanding. We will add them to the final version of the paper:\n\n- The additive term $\\frac{k}{T} \\cdot L_T^{\\min}$ (given in Observation 2.3, and in particular right after line 150) impacts 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 $\\frac{-\\eta L_T^{\\min}}{1-\\eta}$ (in the inequality given right after line 153). This negative term together with the additive $\\frac{k}{T} \\cdot L_T^{\\min}$ term given by best-action queries allows us to set the optimal $\\eta$ to be larger than the usual (order of) $1/\\sqrt{T}$. In other words, the additive impact of the $\\frac{k}{T} \\cdot L_T^{\\min}$ term permits a multiplicative gain in regret as the learning rate $\\eta$ is modified and increased.\n\n- Consider the following natural instance, which provides a simple proof of the $\\Omega(\\sqrt{T})$ 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 $0.5$. Any learning algorithm achieves $T/2$ regret, while the best-fixed arm in hindsight is expected to achieve an extra $\\Theta(\\sqrt T)$ term (this is a simple corollary of the expected distance of a random walk). Now, if the learner is given the power to issue $\\approx \\sqrt{T}$ queries, then its regret naturally drops from $\\sqrt{T}$ to constant.\n\n**Limitation 1.** Are the model assumptions realistic? If not, to what extent do the results crucially hinge on those assumptions? \n\n**Answer.** We thank the reviewer for pointing this out. The model is realistic as the number of queries we can issue is limited. Moreover, if we supposed that the external queries come from a human (which is true in many applicative settings), we could assume that these reviews are noiseless (see, e.g., https://transparency.meta.com/en-gb/policies/improving/content-actioned-metric/, https://transparency.meta.com/en-gb/policies/improving/prioritizing-content-review/, https://support.google.com/adspolicy/answer/13584894). This, of course, no longer holds when queries are generated by an external learning algorithm, where some degree of noise is inevitable. Our current model does not account for potentially faulty queries. Indeed, our algorithms ``blindly trust'' the query, and the bounds presented in our analysis rely on the assumption that the query is perfect. For example, the former intuitive explanation of the multiplicative power given by best-action queries no longer holds. However, we believe that our work, which provides a nearly complete understanding of the perfect prediction case, may work as a natural starting point along such direction. We will expand on this in the final version of the paper, and we will update the checklist accordingly.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "IQ3rq9nzpp", "reviewer": "Reviewer_tNck", "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.\n\nThe 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.\n\nWIthin 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.", "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.\n\nAs 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.\n\nSome minor details I would like to point out:\n- Line 35-36: \"... the platform's task **consists in** deciding whether...\" should be \"consists of\"?\n- Line 49: \"...with full feedback, **an** in the **stochastich** multi-armed...\".\n- 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?\n- Line 102-103: The entire sentence within the parenthesis does not make sense to me, grammatically.\n- 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.\n- 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.\n- 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.\n- Line 196: \"...we construct two **of** randomized instances...\".\n- 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": "The authors sufficiently discuss limitations and future directions in the conclusion section. There are no ethical concerns or limitations in their work.", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The paper studies the intersection between two interesting fields, learning-augmented algorithms and online learning, and contributes positively to its literature. Traditionally, the field of learning-augmented algorithms places a focus on (classical) online algorithms, so whether external predictions can help online learning is a very natural extensional question to ask. Over prior works in the same intersection, the authors additionally showcase the power of such predictions by showing that a small number of queries or predictions can  significantly decrease the loss/regret of online learning algorithms, which is a very surprising and exciting discovery.", "weaknesses": "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.\n\nMore 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.\n\nOverall I still believe that this is an exciting paper; but its presentation can be improved significantly.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the questions and suggestions, which we will implement in the final version of the paper.\n\n**Question.** The authors discuss the potential of using noisy ... prediction regime in the future.\n\n**Answer.**  We thank the reviewer for this suggestion. In several applications, these external queries come from human reviewers, for whom we can assume that query responses are perfect (see, e.g., https://transparency.meta.com/en-gb/policies/improving/content-actioned-metric/, https://transparency.meta.com/en-gb/policies/improving/prioritizing-content-review/, https://support.google.com/adspolicy/answer/13584894). However, if the queries are generated by external learning algorithms (even if trained on this specific task), they may be erroneous. The suggested direction is an interesting open questions, which we defer to future work. We believe that our paper's techniques for perfect best-action queries will be essential to analyze these model's extensions.\n\n**Question (cont'd).** As illustrated above, I believe ... follow the flow of logic.\n\n**Answer.** We appreciate the reviewer’s feedback. We tried to add an overarching explanation of the proof strategy in Section 1.3. We detailed the technical challenges and how we addressed them, providing an explicit explanation of how the various components fit together and highlighting the advantages of uniform querying. We will expand the explanation in the technical sections, exploiting the extra page that is provided for the final version of the paper. \n\n**Minor details.** \n\n- Q - 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?\n\n- A - Yes, the algorithm receives the \\textit{entire} loss vector when it makes the query, and before choosing the action. So, the loss vector itself is given as a whole (and, thus, not partial) but only when a query is issued.\n\n- Q - 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.\n\n- A - Yes, it indeed is the classical Hedge algorithm. However, for the purposes of an easier analysis and calculations, we subtract $\\ell_t(i^*_t)$ in the update rule exponent. As the reviewer points out, however, the distribution from which we sample the next action is the same as in classical Hedge (the exponent cancels out in numerator vs. denominator).\n\n- Q - 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.\n\n- A - We actually meant \"We have the following algorithm.\" Thank you for pointing this out.\n\n- Q - Line 199-200: ``(and is an empty $n$-dimensional vector)''. Is the scope of this sentence if a query is not issued at time?\n\n- A - Thank you for pointing this out. We meant to write ``(and is an empty $n$-dimensional vector otherwise)'', i.e., we do not observe anything if we do not query. We will edit it in the final version of the paper.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: 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.
Decision: Accept (poster)