Multiclass Transductive Online Learning
Authors: Steve Hanneke
Vinod Raman
Amirreza Shaeiri
Unique Subedi
Venue: NeurIPS 2024 spotlight (2024)
Keywords: Online Learning
Transductive Online Learning
Multiclass Classification
Primary area: online_learning
TL;DR: We give a combinatorial characterization of multi class transductive online learnability even when the label space may not be bounded.

Abstract: We consider the problem of multiclass transductive online learning when the number of labels can be unbounded. Previous works by  Ben-David et al. [1997] and Hanneke et al. [2024] only consider the case of binary and finite label spaces respectively.  The latter work determined that their techniques fail to extend to the case of unbounded label spaces, and they pose the question of characterizing the optimal mistake bound for unbounded label spaces. We answer this question, by showing that a new dimension, termed the Level-constrained Littlestone dimension, characterizes online learnability in this setting. Along the way, we show that the trichotomy of possible minimax rates established by Hanneke et al. [2024] for finite label spaces in the realizable setting continues to hold even when the label space is unbounded. In particular, if the learner plays for $T \in \mathbb{N}$ rounds, its minimax expected number of mistakes can only grow like $\Theta(T)$, $\Theta(\log T)$, or $\Theta(1)$. To prove this result, we give another combinatorial dimension, termed the Level-constrained Branching dimension, and show that its finiteness characterizes constant minimax expected mistake-bounds. The trichotomy is then determined by a combination of the Level-constrained Littlestone and Branching dimensions. Quantitatively, our upper bounds improve upon existing multiclass upper bounds in Hanneke et al. [2024] by removing the dependence on the label set size. In doing so, we explicitly construct learning algorithms that can handle extremely large or unbounded label spaces. A key component of our algorithm is a new notion of shattering that exploits the sequential nature of transductive online learning. Finally, we complete our results by proving expected regret bounds in the agnostic setting, extending the result of Hanneke et al. [2024].

Multiclass Transductive Online Learning
Steve Hanneke
Vinod Raman Department Statistics University of Michigan Ann Arbor, MI 48104 vkraman@umii ch. edu
Department of Computer Science
Purdue University
West Lafayette, IN 47907
steve. www.ekeegmail................aae com
Amirreza Shaeri
Unqiue Subedi
Department of Computer Science
Department of Statistics University of Michigan Ann Arbor, MI 48 104 subedi @umi ch. edu
Purdue University
West Lafayette, IN 47907
amirreza, wwweeaaeaaeaeaaeia.aaaaaaamaaaae
Abstract
We consider the problem of multiclass transductive online learning when the number of labels can be unbounded. Previous works by Benn-Dvvdd et [1997] and Hanneke et al. (0036) only consider the case of binary and finite label spaces, respectively. The latter work determined that their techniques fail to extend to the case of unbounded label spaces, and they pose the question of characterizing the optimal mistake bound for unbounded label spaces. We answer this question by showing that new dimension, termed the Level-constrained Littlestone dimension, characterizes online learnability in this setting. Along the way, we show that the trichotomy of possible minimax rates of the expected number of mistakes established by Hanneke et al. 2023b) for finite label spaces in the realizable setting continues to hold even when the label space is unbounded. In particular, if the Iearner plays for T N rounds, its minimax expected number mistakes can only grow like (T)), ((log T'), (1)) To prove this result, give another combinatorial dimension, termed the Level-constrained Branching dimension, and show that its finiteness characterizes constant minimax expected The trichotomy is then determined by a combination of the Level-constrained Littlestone and Branching dimensions. Quantitatively, our upper bounds improve upon existing multiclass upper bounds in Hanneke et al. [2023b] by removing the dependence the label set size. In doing so, explicitly construct learning algorithms that can handle extremely large or unbounded label spaces. A key and novel component of our algorithm is new notion of shattering that exploits the sequential nature of transductive online learning. Finally, we complete our results by proving expected regret bounds in the agnostic setting, extending the result of Hanneke et al. 2023b
Introduction
Imagine you are famous musician who has released K songs. You are now on tour visiting T E N cities worldwide based on the pre-specified plan, each with unique musical preferences that you have some understanding of. At each city, you can perform only one song in your concert, and following each performance, the audience provides feedback indicating their preferred song from your repertoire. Your goal is to select the song that aligns with the majority's 's taste in each
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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city to maximize satisfaction. How can you effectively select songs to ensure the highest audience satisfaction across most cities having minimal assumptions?
The above example and similar real-world situations, where entities operate according to a possibly adversarially chosen pre-specified schedule, can be formulated in a framework called Multiclass Transductive Online Learning. Formally, in this setting, an adversary plays repeated game against the leamner over some T E IN rounds. Before the game begins, the adversary selects sequence of T instances (#1,..., IT) E x from some non-empty instance space X' (e.g. images) and reveals it to the learner. Subsequently, during each round E (1..... ........... ,T) the learner predicts a label ie E y from some non-empty label space y (e.g. categories of images), the adversary reveals the true label yt V, and the learner suffers the 0-1 loss, namely (b) * y. Importantly, the label space y is not required even to be countable; we assume only standard measure theoretic properties for it. Following the well-established frameworks in learning theory, given a concept class C S y of functions c: X' -> )/, the goal of the learner is to minimize the number of mistakes relative to the best-fixed concept in C. If there exists e E such that c(ax)) = ye for all E {l....; T), we say we are in the realizable setting, and otherwise in the agnostic setting. We briefly note that if the learner's predictions are randomized, we focus on the expected value of the mentioned objective.
In this paper, our main contribution is algorithmically answering the following question in the multiclass transductive online learning framework:
Given a concept class C C y what is the minimum expected number of mistakes achievable by a learner against any realizable adversary?
For the special case of binary classification ([>) = 2), this question was first considered by Ben-David
et al. a [1997] and then later fully resolved by Hanneke et al. [2023b). Additionally, Hanneke et al.
2023bb considered the case where [ > 2, but did not resolve this question when y is unbounded.
In fact, the bounds by Hanneke et al. [2033)) break down even when [ 2 2 As a result, Hanneke
ct al. 2023b) posed the characterization of the minimax expected number of mistakes in the multiclass setting with an infinite label set as open question, which we resolve in this paper.
1.1 Online Learning and Multiclass Classification
In this work, we study transductive online learning framework, where the adversary reveals the entire sequence of instances (21...., .r)) to the learner before the game begins. In the traditional online learning framework, the sequence of instances (æ1..... Pr) are revealed to the learner sequentially, one at a time. That is, on round E (1..... . the learner would have only observed .i.... Zt. The celebrated work of Litllesonne [1988] introduced this framework for binary classification and quantified the best achievable number of mistakes against any realizable adversary for concept class C f0. 1122 in terms of a combinatorial parameter called the Littlestone dimension. Later, the work of David et al. [2009] showed that the Littlestone dimension of concept class C C {0, 1) continues to quantify the expected relative mistakes (i.c expected regret) for the mentioned framework in the more general agnostic setting, More recently, Daniely et al. [2012] and Hanneke et al. [2023a] extended these results to multiclass online learning in the realizable and agnostic settings, respectively. See Section for more details.
In traditional online classification, there are two sources of uncertainty: one associated with the sequence of instances, and the other with respect to the true labels. [Ben-David et al. (1997) initiated the study of transductive online classification with the aim of understanding how exclusively label uncertainty impacts the optimal number of mistakes. Furthermore, removing the uncertainty with respect to the instances can significantly reduce the optimal number of mistakes. For example, for the concept class of halfspaces in the realizable setting. the optimal number of mistakes grows linearly with the time horizon 7 in the traditional online binary classification framework, while only growing as (log 7') in the transductive online binary classification framework. So, it is natural to reduce the optimal number of mistakes or extend learnable classes whenever we have additional assumptions. Notably, [Be--David et al. [1997] initially called this setting "offline learning", but it was later renamed "Transductive Online Learning" by Hanneke et al. [2023b due to its close resemblance to transductive PAC learning (Vapnik and Chervonenkis 1974 Vapnik 1982 1998). See Section A for more details.
While Ben-David et al. [1977 and Hanneke et al. (003b) mainly focused on binary classification, in this work, we focus on the more general multiclass classification setting. Natarajan and Tadepalli

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1988 Natarajan [989 and Daniely et al. [2012] initiated the study of multiclass prediction within the foundational PAC framework and tracitional online framework, respectively. More recently, following the work by [Brukhim et al. 2. there has been growing interest in understanding multiclass learning when the size the label space is unbounded, including [Hanneke includinggHanneke et al. [2023c|a]. Raman et al.] [2023]. This interest is driven by several motivations. Firstly, guarantees for the multiclass setting should not inherently depend on the number of labels, even when it is finite. Secondly, in mathematics, concepts involving infinities often provide cleaner insights. Thirdly, insights from this problem might also advance understanding of real-valued regression problems [Attias et al. 2023]. Finally, a practical front, many crucial machine learning tasks involve classification into extremely large label spaces. For instance, in image object recognition, the number of classes corresponds to the variety of recognizable objects, and in language models, the class count expands with the dictionary size. See Section A for more details.
1.2
Main Results and Techniques
In the following subsection, we present an overview of our main findings along with summary of our proof techniques.
1.2.1 Realizable Setting
In the realizable setting, we assume that the sequence of labeled instances (41, Vi)..... (IT, VT) played by
b the adversary, consistent with at least one concept in C. Here, our objective the well-known notion of the expected number of mistakes. We provide upper and lower bounds on the best achievable worst-case expected number of mistakes by the learner as a function of T and C, which we denote by M" MT,C)) (T,C).
minimize
Hanneke et al. [203bb] established trichotomy of rates in the case of binary classification. That is, for every C C (0, 1Fa we have that M* MT, (T.C) can only grow like ((T), O(log T), or (()); where the Littlestone and Sopnik-CChevvonnniie (VC) dimensions of C characterize the possible rate. In this work, we extend this trichotomy to the multiclass classification setting, even when y is unbounded. To do so, we introduce two new combinatorial parameters, termed the Level-constrained Littlestone dimension and Level-constrained Branching dimension.
To define the Level-constrained Littlestone dimension, we first need to define the Level-constrained Littlestone tree. A constraend Littlestone tree is a Littlestone tree with the additional re- quirement that the same instance has to label all the internal nodes across given level. Then, the Level-constrained Littlestone dimension is just the largest natural number d E N. such that there exists shattered Level-constrained Littlestone tree T of depth d. To define the Level-constrained Branching dimension, we first need to define the Level-constrained Branching tree. The Level-constrained Branching tree is Level-constrained Littlestone tree without the restriction that the labels the two outgoing edges are distinct. Then, the Level-cone devel-constrained Branching dimension is then the smallest natural number d such that for every shattered Level-constrained Branching tree T', there exists a path down T such that the number of nodes whose outgoing edges are labeled by different elements of y is at most d. The Level-constrained Littlestone dimension reduces to the VC dimension when Branching and Littlestone dimension
|D = 2. Additionally, the finiteness of the Level-constraing coincide when | 2. Finally, we note that the Level-constrained Branching dimension is exactly equal to the notion of rank in the work of Ben- Ben-David et al. [1997]. However, we believe it is simpler to understated. Using the Level-constrained Littlestone and Branching dimension, we establish the following trichotomy.
Theorem 1. (Trichotomy) Let C C y be concept class. Then, we have:
(()),
BCC 3c. (log if D(C) < co and BCC)= = DC.
M((,C) E
(TT),
if D(C) = xc.
Here, B(C) is Level-constrained Branching dimension, and D(C) is Level-constrained Littlesonne dimension defined in Section 2
To prove the O(log T) upper bound for binary online classification, Hanneke et al. 2023b run the Halving algorithm the projection of C onto 21, ..., ZT and use the Sauer-Shele P (SSP)
3

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lemma to bound the size of this projection by ((T)V) However, this approach is not applicable when Y/ is unbounded. For example, when C= + E N] is the of all constant functions over N, the size of the projection of C. onto even single x E X is infinity. Moreover, the mentioned class can be learned with at most one number of mistakes. Thus, fundamentally new techniques are required. To this end, we define a new notion of shattering which makes it possible to apply an analog of the Halving algorithm. Additionally, while the proof of the O(1) upper bound in Hanneke et al. [223bb follows immediately from the guarantee of Standard Optimal Algorithm (SOA) by Littlestone 1988), our O(1) upper bound in terms of the Level-constrained Branching dimension requires a modification of the SOA. We complement our results by presenting matching lower bounds. See Section 3 for more details.
In Section[F we provide comprehensive comparison between our dimensions and existing multiclass combinatorial complexity parameters.
1.2.2 Agnostic Setting
In the agnostic setting, we make no assumptions about the sequence (I1, 31), (I2, y2)..... (2T, yr played by the adversary. Here, our focus shifts to the well-established notion expected regret, which compares the expected number of mistakes made by the algorithm to that made by the best concept in the concept class over the sequence. As in the realizable setting, we aim establish both upper and lower bounds on the optimal worst-case expected regret achievable by the learner, expressed as a function of T' and the concept class C, denoted by R (T,C)..
The prior work by Hanneke et al. 2023b) showed that in the case of binary classification, (T CT) is Ô T) whenever VC(C) oo and ((T) otherwise, where Ô hides logarithmic factors in T. Using the Level-constrained Littlestone dimension in hand, we extend these results to multiclass classification.
Theorem 2. For every concept class C yxx and T 2 D(C), we have the following:
T D(C) R'(T.C) TT log ET
(D(C):
8
where D(C) is Level-construined Littlestone dimension defined in Section 2
Our results in the agnostic setting be proved using core ideas in the proof of the agnostic results
from BBen-aavid et al. (2009. Hanneke et al. 2023a]. and Hanneke et al. 2023b) See Section F for
more details.
2 Preliminaries
2.1 Notation
Let X' denote an example space and Y' denote the label space. We make no assumptions about ), so can be unbounded and even uncountable (e.g. y = IR. Following the work of Hanneke let al. [2023a], if consider randomized learning algorithms, the associated -lalgebra is of little consequence, except that singleton sets {y} should be measurable. Let C C y denote concept class. We abbreviate sequence 21, .... ZT by 21:T. Moreover, we also define 2<t and žςイ = (21...., 2t). Finally for 7 E N, we let [n] == [I....,
2.2 Transductive Online Classification
In the transductive online classification setting. learner A plays repeated game against adver- sary over rounds. Before the game begins, the adversary picks a sequence of labeled instances (1),8)),..., (xT, yr) (X' x y))² and reveals æ1.7 to the learner. Then, in each round E (T), using d1.r and 1/ the learner makes a potentially randomized prediction A(æa) E y. Finally, the adversary reveals the true label yt, and the learner suffers the loss (AA(;)) # ye ). Given concept class C C y the goal of the learner is to output predictions such that its expected regret,
T
T
(T(TC) ==
sup (211 I.... (FT.
E
Σ
1{A(xt) + vm)
inf CEC
(c(x,)) + wr
t=1

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is small. Moreover, we define R (T, := inf RA(T,C) where the infimum is taken over all transductive online algorithms. We say that concept class is transductive online learnable in the agnostic setting if R (T, o(T)
If the learner is guaranteed to observe sequence of examples laheled by some concept C, then we say we are in the realizable setting, and the goal of the learner is to minimize its expected cumulative mistakes
MA(T,C) :- sup sup E cEC F1.7
Σ [(A())) + c(xx))]
Similarly, define (T,CC) A MT(T,C), and analogous definition of transductive online learnability in the realizable setting holds.
2.3 Combinatorial Dimensions
Combinatorial dimensions play an important role in providing a tight quantitative characterization of learnability in learning theory. In this section, we review existing combinatorial dimension in online classification and propose two new dimensions that help us establish the minimax rates for transductive online classification. We start by defining the Littlestone dimension which characterizes multiclass online learnability.
Definition 1 (Littlestone dimension). The Littlestone dimension of denoted L(C), the largest d N such that there exists sequences of functions {X+) where Xt {0, [0,1)'-1 X and (Yi)a where Y, {0, 1}¹ y such that for every E {0, 1), the following holds:
(i) Yı((ozı;0) # Yi((oe,, 1)) for all [d].
(ii) 3cg E C such that co (X)(o=z)) = Yi(oss)) all t E [d].
If for every d N, there exists sequences {Xe] and (Yi)e satisfying (i) and (ii), we let L(C) = co.
On the other hand, in this paper, we show that different dimension, termed the Level-constrained Littlestone dimension, characterizes transductive online classification.
Definition (Level-constrained Littlestone dimension). The Level-constrained Littlestone dimension Cf denoted D(C), is the largest d IN such that there exists sequence of instances I Fi, :. Fa E x and a sequence of functions (Yr) where Y₁ 10, V, such that for every {0,1)4 the following holds:
(i) Yi((ocr,)) # Yi(lo 1)) for all e [d].
(ii) 3eg E such that co (ax) Y((ag)) for all all e [d].
If for every E N, there exist sequences { and [Y,)²= satisfying (i) and (ii), we let D(C)
The Littlestone and Level-constrained Littlestone dimensions can also be defined in terms of complete binary trees. Littlestone tree 7 of depth d is complete binary tree of depth d where the internal nodes are labeled by elements X' and for every internal node, its two outgoing edges are labeled by distinct elements in y. Such
is shattered by if for every root-to-leaf path or E {0, 1)d there exists concept co E C' consistent with the sequence of instance-label pairs obtained by traversing down T along a. The Littlestone dimension is then the largest shattered Littlestone tree of depth d. From this perspective, the functions {X, and {Y/}_ in Definition Analogously, Level-constraince Littlestone tree is simply Littlestone tree with the additional requirement that the instances labeling the internal nodes are same across each level. Definition 2. labels all the internal nodes on level one, labels all the internal nodes on level two, and so forth. The functions {Y:]{] provide the labels on the outgoing edges of a Level-constrained Littlestone tree. Then, the Level-constrained Littlestone dimension is the largest d N for which
for which there exists
! provide the labels on the internal nodes and the outgoing edges of T respectively.
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there exists shattered covss-contt Littlestone tree of depth d. We will use the dinction-based and tree-based definitions of these dimensions interchangeably.
Moreover, we show that the Level-constrained Branching dimension characterizes when constant minimax rates possible in transductive online classification.
Definition constalcon Branching dimension). The of C, denoted B(C),, is the smallest E N such that for every d E N, every sequence of instances U1, Zd E 2X, and every sequence of functions {Ye]== where Y; {0, + yy
Branching dimension
Vo E [0,1]" Eco E such that co (zz) Yi(og)) for all E [d]
arg min Σ (E(0.1) t=1
1 0)) + Ye((o<e sp.
If no such E N exist, we let B(C)- = oc.
For given path a E {0, 1]d, we refer to Ei= ((Ya((o<s,0) 0)) + Yi(lo 1))} as the branching factor of the path. In terms of trees, Level-constrainee Branching tree is a Level-constrained Littlestone tree without the restriction that the labels on the outgoing edges of any internal node need to be distinet. Given path in such tree, the branching factor of a path counts the number of nodes in the path whose two outgoing edges are labeled by distinct labels in V. Finally, the Level-constrained Branching dimension can be equivalently defined as the smallest p N such that every shattered Level-constrained Branching tree T of depth E N must have at least one path with branching factor most p.
The following proposition, whose proof is in Appendix B establishes the relationship between the three dimensions.
Proposition For every C S y we have that D(C) s B(C) < L(C).
We also compare our dimensions to other existing dimensions in multiclass learning in Section F
3 A Trichotomy in the Realizable Setting
We start by establishing upper and lower bounds on the minimax expected number of mistakes in the realizable setting in terms of the cevet-constrained Littlestone dimension and the Level-constrained Branching dimension.
Theorem (Mistake bound). For every concept class C S y we have
min max D(C), [logT] -[B(C) oe1) < (T < min BBC),, D(C) log
el D(C)
:T.
2
One can trivially upper bound M C) by L(C). However, by Proposition I our upper bound in terms of B(C) is sharper. We can also infer from the proof in Section 3.3 that when T is large enough (namely T >> 2B(C)), the lower bound in the realizable setting is also BiC)
Given Theorem 3 we immediately infer a trichotomy in minimax rates.
Corollary (Trichotomy). For every concept class C C y*, we have
((),
BCC < oc. DC) < oo and B(C) = xc.
MT
O(logT)
O(T),
if DCC) co.
Proof. (of Corollary When BCC) < 0o, Theorem gives that 1 D(C) & MT, S B(C) for T 2 D(C). When B(C) but DCC) < co, Theorem gives that 2 [log T M' (T,C) S D(C) log
cT for [log T] > D(C). Finally, when D(C) co, Theorem gives that T D(C
(T ST
The remainder of this Section dedicated to proving Theorem The proof of the lowerbound DCC)/2 /2 follows from standard techniques, defer it to Appendix

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3.1 Proof of Upperbound B(C)
Proof. Fix n E N. a sequence of instances Fim := (21,..., In)) X", sequence of functions
Y1.1- (Yi..... Yn) such that Y2 {0, 1)"h + V. and set of concepts V S C. If Vo E [0.1]".
there exists Co E V such that co (Zx) arg (o.1"" Et= ((V(( 0)) * Vi((T<r,, 1)). Otherwise, define B(V, Ni:,, Vim) == 0. Recall that we can represent Tim and Yi with level-constrained trees T of depth n. With the tree representation. B(V F1mm Ym == 0 if V does not shatter T. If T is shattered by V. then B(,, Y1s)) is the minimum branching factor across all the root-to-leaf paths in T. Recall that the branching factor of a path is the number of nodes in this path whose left and right outgoing edges are labeled by two distinct elements of V/ In this proof, we work with an instance-dependen complexity measure of V defined as
Timm: (((ss)) for all t E [n], then define B(V, Y1.) :=
B(V, Fi:n) = sup B(V,T1 V1.)).
YI..
We now define a learning algorithm that obtains the claimed mistake bound of B(C). Fix a time horizon 7 E N, and let I1. = (I), .2,..., TT) denote the sequence of instances revealed by the adversary. Initialize V1 :- C. For every & {l,..., T}. if we have {c(œx) then predict Dt y. Otherwise, define V/2 = {c E V c(x,) = y) for all y E Y and predict Jt = arg max, yey B(V x1+1:T). Finally, the learner receives a feedback yr E D, and updates Vi+1 < V When = T, the sequence 2T41.T is null and we define ÿr = arg max B(V)). For this learning algorithm, we prove that
V/} {v}.
B(Ve+1, T(41)) N B(V,, Ter) - {y + de
(I)
Rearranging and summing over E [T] rounds, we obtain:
T
T
Σ Live + 0.) S
Σ
(( - B(KE)))441)) B((i, - B(VTH),
2=1
S 2B((, S BCC)
The equality above follows because the sum telescopes. The final inequality follows be- cause V C and the level-constrained branching dimension of C is defined as B(C) = supres suP B1C,
We now prove inequality 11. There are two cases to consider: (a) y2 On and (b) yt + 9.- Starting with (a), let y i.- Recall that B(V++1, I++1:7) B(V," F-+1:7)- Since c(x2) yt for all h E V. we must have B(V"". .I2+1TT) S B(V," .T.T). Finally, using the fact that V. C V,, we have B(V"", FT.T) I B(V, F.T)). This establishes (I) for this case.
Moving to (b), let ye * dt. Note that we must have B(V, > 0. Otherwise, B(V, FeT) = O, then we have {c(x2) c Vi}= {y}. So, by our prediction rule, cannot have yr # in under realizability. To establish ( we want to show that B(V+1, xt+1::)) < B(V, TT.T). Suppose, for the sake of contradiction, d dd=B(C
BVV, ZTrr). Then, let us define If dd= O, then our proof is complete because < B(V,, (tr) - 1 (3++ + 9.)- Assume that d > O and recall that Vz+1 V By definition of B(V 32+1:1) and its equivalent shattered-trees representation, there exists a level-constrained tree Ty, of depth T - whose internal nodes are labeled by Tt,..... 27 and is shattered by V.t Moreover, every path down Ty, has branching factor 2 d.
instead have B(V I(+11))
Next, as h.- = arg max B(V", 11+1r)), we further have B(V #(+1:)) 2 B(V 2141:) 2 d. Thus, there exists another lenst-cont tree Ty, of depth T whose internal nodes are labeled by st,..., 2r,, that is shattered by ViS, and every path down Ty. has branching factor 2 d. Finally, consider new tree T with root-node labeled by 24, the left-outgoing edge from the root node is labeled by St. and the right outgoing edge is labeled by ÿ.. Moreover, the subtree following the outgoing edge labeled by Ye is Ty. and the subtree following the outgoing edge labeled by d: is Ty. Since both Ty. and Ty. are valid level-constrained trees each with internal nodes labeled by .t-1...., Fr, the newly constructed tree T is also level-constrained trees of depth T - -t + 1 with internal nodes labeled by ,t,..., .T- In addition, as Ty, and Ty, are shattered by V." and v respectively, the tree T must be shattered by V U V. Finally, every path down each
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sub-trees Ty. and Tg. has branching factor 2 d and * Dr. every path of T must have branching factor 2 d 1. This shows that B( V. I.TT 2 d + And since V V S V., we have d + < B(V UV FT.T) S B(V, (er) by monotonicity. This contradicts our assumption that d= B(V++1 ((+ B(V,, (..)). Therefore, we must have
< B(V, F.T)).
This establishes ( completing our proof.
3.2 Proof of Upperbound D(C) log
TT
D(C)
Proof. Fix the time horizon 7' E NN and let Z1. == (x1,..,, .T) E XT be the sequence of T instances revealed to the learner at the beginning of the game. We say subsequence 2/ == ( z',). preserving the same order as in #1.7, is shattered by V C C if there exists sequence of functions {Y+)==I, where Y; {0, 1] + V, such that for every a E (0, 1)", we have that
(i) Yi(oce, 0) & (((o e 1) for all E [n].
(ii 3e, E V such that (x²) (o))))) for all & [n]]
For every V C. let S(V) be the number of subsequences of æ1.7 shattered by V. In addition, for every (z, v) E X' x D. let V(a,)) := {c V c(x) = y}. Consider the following online learner. At the beginning of the game, the learner initializes V = C. Then, in every round E [T], the learner predicts Ot E arg max (x.a))) receives y/ y and updates y*+1 Vt. (F(a)))
For this learning algorithm, we claim that
S(V++) max{ Li úch s
for every round E [T]. This implies the stated mistake bound since S(C) I DD(C) S D(C)
and the learner can make at most log(S(C)) mistakes before S(V*) 1. We now prove this claim by considering the case where &t Yt and ÿt # Y1 separately.
D(C)
Let & E [T] be round where ý = ye- Then, 1(V++1) s S(V*) since V (2r..))) S V*. Now, let t e [T] be round where de yı- We need to show that S(V < S(V*). For any V S C.
let Sh(V (V) be the set of all subsequences of æ1.7 that are shattered by V. Then, for any subsequence
q E Sh(V*), only one of the following properties must be true:
(1) g $ Sh(Vi, (I.))) and & (h(V(e..))))
(2) g Sh(Vi (() ) A ())..)))
(3) q Sh(V) (o)))) Sh(V ((.))))))))
where A denotes the symmetric difference. For every i E (1,2,3), let Sh' c Sh(V*) be the subset of Sh(V') that satisfies property (i). Note that Sh(V") = ,
Sh (V') and (Sh'(V')
are pairwise disjoint. Therefore, (Sh' [2 forms partition of Sh(V*). For each E {1, 3}. we compute how many elements Sh ( (Vt) we drop when going from Sh(V)) to Sh(V++1). We can S(V++1) by lower bounding [Sh(V") Sh(V++1)) }], the number of elements drop across all of the subsets (Sh' (V")) when going from V to V++1
then upperbound |Sh(V
Starting with i = 1, observe that for every q E Sh (V*), we have that & Sh(V Sh(V), (2..))) Therefore, Sh (V') n Sh(Vi (((()))))))) )1 = 0, implying that we drop all the elements from Sh (V*) when going from Sh(V') Sh(V++1).
For the case where 2, note that Sh(Vt (z. (e)...)): Sh (2). S Sh(V") and s(v >
S(Vt ())))) where the latter inequality is true by the definition of the prediction rule. Moreover, using
8

---

the fact that {Sh' (2)) forms partition of Sh(V),, we can write
())))))))) (Sh | [ (S (c..
and
S(V |Sh (v') n nSh(V (a.
Since S(V ((()))))) 2 (()))))) we get that |Sh (V*) Sh(V (a)))))))))
[Sh?(v) n
Sh(V%, (o))))))
M1 This implies that (S*) n Sh(V (m..))) I (1Sh((()) since
(Sh?(Ve)
Sh(V (ar 0i)
) (V 20)))) Sh Sh
(V*) and (Sh)(V')) (Vt) S (m.. (())
Shh (V*))
Sh(V (o 0. Thus, we drop at least half the elements from Sh (Vt) when going from (Vt)
Sh(V++1).
Finally, consider when i = 3. Fix E Sh We claim that I1,..., é This is because every t- In addition, 2, & q because q E Sh(V .o) (nSh(V (o.)))) and every concept in V(e....) and V( (i...S.)) outputs y and jt on respectively. Thus, the sequence T, q, obtained by concatenating I2 to the front of g. is valid subsequence of Ti.7- Since Di # yt. we also have that ze o is shattered by V Using the fact that x, o q & Sh(V(r....) (r...e)) and ry o q & Sh(Viin.su)) gives that 2, og E Sh (V)). Since our choice of q was arbitrary, this implies that for every q E Sh² (V)), there exists subsequence d' 22 q Sh (V*), ultimately giving that (V")(V 2 (V)))
c E V² outputs 5/ on z, for all j S
To complete the proof, we lowerbound the total number of dropped elements when going from Sh(V')) to Sh(V by
Sh (V*)(V
SS((V') \ (Sh(V( (or))))) 2 Sh (V*) +
2
Sh (Vo)
|Sh\(V9)
2
2
|Sh(V))
S(V))
2
2
The number of remaining elements is then S(V(-1) = SS (2. [Sh(V") (Sh(V*)
Sh(V (w. (R. S 1/2 S(V'), as needed.
We end this section by noting that the algorithm in the proof of Theorem 3 can be made conservative (i.e. does not update when is correct) with the same mistake bound. This conservative-version of the realizable learner will be used when proving regret bounds in the agnostic setting (see Section E.
3.3 Proof of Lowerbound log
IB(C) = oo]
If B(C)= oc, then for every g E N. Definition 3 ggaarantees the existence of d N, sequence of instances Tı,-.. Ttd, and a sequence of functions Yi...., Ya where Y; (0, 13 y such that the following holds: (i) Yo E 10, there exists Go E C such that (xx) = () (ii) Vo E {0, 1]d, we have Σde ((Yi((o 0)) A Yi((o 1))} 2 q. Equivalently, there exists a shattered level- constrained branching tree T of depth with internal nodes labeled by instances Zi..... za such that every path down the tree T has 2 q branching factor. Recall that the branching factor of path is the number of nodes in the path whose two outgoing edges are labeled by two distinct elements of Y. We say that an internal node has branching if the left and right outgoing edges from the node are labeled by two distinct elements of y
Without loss of generality, we will assume that the T guaranteed by Definition 3aas the following properties: (a) every path in T has exactly q branching factor and (b) T is symmetric along its non-branching nodes- that is, for every node in T that has no branching, the subtrees on its left and right outgoing edges are identical. There no loss generality because given without property (a), we can traverse down every path in T, and once the path has branching factor g, label all the

---

subsequent outgoing edges down the path by a concept in C. that shatters any completion of that path. For property (b), any non-branching node has two different subtrees, then replace the right subtree with the left subtree. Given such tree T, let BN(T) denote the number of levels in T with at least one branching node. 'he following Lemma, whose proof can be found in Appendix D provides an upperbound on BN(T).
Lemma 1. Let T be any level-constranee branching tree shattered by C such that: (a) every path in has exactly q E N branching and (b) for every node T without branching, the subtrees on its left and right outgoing edges are identical. Then, BN(T) S 29 -1.
xo|. Assume B(C) = do and take g = [log | Definition Lugaaatttenngaaeneeeteeee the existence of shattered Level-constrained branching tree T that satisfies property (a) and (b) specified in Lemma T Next, Lemma I implies that (N(()
Given this Lemma, we now prove the claimed lowerbound of [o 1 [
1 - T' 1. Let d be the depth T and S C {1 ...., d3 be the levels in T with at least one branching node. By definition, we have |S| (N(T) < T 1. Recall that T can be identified by sequence of instances 21...... 2d and sequence of functions Vi...., where Y, {0, 1 + for every E [d]. For any path gives the labels along this path. Moreover, as all the branching on T occurs on levels in S, we have Ei= 1
2|log
{0, 1 down T, the set {
(( 0)) * Yi((a 1))} = Eles ((V(((<<r, 0)) + Yi((oe,, = [log T for every path a E 13
We now specify the stream to be observed by the learner A. Draw a Uniform( (0, 13 and consider the stream {xx, (((o))))ee Repeat (Zm, Y (m)g)) for remaining T - [S] timepoints, where m is the largest index in set Since this stream is a sequence of instance-label pairs along the path in the shattered tree T. there exists co E C consistent with the stream. However, using similar arguments as in the proof of the lowerbound D(C)/2, we can establish
E
1 (A(x)) # vi((g=))
E
Σ
((A(x)) # +Y(oss))
LES
E
Σ
((Y((((css))) 0)) + (r((ocn, 1))}
2
LIES
1
log T|.
2
This completes our proof of lower bound.
4 Discussion
In this paper, we study the problem of multiclass transductive online learning with possibly arbitrary label space. In the realizable setting, we establish a trichotomy in the possible minimax rates of the expected number of mistakes. Furthermore, we show near-tight upper and lower bounds on the optimal expected regret in the agnostic setting. Along the way, we introduce two new combinatorial complexity parameters, called the Level-constrained Littlestone dimension and the Level-constrained Branching dimension.
Finally, we highlight some future directions of this work. First, can we extend our results settings such as transductive online learning under bandit feedback, list transductive online learning, and transductive online real-valued regression? Moreover, use similar ideas to establish the possible minimax rates of the number of mistakes in the self-directed and the best-order settings initially studied in [Ben-Davis al.
shattering technique is general,
1995 1997
Acknowledgments and Disclosure of Funding
VR acknowledges support from the NSF Graduate Research Fellowship Program.
10

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References
Scott Aaronson, Xinyi Chen, Elad Hazan, Satyen Kale, and Ashwin Nayak. Online learning of quantum states. Advances in neural information processing systems, 31, 2018.
Noga Alon, Roi Livni, Maryanthe Malliaris, and Shay Moran. Private pac learning implies finite littlestone dimension. In Proceedings of the st Annual ACM SIGACT Symposium on Theory of Computing, pages 852-860, 2019.
Noga Alon, Mark Bun, Roi Livni, Maryanthe Malliaris, and Shay Moran. Private and online learnability are equivalent. ACM Journal of the ACM (JACM), 6-44),134, 2022.
Idan Attias, Steve Hanneke, Alkis Kalavasis, Amin Karbasi, and Grigoris Velegkas. Optimal learners for realizable regression: Pac learning and online learning. arXiv preprint WxXiv:2307.03845 2023,
Shai Ben-David and Nadav Eiron. Self-directed learning and its relation to the ve-dimension and to teacher-directed learning. Machine Learning, 33:87-104, 1998.
Shai Ben-David, Nicolò Cesa-Bianchi, and Philip M. Long. Characterizations of learnability for classes of O,..., n-valued functions, 1992.
Shai Ben-BenDDDaa Ben-David, Nadav Eiron, and Eyal Kushilevitz. On self-directed learning. In Proceedings of the eighth annual conference on Computational learning theory, pages 133, 1995.
Shai Ben-David, Eyal Kushilevitz, and Yishay Mansour. Online learning versus offline learning. Machine Learning, 29:45-63, 1997.
Shai Ben-David, Dávid Pál, and Shai Shalev-Shwartz. Agnostic online learning, In Conference Learning Theory, volume 3, page 1, 2009.
Olivier Bousquet, Steve Hanneke, Shay Moran, Ramon Van Handel, and Amir Yehudayoff. A theory of universal learning. In Proceedings of the 53rd Annual ACM SIGACT Symposium Theory of Computing, pages 532-541, 2021.
Nataly Brukhim, Elad Hazan, Shay Moran, Indraneel Mukherjee, and Robert E Schapire. Multiclass boosting and the cost of weak learning. Advances in Neural Information Processing Systems, 34: 3057-3067, 2021.
Nataly Brukhim, Daniel Carmon, Irit Dinur, Shay Moran, and Amir Yehudayoff. A characterization of multiclass learnability, 2022.
Mark Bun, Roi Livni, and Shay Moran. An equivalence between private classification and online prediction. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 389-402. IEEE, 2020.
Amit Daniely and Tom Helbertal. The price of bandit information in multiclass online classification. In Conference on Learning Theory, pages 93-104. PMLR, 2013.
Amit Daniely and Shai Shalev-Shwartz. Optimal learners for multiclass problems. In Conference on Learning Theory, pages 287-316. PMLR, 2014.
Amit Daniely, Sivan Sabato, Shai Ben-David, and Shai Shalev-Shwartz. Multiclass learnability and the erm principle. In Proceedings of the 24th Annual Conference on Learning Theory, pages 207-232. JMLR Workshop and Conference Proceedings, 2011.
Amit Daniely, Sivan Sabato, and Shai Shwartz. Multiclass learning approaches: A theoretical comparison with implications. Advances in Neural Information Processing Systems, 25, 2012.
Ofir David, Shay Moran, and Amir Yehudayoff. Supervised learning through the lens of compression. Advances in Neural Information Processing Systems, 29, 2016.
Pramith Devulapalli and Steve Hanneke. The dimension of self-directed learning, 2024.
Jesse Geneson. A note on the price of bandit feedhack for mistake-bounded online learning. Theoret- ical Computer Science, 874:42-45, 2021.

---

Sally A Goldman and Robert H Sloan. The power of self-directed learning, Machine Learning, 14: 271-294, 1994.
Steve Hanneke and Liu Yang. Bandit learnability can be undecidable. The Thirty Sixth Annual Conference Leurning Theory, pages 5813-5849, PMLR, 2023,
Steve Hanneke, Shay Moran, Vinod Raman, Unique Subedi, and Ambuj Tewari. Multiclass online learning and uniform convergence. In The Thirty Sixth Annual Conference on Learning Theory, pages 6696-5696. PMLR, 2023a.
Steve Hanneke, Shay Moran, and Jonathan Shafer. trichotomy for transductive online learning. Advances in Neural Information Processing Systems, 36, 2023b.
Steve Hanneke, Shay Moran, and Qian Zhang. Universal rates for multiclass learning. In The Thirty Sixth Annual Conference on Learning Theory, pages 5615--681. PMLR, 2023c.
David Haussler and Philip M Long. generalization of sauer's lemma. Journal of Combinatorial Theory, Series A, 71(2):219-22) 240, 1995.
Wilfrid Hodges. A shorter model theory. Cambridge university press, 1997.
Nick Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine learning, 2:285-318, 1988.
Philip M Long. New bounds on the price of bandit feedback for learning. In International Conference on Algorithmic Learning Theory, pages 3-10. PMLR, 2017.
online multiclass
Preetham Mohan and Ambuj Tewari. Quantum learning theory beyond batch binary classification. arXiv preprint arXiv:330..07409 2023.
Balas K Natarajan. On learning and functions. Machine Learning, 4:67-97, 1989.
Balas K Natarajan and Prasad Tadepalli. Two frameworks for learning. In Machine Learning Proceedings 1988, pages 402-415. Elsevier, 1988.
Ananth Raman, Vinod Raman, Unique Subedi, and Ambuj Tewari. Multiclass online learnability under bandit feedback, 2023.
Benjamin Rubinstein, Peter Bartlett, and Rubinstein. Shifting, one-inclusion mistake bounds and tight multiclass expected risk bounds. Advances in Neural Information Processing Systems, 19, 2006.
Saharon Shelah. Classification theory: and the number of non-isomorphic models. Elsevier, 1990.
Vladimir Vapnik and Alexey Chervonenkis. Theory pattern recognition, 1974.
Vladimir N. Vapnik. Estimation of Dependencies Based on Empirical Data. Springer-Verlag, New York, NY, 1982.
Vladimir N. Vapnik. Statistical Learning Theory. Wiley, 1998.
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A Related Work
Online Learning. Online learning has been a subject of study for more than half century. Moreover, the seminal work by [ittlestton [988]] initiated this line of research within the computer science community. Since that pivotal contribution, online learning has been explored in various settings, from learning under the bandit feedback Daniely et al. [201T], Daniely and Helbertal (2013), Long
[2077]. Geneson [202T]. |aaaan et al. [2023]. Hanneke and Yang [2023] to quantum settings Aaronson et al. 2018J. Mohan and Tewari [2023]. Furthermore, it is also linked to a broad set of
problems, such as differential privacy, highlighted in studies by |Alon et al. (2019), BBun et al. (2020),
Alon et al. 2002). Further, given its fundamental nature, it is not surprising that online learning has
found numerous practical applications.
Transductive and other Online Learning Frameworks. The concept of the transductive learning
model traces its origins to seminal works by Vapnik Vapnik/Vapnik and Chervonenkis) (1974), Vapnik] [974),Vapnikk [1982]
1998]. where it was explored within the PAC learning framework. Subsequently, [Bnn-Dvid et al.
1997] initiated the study of this model under the umbrella of online learning, referring to it as "offline learning". They utilizes notion of rank based dimension to prove their results. Notably, as we mentioned in the introduction, our Level-constrained Branching dimension is exactly equal to their rank based dimension. A significant advancement came recently with the work of Hanneke ct al. [2023b]. Furthermore. other conceptually related models have also been rigorously studied, as seen in works by Goldman and Sloan [1994], Bev--David et al. 1995], Ben- and Eiron 1998). Devulapalli and Hanneke| (2024). These studies notably include the self-directed online learning framework, which allows the learning algorithm select the next instance for prediction from the remaining set of instances in each round, and additionally, the best order, which allows the learner
(instead of adversary) select the order at the beginning of the game.
Multiclass Classification. A substantial volume of theoretical research has been conducted on various aspects of multiclass classification, as demonstrated by studies Natarajan and Tadepalli
1988 Natarajan [1989]. Ben- David et al. [1992]. Haussler and Long [195]] Rubinstein et al.
2012]. Despite this extensive body of work, combinatorial characterization of multiclass classification
2006 Daniely et al. [011 2012). Daniely and Shalev-Shwartz/ (2014), Brukhim et al. [2021].
with an infinite number of classes under Valiant's s PAC learning framework in the realizable setting remained open until recently. In pursuit, the seminal paper by |Brukhim et al. [2022] provided combinatorial characterization in the mentioned setting. A key innovation in this breakthrough was the utilization of list learners. This dimension also serves to characterize the agnostic variant of this problem David et al.] [2066]. For standard multiclass online learning with potentially unbounded label Daaielyy al. [[011] presented characterization for the realizable setting. Building on this, Hanneke et al. [2023a] extended Aien-David et al. (2009) technique to the agnostic setting with infinite label space. Notably, similar trend can also be observed the online learning under bandit
feedback in the work of Daniely et al. 2011] followed by Raman et al. 2023
B Proof of Proposition 1
Let C C y be any concept class. To see that D(C) < B(C), note that if D(C) = d, there exists a Level-constrained Littlestone tree T of depth d with branching factor d. Thus, it must be the case that (CC) 2 d.
To prove that B(C) < L(C), it suffices to show that for every shattered Level-constrainee Branching tree T with branching factor n E IN there exists a shattered Littlestone tree of depth n. In particular, we will prove via induction the following claim: if T is a Level-constrained Branching tree with branching factor n shattered by some C' C C, then there exists a Littlestone tree T of depth n shattered by C².
For the base case let T be a Level-constrained Branching tree with branching factor shattered by some C' C C. Without loss of generality (see Lemma [ suppose that branching occurs on the root node of T. Then, it is clear that just the root node 7 along with its two outgoing edges is Littlestone tree of depth 1 shattered by C².
13

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Now for the induction step, suppose the induction hypothesis is true for some S B(C) 1. Let be Level-constrained Branching tree with branching factor n - shattered by C' C C. Again. without loss of generality, suppose branching occurs the root node of T. Let To and Ti be the left and right subtrees of T respectively shattered by C. C' and C₁ C C' respectively. Then, note that To and Ti must both have branching factor exactly Then, by the induction hypothesis, there exist Littlestone trees Ty and TI' of depth n shattered by Cj and C₁ respectively. Since branching occurs at the root node, the tree T' obtained by keeping the root node and its two outgoing edges of T, but replacing To and Ti with To and 7I respectively, is a Littlestone tree depth n + shattered by C U.1 C
C Proof of Lowerbound D(C) 2
Proof. Fix a transductive online learner A. We will construct randomized, hard realizable stream such that the expected number of mistakes made by AA is at least D(C) Using the probabilistic method then gives the stated lowerbound.
2
Let == D(C). Then, by Definition 2 there exists a sequence of instances 21, -.: Ea E X and
sequence of functions {Yi)= where Y, [0, y such that for every o E {0, 13dd the following holds:
i) ca, 0) Yi(o for all t [d].
(ii) 3cr E C such that co (xx) (a(s))) for all & E [d].
Let σ ~ f0, 1] a denote bitstring of length d sampled uniformly at random and consider the stream (I1, Y1 (o (za, Ya(o<a)). By the definition, there exists concept co E C such that co (1x) = ((() for all / E [d]. Moreover, observe that
E
1 (A(x)) + Y((osı))
= 1=1
E ((())) + Y((ss))]
Σ
E E 1 {AA (3:) # Vo((<<))))))) [ox]]
R (Yi((o<rs,) # ((((e11))) Oct
E
Σ 1 [Yi(la<s, 0)) # ( 1))} =
where the first inequality follows from the fact that o; ~ (niform((0,11)) This completes the proof.
D Proof of Lemma 1
Proof. (of Lemma] If depth(T) S 29 - 1, the claim holds trivially. So, we assume depth
29 1. We now proceed by induction on q. For the base case q = 1, T denote level-constrained tree of depth(T)= > 1 shattered by C that satisfies property (a) and (b) specified in Lemma | First, consider the case where the root node of T has branching- Since every path T can have exactly 1 branching, there can be no further branching in T. Next, consider the case when the root node of T not the branching node and is the first level in T with branching. There must be 2 nodes in this level, henceforth denoted by (v: )²- Moreover, denote Te, to be the corresponding subtree in T with vi the root node. Since T satisfy property (b) and there are branching nodes before level f, the subtrees (Tn, 1²=1 must be identical. Since all subtrees {T,, 1 have branching on the root node, there can be no further branching in these subtrees beyond the root node. Therefore, there
cannot be any other levels l' > in T with branching node. This establishes that E the only level in
T with at least one branching node. In either case, we have BN(T) 1 29 1.
14

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Assume that Lemma is true for some To that end, let 7 be level-constrained tree with branching factor q = n + shattered by that satisfies (a) and (b). Let 8 2 1 be the first level in T with at least one branching node, and {T: /=1 be all the subtrees with its root node being a node on level É. As argued the base case, all these subtrees must be identical. Thus, branching occurs on the same set of levels on all these subtrees, which implies that (T) = (T.) for subtree following the two outgoing edges from the root node of Ti. Since, there is branching on the root-node of TI we must have (N - 1+ (T) + BN (N, ). For each i E {0, 1]. the subtree Ti is level-constrained tree shattered by C that satisfies properties (a) and (b) for q
E N. We now establish Lemma for = + 1.
2-
[2 Let T and T, denote the left and right
Using the
inductive concept, we have (T)) I 2" for i {0, 13. Therefore, combining everything
BN( = (T & 1 ++ (N( + BN( 11+2" - 1+ 2" - = 2n+ -1.
This completes our induction step.
E Minimax Rates in the Agnostic Setting
We go beyond the realizable setting, and establish the minimax regret in the agnostic setting in terms of the Level-constrained Littlestone dimension.
Theorem 4 (Regret bound). For every concept class C C yi and T D(C), we have
T D(C) < DCC) log
eT D(C))
We note that the upper- and lower-bounds in Theorem 4
are only off only by factor logarithmic in T. We leave it as open question to establish matching upper- and lower-bounds.
Proof. (of upper bound in Theorem 4 To prove the upper bound, we will use the agnostic-to- realizable reduction from Hanneke et al. [023a to convert our realizable learner in Section 33.2 to agnostic learner with the claimed upper bound expected regret. By Theorem in (Hanneke et al. 2023a any conservative deterministic learner A with mistake bound M can be converted into
an agnostic learner with expected regret at most
T log
c7 M
Although the proof by Hanneke
ct al. [023a only coverts the conservative Standard Optimal Algorithm to agnostic learner, the arguments are general enough such that the conversion can be adapted for any conservative deterministic mistake-bound learner. By Theorem 3 and the proof in Section 3.2, there exists
conservative deterministic realizable learner with mistake bound at most (CC) log
eT Using the
D(C)
velizable-to-agnostiie conversion from Theorem in [Hanneke et al. 2023a] with the conservative- version of the realizable learner in Section 3.2 gives an agnostic learner with expected regret at most
T D(C) log
eT ((C)
log
eT
cT S DCC) log D(C))
)(C)
D(C) log
cT DIeT
completing the proof.
Proof. (of lower bound in Theorem 4) Our proof of lower bound is identical to the lower bound for the binary setting proved in (Hanneke et al. 2023b Theorem 6.1], which is just simple adaptation of standard lower bound technique from Ben-David et al. 2009). Thus, we only outline the sketch of the proof here.
Let = DCC. Consider a sequence of instances {øt.... #t} C. X and a sequence of functions 1d { that is shattered by C according to Definition 2 Pick the largest odd number & E N such that kd T. First, the adversary reveals the instances {01 .... xT) such that = x for = 1, k. followed by x, = 2 for = + I,... , 2k, and so forth. If kd, take x, = x; for all > kd. Uniform([O, 1)²). Then, for
As for labels, the adversary will first sample (0), .2. .T) .., 2k, the labels are selected as t = I,..., k the labels are selected yı (i())). For t = k + 1,...
15

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yr (i),, where o1 = { 1|o, =0 =0 E = is the majority bit in the first block / = 1, k. One can define Y/: for all <>> 2k analogously. For this stream, the label yr is essentially equivalent to the bit o, E {0, 1}. Therefore, following the exact same arguments as in [Hanneke et al. 2023b Theorem 6.1] establishes the lower bound of VTd/8. This completes the sketch of our proot.
Remark 1. Let & E N be the number of classes. Lei C C (1.2..... .k} be concept class. It is notable that for small number of classes k. (i.e. k << 2(logTy*), the Natarajan bound that can be proved using the technique of Hanneke el al. (2023b) can be smaller than the upper bound in terms of the Level-constrained Littlestone dimension. However, for large ls (i.e. k >> 20ooyT ), our upper bound terms of (CC) can be better.
F
Comparisons to Existing Combinatorial Dimensions
In this section, compare the Level-constrained Littlestone dimension 2 and the Level-constrained Branching dimension to existing combinatorial dimensions in multiclass learning.
F1 Existing Combinatorial Dimensions
Definition& 4 (i-neighbour), Let f.g E ydh for some d E N. For every E [d], we say that f and are i-neighbours if f: + 9i and jE(d)() fj =9-
9j-
Definition 5 (DS dimension Daniely and Shalev-Shwe Shwwrtz (2114). Ler C C yo be a concept class. N. We say that S DS-shattered by C. if there exists
Let S E X be a sequence for some F C,I, < oo such that for all f E {g I g yd, 3ger ViEld = f(S))} and for all i E [d], I has at least one i-meighbor. The DS dimension of C, denoted DS(C), is the maximal size of sequence E x² for some d E N that is DS s-attteeedd by C.
Definition 6 (Graph dimension). Let C yx be a concept class. Let S X. We say that S is G-shattered by C, if there exists an f S + y such that for every T C S there is E C such that:
Yeet g(z) ) = f (x) and Vaes-T g(x) + f(x)
The graph dimension C,, denoted G(C), is the maximal cardinality of a set S C X that is G-shattered by
Definition 7 (Natarajan Threshold dimension). Let C C. yx be concept class. Let S E X be sequence, for some d E N. We say that S is NT-shattered by C, if there exist f.g [d] + y such that
Wield f(i) A g()), and there exists (Co, C1. .2.....CC.. ca) E cd+1 such that for every E [d+1],j j e [d]:
f((), isi (g(),, j2i
c-1(S)))
The Natarajan Threshold dimension of C, denoted NT(C), is the maximal size of a sequence S E Xd for some d E N that is NT-shattrred by C.
F.2 Comparison
is easy to show that for every concept class C C y* its Natarajan dimension is always less than or equal to its DS dimension. Moreover, the work of Brukhim et al. [2022] demonstrated there exists concept class C C yx for which the Natarajan dimension is but DS(C)) = Do. Here, we show that for every concept class C y its DS dimension is always less than equal its Level-constrained Littlestone dimension. Furthermore, we demonstrate there exists a concept class C' C y such that DS(C') 1 but D(C") so. These two results are shown in Proposition
Proposition 2. For every concept class C C yx, we have: DS(C) x D(C). Moreover, there exists concept class C¹ yx such that DS(C") but D(C") oc.
Proof. First, prove that for every concept class C C )xt we have: DS(C) < D(C). Let C C y be a concept class such that DS(C) is finite. Subsequently, we show that we can construct a Level- constrained Littlestone tree T of depth DS(C), which is shattered by C. Thus, DS(C) s D(C).
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Let S E XDS((C) be sequence of instances, which is DS-shattered by C. We show that we can construct Level-constrained Littlestone tree T of depth DS(C), having members of S as its nodes in order with the first member being its root and so on, which is shattered by C. To show the construction, we use induction. If DS(C)= = 1, it is clear that we can construct a Level-constrained Littlestone tree 'I of depth 1, which is shattered by C. This is because there must be two concepts in C, which disagree on one member of S. We assume that DS(C) = d, we can construct a Level-constrained Littlesooee tree T of depth d. having members of S as its nodes in order with the first member being its root and so on, which is shattered by C, where E X is a sequence of size d witnessing DS(C)= d. Now, we prove that if DS(C) = d + 1, we can construct Level-constrained Littlestone tree T of depth d + 1. having members of S as its nodes in order with the first member being its root and so on, which is shattered by C. where S E x is sequence of size d + witnessing DS(C)= = d + 1. Let F C C be a set witnessing DS(C) = d + 1. Take any two distinct concepts 02 E F. Define F' as follows: F' == {f I E F, ((Si) = (Si)). In addition, define F" as follows: F" : (f I F, f(Si) = (2())) Observe that (S2, S3. ................................ Sa-1) and F" can witness DS(C)= 2 d. Similarly, observe that (S2, Ss....;. S4411) and F" can witness DS(C)= 2 d. Now, set the root of T as S1 and branches with c1 (Si) and 02 (2))) labels. Based on the inductive assumption combined with the facts that we mentioned, we can complete the construction of Level-constrained Littlestone tree T of depth DS(C), having members of S as its nodes in order with the first member being its root and so on, which is shattered by C.
Finally, we note that if DS(C)= oc, as we can do the construction for every depth d E N. we should have D(C)= = 00.
Second, we prove that there exists a concept class C' C y such that DS(C") = 1 but D(C") = co. To show this, we use our next proposition, namely 3 combined with the well-known fact that for every C _ y*x we have: DS(C) S G(C).
Next, we show there that exists a concept class C C y such that G(C) and D(C) co. On the other hand, we also prove the existence of a concept class C' C ydx such that G(C) = oo and D(C) = 1. These two results, shown in Proposition 3 imply that the Level-constraneed Littlestone dimension and the Graph dimension are not comparable. Moreover, our first claim has an interesting consequence. In particular, it illustrates that having a finite eevel-constrained Littlestone dimension not necessary for having a bounded size sample compression scheme. This follows from the fact that having finite Graph dimension is sufficient for having a bounded size sample compression scheme [David et al. (0166). We also remark that for every concept class C C y its DS dimension is always less than or equal to Graph dimension.
Proposition 3. There exists concept class C. C yxx such that G(C) = and D(C) = 00. Moreover, there exists concept class C² C yd such that G(C') = oa and D(C')= = 1.
Proof First, we prove the second claim. To show that, we rely on Example in Hanneke et al. 2023a In particular, they showed there exists a concept class C² C ydd such that G(C") = 00 and I((C) = 1. As we know D(C') I L(C), we conclude there exists a concept class C' C y such that G(C') xc and D(C") 1.
Second, we prove that there exists concept class C C y such that G(C) and D(C) oo. Let T be an infinite depth rooted perfect binary tree so that all of its levels and edges are labeled by distinct elements. The definition of such a tree is similar to Definition 1.7 in the work of Bousquet et al. [202I]. Let X' be the elements on the levels of T and y be the elements on the edges of T. Also, define the concept class C C y as follows: C only contains all concepts consistent with a branch of 1. We prove this by contradiction. Assume G(C) 2 Thus, there exist S (z), x2) C X of size 2 and f S + y' witnessing the fact that G(C)= = 2. Without loss of generality, we assume that x1 is above 22 in T. Using the fact that the edges of T are labeled with distinct elements of V, there cannot exist both e, E C and
T. Thus, clearly, we have: D(C)- co. Now, we show that G(C)
C2 E C such that ei (x1) = =f(x)) ei (x2) ((22), (2(x1) # f(x1), and c2(22) f(22). This is contradiction, thus G(C)- = 1.
It is well-known that for every concept class C C y its Littlestone dimension is always less than equal to its sequential graph dimension. Moreover, the work of Hanneke et al. [2023]] demonstrated there exists a concept class C Ç y such that L(C)= = 1 and SG(C) = cc. Here, we show that for
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every concept class C _ its Level-constraineed Branching dimension is always less than equal to its Littlestone dimension. Furthermore, we demonstrate that there exists a concept class C' y
such that B(C") S 2 and T(C')) = oc. These two results are shown in Proposition 4
Proposition 4. For every concept class C C y we have: B(C) - L(C). Moreover, there exists concept class C' s y such that B(C") I and T((')= = oo.
Proof. The proof of the following claim: for every concept class C. S y we have that B(C) < L(C) given by Proposition| Therefore, we focus on showing that there exists a concept class C' y such that B(C²) I 2 and L(C")= = cc. Let T be an infinite depth rooted perfect binary tree so that all of its nodes are labeled by distinct elements, all of its left edges are labeled by 0. and all of its right edges are labeled by 1. The definition of such a tree is similar to Definition 1.7 in the work of Bousquet et al. [202T]. Let X' be the elements on the nodes of T. Also, define the concept class C' as follows: C" contains only the concepts consistent with branch of T. Further, each of these concepts predicts a unique label for all instances outside its associated branch. In addition, define y as the union of (0. }} and all unique labels used in the definition of C'. Thus, we have: T.(C') = co.
Now, we show that B(C)) < 2. To prove this, we demonstrate that for every T E N, we have:
infeemeministic A MT C') S 2. As result, we can then conclude that B(C") S 2.
To see why inf Desermiiitt A (T C*) I 2 for every T E N implies that B(C") S 2, suppose for the sake of contradiction that B(C²) 2 3. So, there exists Level-constrained Branching tree of depth d E N such that its branching factor is at least 3. Let T d. It is not hard to see that there exists a sequence of instances of size T" such that for every deterministic learner, there exists a realizable labeling of instances that forces the learner to make at least mistakes over T' rounds. This leads to a contradiction. Thus, we conclude that (CC") < 2.
We now construct deterministic learner A such that MA(I,C") S for every T E N. Let e N. Let SXx be the sequence chosen by the adversary at the beginning of the game. Also, let c* E C be the target concept chosen by the adversary. Further, let be the root-to-leaf path in T associated with the concept c*. In addition, for every i E [T], let vú be root-to-leaf path in T containing first i members of S, if it exists. Finally, let i* be the smallest positive integer such that 0:- does not exist. If i" itself does not exist, let i* = T + 1.
the (0, 1} labels associated with the path -1- for the first i* - points in S. Furthermore, if the adversary ever reveals a unique label, we use its corresponding E C' to make predictions in all future rounds. For the i* 'th member of S, if it exists, we predict arbitrarily. To see that this algorithm makes at most 2 mistakes, we consider two cases. (1) If 8- T 1, then our algorithm makes at most one mistake. In fact, our algorithm makes mistake: (a) if the adversary switches the label from a bit in 10.1) to a unique label corresponding to the target concept c². (b) perhaps on the last instance. (2) Otherwise, the algorithm makes most two mistakes; the first mistake can be on round i* - 1, and the second mistake can be on round i*, after which the true c* is known to the learner from its unique label. Indeed, if the adversary switches the label from a bit in {0. 1} to unique label corresponding to the target concept c* before round i* 1. we only make one mistake. This completes the proof.
Our algorithm A predicts according
Finally, the works of Shelah [1990], , Hodges [997] showed that the finiteness of the Littlestone and Threshold dimensions coincide in the binary setting. Here, we show that this is not the case between the conss-con Branching dimension and the Natarajan Threshold dimension. More specifically, we show that for every concept class C C y its Level-constrained Branching dimension is always greater than or equal to the log of its Natarajan Threshold dimension. However, we give a concept class C" C )x such that NT'C") = and B(C") = Do. These two results are shown in Proposition Notably, the lower bound of Hanneke et al. [2023b), based on the threshold dimension, can be easily generalized to our setting for the Natarajan Threshold dimension.
Proposition 5. For every concept class C C y*x we have: log(NT(C)) S B(C). Moreover, there exists a concept class C' y such that NC(C") = and B(C')= = 00.
Proof. First, we prove that for every concept class C C y we have: log(NT(C)) S B(C). Let C C )d be a concept class such that (C(C) = d for some d E N. 7et d. On the one hand, by presenting the sequences of instances that are NT-shattered by C to the learner, we can use a similar technique as [Hanneke et al. 2023b Claim 3.4], to prove a lower bound of (og(N((C)) on M (T,C).
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On the other hand, based on Section we can prove an upper bound of B(C) on M (T.C). Thus, we have (NT())(C) & B(C).
Second, we prove that there exists a concept class C² C ya such that NT(C)) = and B(C²) = co. Let T be rooted binary tree that it has the following three properties: (1) all of its levels and edges are labeled by distinct elements. (2) each level only contains one node with two children (3) its branching factor is infinite. It is not hard to see that such a tree exists. The definition of such a tree is similar to Definition 1.7 in the work of Bousquet et al. the levels of and be the elements on the edges of Also, define the concept class C' y as follows: C' only contains all concepts consistent with branch of T. Thus, clearly, we have: B(C") = oc. Now, we show that NT(C') 1. We prove this by contradiction. Assume NT(C") 2
[202 I2. Let A' be the elements on
Then, there exist S (I1; x2) E X² and (co, C1, 02) E C²3 witnessing NT(C)) = Without loss of generality, we assume that #1 is above 12 in T. Based on constriction of 'T, is simple to see that (x() + ei (22) and (x2) o (2 and e1 (222 # (2(22). Thus, NT(C") can not be even which completes our contradiction-be baaddd proof.
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Reviews:
[{"review_id": "pBoS8HG71U", "reviewer": "Reviewer_i2TD", "summary": "This paper addresses the problem of multiclass transductive online learning with unbounded label spaces. The paper extends previous work on binary and finite label spaces to the more general case of unbounded label spaces. The authors introduce two new combinatorial dimensions - the Level-constrained Littlestone dimension and the Level-constrained Branching dimension - to characterize the optimal mistake bounds in this setting. They establish a trichotomy of possible minimax rates in the realizable setting, showing that the expected number of mistakes can only grow like \\theta(T), \\theta(log T), or \\theta(1).", "questions": "- Can the authors provide more more intuitive explanation of the Level-constrained Littlestone and Branching dimensions.\n- Can the authors discuss the computational complexity of their algorithm for this setting?", "limitations": "Yes", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper solves an open problem in online learning theory by characterizing optimal mistake bounds for unbounded label spaces.\n- The paper is very well written and easy to understand.", "weaknesses": "- This paper extends the results of multi class transductive learning to infinite label space but the not clear how important is this setting.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for noting that our work resolves an open problem in online learning theory and that our paper is very well written and easy to understand. We address each weakness and question below. \n\n- Multiclass learning with unbounded label spaces is a fundamental setting that has been under study for nearly 40 years, starting with [1,2] and more recently in [3,4,5,6]. Studying infinite label spaces is important as guarantees for multiclass learning should not inherently depend on the number of labels, even when it is finite. This is quite a practical concern as many modern machine learning paradigms have massive label space, such as in face recognition, next word prediction, and protein structure prediction, where the dependence of label size in learning bounds would be undesirable. Beyond being of practical interest, multiclass learning with infinite labels might also advance the understanding of real-valued regression problems [7]. Finally, in mathematics, concepts involving infinities often provide clearer insights into the true hardness of a problem. These motivations have been highlighted in lines 78-85. \n- To define the Level-constrained Littlestone dimension, we first need to define the  Level-constrained Littlestone tree. A Level-constrained Littlestone tree is a Littlestone tree with the additional requirement that the same instance has to label all the internal nodes across a given level. The  Level-constrained Littlestone dimension is just the largest natural number $d \\in \\mathbb{N}$, such that there exists a shattered  Level-constrained Littlestone tree of depth $d$. We will add a more intuitive explanation of this dimension in the final version. \n- To define the Level-constrained Branching dimension, we first need to define the  Level-constrained Branching tree.  The Level-constrained Branching tree is a Level-constrained Littlestone tree without the restriction that the labels on the two outgoing edges are distinct. The Level-constrained Branching dimension is then the smallest natural number $d \\in \\mathbb{N}$ that satisfies the following condition: for every shattered Level-constrained Branching tree $\\mathcal{T}$, there exists a path down $\\mathcal{T}$ such that the number of nodes on this path whose outgoing edges are labeled by different elements of $Y$  is at most $d$. We will add a more intuitive explanation of this dimension in Section 1.2 of the final version. \n- Our algorithms are not computationally efficient. Indeed, our algorithms require calculating the level-constrained Littlestone dimension or level-constrained branching dimension for concept classes. In the binary setting, the level-constrained Littlestone dimension equals the VC dimension, which is computationally hard to compute for general concept classes. That said, most online learning algorithms in online learning theory are not computationally efficient. For instance, in the case of adversarial online learning, SOA involves computing the Littlestone dimension of concept classes defined by the online learner in the course of its interaction with the adversary, which are challenging computations, even when the concept class and the set of features are finite [8]. Notably, no efficient algorithm can achieve finite mistake bounds for general Littlestone classes [9]. \n\n[1] Balas K. Natarajan. Some results on learning. 1988.\n\n[2] B. K. Natarajan. On learning sets and functions. Machine Learning, 4:67–97, 1989.\n\n[3] A. Daniely, S. Sabato, S. Ben-David, and S. Shalev-Shwartz. Multiclass learnability and the ERM principle. 24th Conference on Learning Theory, 2011.\n\n[4] A. Daniely and S. Shalev-Shwartz. Optimal learners for multiclass problems. 27th Conference on Learning Theory, 2014.\n\n[5] N. Brukhim, D. Carmon, I. Dinur, S. Moran, and A. Yehudayoff. A characterization of multiclass learnability.  63rd Annual IEEE Symposium on Foundations of Computer Science, 2022.\n\n[6] S. Hanneke, S. Moran, V. Raman, U. Subedi, A. Tewari. Multiclass Online Learning and Uniform Convergence. 36th Conference on Learning Theory, 2023.\n\n[7] I. Attias, S.Hanneke, A.Kalavasis, A. Karbasi, G. Velegkas. Optimal learners for realizable regression: Pac learning and online learning. Advances in Neural Information Processing Systems 36 (2023).\n\n[8] P.Manurangsi, A. Rubinstein. Inapproximability of VC dimension and littlestone’s dimension. 30th Conference on Learning Theory, 2017.\n\n[9] A. Assos, I. Attias, Y. Dagan, C.Daskalakis,  M. K. Fishelson. Online learning and solving infinite games with an erm oracle. 36th Conference on Learning Theory, 2017.", "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": "r6gDhiaefg", "reviewer": "Reviewer_Cx7F", "summary": "This work continues the study of transductive online classification (a learning setting from the 90s recently reviewed by Hanneke et al. [NeurIPS 2023]. The main result is a trichotomy of possible rates for the general multi-class case (even for the infinite label case) in the realizable setting; answering an open question by Hanneke et al. The three cases are characterized by novel combinatorial dimensions (variant of Littlestone dimension called level-wise Littlestone and ).\n\nAdditionally they achieve optimal $\\tilde{\\Theta}(\\sqrt{TD(\\mathcal{C})})$ (up to log factors) rates in the agnostic setting again determined by the level-wise Littlestone dim. $D(\\mathcal{C})$.", "questions": "In the agnostic setting for $|Y|=k$ one would expect something like $O(\\sqrt{T\\mathrm{Ndim}(C)k})$ up to log factors, similar to the bound in the realizable setting in Hanneke et al. 2023  (Theorem B.3). Please relate this to your agnostic bound.\n\nCan the gap between $D(C)$ and $B(C)$ be arbitrary? More generally how far can $D(C)$,$B(C)$, and $L(C)$ be from each other. I agree that stating the rate as $O(1)$ makes sense, but it would be nice to have some quantity explicitly giving the rate in this case some Xdim s.t. $O(\\mathrm{Xdim}(H))$ (as Ldim does for standard online classification). Note that the tree ranks by Ben-David et al [1997] (see limitations, as well), achieves this specifically also for the worst-sequence/transductive setting (at least for the binary case).", "limitations": "Please discuss previous work more thoroughly. The tree ranks from the \"Online learning versus offline learning\" (self-directed, worst-case sequence, etc.) [Ben-David et al. 1997] and similar papers are very much related to the proposed dimensions here (which is only very briefly acknowledged in a short sentence here). E.g., the \"level-constrained\" variant of the trees in these papers also become the VC-dim for $|Y|=2$. Also similarities to Devulapalli and Hanneke could be discussed more (e.g., the lower bounds there probably apply here too).", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "Timely and interesting paper continuing the recent interest in transductive online classification and related models. Almost tightly characterizes the possible rates for agnostic and realizable learning.\n\n--- rebuttal ---\nchanged from 6 to 7", "weaknesses": "Not in-depth discussion of previous related work.\nThe gap between $D(C)$ and $B(C)$ in Theorem 3 (in the $B(C)<\\infty$ case) is somewhat unsatisfactory.\n\nSee also questions and limitations below.\n\nMinor:\n* You might want to fix \"Hanneke et al. [2024]\" to [2023], otherwise you cite a NeurIPS paper in the future.\n* typo in line 37: should be $c:X\\to Y$ not $c:X\\mapsto Y$.\n* typo in line 207: \"upper=\"\n* Perhaps hint the additional results (Prop 4, comparison to DS, graph-dim) in the main paper, at least with some short sentences.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for finding our work timely and interesting.  All minor typos and suggestions will be incorporated in the final version. We address each weakness and question below. \n\n- For small $k$ (i.e. $k << 2^{(\\log{T})^2}$), the Natarajan bound can be smaller than the upper bound in terms of the Level-constrained Littlestone dimension. However, for large $k$ (i.e. $k > 2^{(\\log{T})^2}$), our upper bound in terms of $D(C)$ can be better. We will make sure to point this out in the camera ready version. \n- There can indeed be an arbitrary gap between $B(C)$ and $D(C)$. For example, for the class of thresholds $C = \\lbrace{x \\mapsto 1\\lbrace{x \\geq a \\rbrace}: a \\in \\mathbb{R} \\rbrace}$, we have that $D(C) = VC(C) = 1$, however $B(C) = \\infty$ using the lower bound from Hanneke et al. [2023]. We will make sure to point this out in the main text. With regards to $B(C)$ and $L(C)$, there can also be an arbitrary gap. Proposition 4 in Appendix F gives a class where $B(C) \\leq 2$ but $L(C) = \\infty$. We will make this explicit in the main text again. \n- Regarding the case when the rate is $O(1)$, Theorem 3 shows that when $B(C) < \\infty$,  the minimax rate is at most $B(C)$ in the realizable setting. With regards to lower bounds, we can also show that when $T$ is large enough (namely $T >> 2^{B(C)}$), the lowerbound in the realizable setting is also $B(C)/2$. We will include this lower bound in the camera-ready version. \n- We thank the reviewer for pointing out these related previous works. We note that we do have a more in-depth discussion of prior work in Appendix A. In the final version, we will relocate this section to the main text. Additionally, we will incorporate a sentence that draws a precise comparison between our $B(C)$ and the rank notion from the paper by [Ben-David et al. 1997]. It is also important to note that while the dimension introduced by Devulapalli and Hanneke provides a lower bound, it is not feasible to establish an upper bound based on it. Specifically, their paper includes a theorem demonstrating the gap between transductive and self-directed online learning, even in the binary case.", "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": "X3rgfmb9WP", "reviewer": "Reviewer_DRE4", "summary": "The paper studies the problem of multiclass transductive online learning where the number of classes can be unbounded.\n\nIn the transductive setting, the learner receives in advance a sequence of instances $(x_1,…,x_T)$ by an adversary. Then, sequentially at each time step t, it needs to decide a label $\\hat{y}_t$, and then the adversary reveals the true label label $y_t$. Given a concept class $C$, in the realizable setting, the adversary must choose the labels according to a concept $c \\in C$.  The goal is to design an online algorithm that minimizes the regret, i.e. the total number of mistakes done by the learner compared to the best concept chosen from C in hindsight. \n\nThe paper fully characterizes the regret for the realizable and the agnostic setting for this problem in the case of an infinite number of classes, extending previous results of Hanneke et al. [2024] limited to a finite number of classes.", "questions": "1. Does the label set Y need to be countable?\n\n2. The DS dimension characterizes the learnability of multiclass learning in the pac setting (Brukhim et al 2022) . Would it be possible to express Theorem 1 using DS and D (i.e., can DS replace the branching dimension, to have a nice parallel between VC and DS?)\n\nTypos\n207 upper=\n211 upper round\n359 lowerbounds", "limitations": "There is no specific section for limitations. This theory paper is self-contained so I do not believe it is needed.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper characterizes the multiclass transductive online learning problem in the case of an infinite number of classes. This is a nice contribution to a fundamental problem. The paper is well written.", "weaknesses": "It is not clear whether the tools presented in this paper can be applied to other settings. Specifically, the definitions of level-constrained branching dimension, and level-constrained littlestone dimension seem specific to solve this (still important) problem. (See also question 2 below).\n\nIn Lines 102-103, it is claimed that finiteness of D(C) and B(C) coincides for |Y| = 2. However, if that’s the case, it seems that Theorem 1 cannot show the Trichotomy of Hanneke et al 2024 for the binary case.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for finding our work to be well written and a nice contribution to a fundamental problem. All minor typos and suggestions will be incorporated in the final version. We address each weakness and question below. \n\n- The algorithm achieving the $\\log{T}$ upper bound in the realizable setting is the most significant contribution of our work, as mentioned in Section 1.2. We believe that the adaptation of this algorithm is applicable to various other settings, including list transductive online learning and transductive online learning with bandit feedback.\n- In lines 102-103, we first stated that $D(C) = VC(C)$ for binary classification. Then, we stated that $B(C) < \\infty$ if and only if $L(C) < \\infty$ for binary classification. We did not make the claim that $B(C) < \\infty$ if and only if $D(C) < \\infty$. \n- The label space $Y$ does not need to be countable. Following the work of [1], we only require that the singleton sets $\\lbrace{y \\rbrace}$ need to be measurable. We will make the assumption needed on $Y$ more clear in the camera-ready version. \n- No, it is not possible to express Theorem 1 using DS and D because $DS(C) \\leq D(C)$ for every $C$. In addition, Claim 5.4 in Hanneke et al. [2023] shows that the DS dimension does not characterize multiclass transductive online learning. Namely, there is a class where $DS(C) = 1$, but $C$ is not transductive online learnable. However, if $|Y| < \\infty$, then the DS dimension does characterize learnability. We also note that we have comparisons to other existing dimensions in Appendices B and F. \n\n[1] S. Hanneke, S. Moran, V. Raman, U. Subedi, A. Tewari. Multiclass Online Learning and Uniform Convergence. 36th Conference on Learning Theory, 2023.", "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": "iqkAFjBts8", "reviewer": "Reviewer_Nc66", "summary": "This paper studies transductive online learning in the multiclass setting, where the label space can be unbounded. In the transductive setting, the adversary commits to a sequence of examples and can only adaptively choose labels for a sequence of instances, unlike the pure online setting where the adversary can adaptively choose both the sequence of examples and the labels, or the pure offline where the adversary commits to both a sequence of examples and labels. \n\nThe main result of this paper extends the results of Hanneke et. al. [2023] by characterizing the optimal mistake bound in the unbounded label case. The proof techniques involve defining and leveraging two new combinatorial parameters termed Level-constrained Branching and Littlestone dimensions. The authors prove that the level-constrained Littlestone dimension characterizes this specific variant of online learning.  \n\n\nHanneke et. al. [2023]: A Trichotomy for Transductive Online Learning.", "questions": "Minor issues\n------------------\n1. It is unclear from the definitions in section 2.2 how optimal regret and optimal mistake bound would be defined differently in the realizable setting. Could the authors expand on this?\n\n2. Equations on lines 210, 285, and 354 should be formatted properly.\n\nFuture Work Discussions\n-----------------------------------\nPlease feel free not to answer these questions if the authors are cramped for time. Not answering these questions will not affect my score.\n\n1. Do the lower bounds of Theorem 3 hold directly for the list transductive online setting?\n2. In the online real-valued regression setting, can one reduce a discretized version of the problem to the unbounded label case setting?", "limitations": "Adequately discussed in the paper.", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper characterizes the optimal mistake bound of transductive online learners in the unbounded label space setting in both realizable and agnostic cases.\n\n2. The paper introduces two new combinatorial parameters: the Level-constrained Littlestone dimension $D(C)$ and the Level-constrained Branching dimension $B(C)$, and show that $D(C)$ characterizes transductive online learning in the unbounded label setting. These new parameters are also compared to existing combinatorial parameters like the Natarajan Dimension, DS dimension, and the Graph dimension. \n\n3. The technical parts of the paper involve modifying the Halving technique to work with  new notion of shattering using the above newly introduced combinatorial parameters,  and a modified version of Littlestone's standard optimal algorithm. Combined with the definition of $D(C)$ and $B(C)$, I feel that these new definitions and techniques could be of independent interest to the learning theory community.\n\n4. This work almost rounds out the literature on regret and mistake bounds in the original transductive online learning setting. Removing the logarithmic factor in the upper bound of Theorem 3 seems to be the final milestone in this setting.", "weaknesses": "1. The definitions of expected regret and expected cumulative mistakes in section 2.2 seem to be incomplete. What is the probability distribution that the expectations are taken with respect to? The only clear choice to me is some probability distribution over the label space $\\mathcal{Y}$. Since the label space is unbounded (a critical point of this work), the notion of a probability distribution over this unbounded label space should be rigorously defined somewhere in the paper (maybe in the Appendix if there are space constraints). This seems to be an inexplicable oversight in an otherwise highly technical and detailed paper.\n\n2. Even though the regret in the agnostic setting is stated in terms of the level-constrained Littlestone dimension, the proof technique heavily borrows from previous works. The authors state this clearly in the text as well. While the result serves a pedagogical purpose and also lends to completeness of the results, Theorem 4 cannot be counted as a significant contribution.\n\nThe following weaknesses do not directly impact my scores but should improve the paper's readability.\n\n3. In reference [Brukhim 2022], the DS dimension characterizes multiclass learnability for unbounded label space in the offline setting. It is not immediately clear from the main text why the DS dimension cannot similarly characterize the transductive online setting and why $D(C)$ and $B(C)$ are required in the first place. The answer is possibly buried in Proposition 2, given in Appendix F.2, and should clearly be stated in the main text (even informal statements suffice), given the fact that the new combinatorial parameters are a core contribution of this paper.\n\n4. The proofs in section 3.1 and 3.2 use the standard Halving technique, but make use of the newly defined combinatorial parameters. Both upper bounds share similar overarching ideas with each other, and also with results in previous works for the constrained label space. These two sections can be rewritten in a manner that highlights the power of the Halving technique. While the proofs themselves appear to be correct, if we take the above weakness into consideration, its also not immediately apparent from the main text why the Halving technique cannot be applied using more standard combinatorial parameters such as the DS dimension. Also, I feel that some parts of the proof can be deferred to the appendix (up to the discretion of the authors).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for pointing out that our techniques could be of independent interest to the learning theory community and that our result almost rounds out the literature on regret and mistake bounds in the original transductive online learning setting. All minor issues and suggestions will be incorporated in the final version. We address each weakness and question below. \n\n- The expectation is taken with respect to only the randomness of the learner. The learner makes predictions by sampling from distributions over $Y$. Thus, the reviewer is correct in the sense that the expectation is taken with respect to distributions over the label space $Y$. Regarding issues of measurability, following the work of [1], we only require that the singleton sets $\\lbrace{y \\rbrace}$ need to be measurable. We will make the assumption needed on $Y$ more clear in the camera-ready version. \n- We agree with the reviewer that our agnostic upper bound uses a pretty standard technique. Our main contribution is in the realizable setting. \n- Hanneke et al. [2023] in Claim 5.3 show that the DS dimension cannot characterize multiclass transductive online learnability. Namely, they give a class $C$ such that $DS(C) = 1$, but $C$ is not transductive online learnable. This necessitates the need for new dimensions since the Littlestone dimension is clearly not necessary. We will make this explicit in the main text of the camera ready version. \n- We provided a brief explanation of the relevance of the Halving algorithm in transductive online learning in lines 110-120. In particular, we noted that a naive adaptation of the Halving algorithm for multiclass learning would not work and that we defined a new notion of shattering that would allow us to apply an analog of the Halving algorithm. Nevertheless, we will make sure to point out that both algorithms in Section 3.1 and 3.2 share similar overarching ideas in the sense that they both use the Halving technique with a particular combinatorial dimension. We note that the Halving technique using the DS dimension cannot work because the finiteness of the DS dimension is not sufficient for online learnability.  \n- In the realizable setting, we evaluate the learner through its mistake bound. On the other hand, in the agnostic setting, we evaluate the learner through its regret bound. In the realizable setting, the regret bound and mistake bound are the same quantities. However, this is not the case in the agnostic setting, where one usually only cares about the regret bound. \n- Regarding the list setting, our lower bound applies to the list transductive online setting. However, it is possible to establish tighter lower bounds by adapting our definitions for $(L+1)$-array trees when the list size is $L$. Regarding the regression setting, one could approach this problem using binary results. We are uncertain whether our results will be directly applicable.\n\n[1] S. Hanneke, S. Moran, V. Raman, U. Subedi, A. Tewari. Multiclass Online Learning and Uniform Convergence. 36th Conference on Learning Theory, 2023.", "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: This is a theory paper on multiclass transductive online learning when the number of labels can be unbounded. All reviewers were very positive about the submission and found it to be a technically solid and valuable contribution to the field.
Decision: Accept (spotlight)