SlowGuess/NeurIPS_2023_OCR huggingface dataset:
steve.hanneke@gmail.com
Faculty of Mathematics,
Faculty of Computer Science, and
Faculty of Data and Decision Sciences
smoran@technion.ac.il
Jonathan Shafer
Computer Science Division
shaferjo@berkeley.edu
We present new upper and lower bounds on the number of learner mistakes in the 'transductive' online learning setting of Ben-David, Kushlevitz and Mansour (1997). This setting is similar to standard online learning, except that the adversary fixes a sequence of instances  $x_{1}, \ldots, x_{n}$  to be labeled at the start of the game, and this sequence is known to the learner. Qualitatively, we prove a trichotomy, stating that the minimal number of mistakes made by the learner as  $n$  grows can take only one of precisely three possible values:  $n, \Theta(\log(n))$ , or  $\Theta(1)$ . Furthermore, this behavior is determined by a combination of the VC dimension and the Littlestone dimension. Quantitatively, we show a variety of bounds relating the number of mistakes to well-known combinatorial dimensions. In particular, we improve the known lower bound on the constant in the  $\Theta(1)$  case from  $\Omega\left(\sqrt{\log(d)}\right)$  to  $\Omega(\log(d))$  where  $d$  is the Littlestone dimension. Finally, we extend our results to cover multiclass classification and the agnostic setting.
In classification tasks like PAC learning and online learning, the learner simultaneously confronts two distinct types of uncertainty: labeling-related uncertainty regarding the best labeling function, and instance-related uncertainty regarding the instances that the learner will be required to classify in the future. To gain insight into the role played by each type of uncertainty, researchers have studied modified classification tasks in which the learner faces only one type of uncertainty, while the other type has been removed.
In the context of PAC learning, [BB11] studied a variant of proper PAC learning in which the true labeling function is known to the learner, and only the distribution over the instances is not known. They show bounds on the sample complexity in this setting, which conceptually quantify the instance-related uncertainty. Conversely, labeling-related uncertainty is captured by PAC learning with respect to a fixed (e.g., uniform) domain distribution [BI91], a setting which has been studied extensively.
In this paper we improve upon the work of [BKM97], who quantified the label-related uncertainty in online learning. They introduced a model of transductive online learning, in which the adversary
[1] [BKM97] call their model 'off-line learning with the worst sequence', but in this paper we opt for 'transductive online learning', a name that has appeared in a number of publications, including [KK05, Pec08, CS13, SKS16]. We remark there are at least two different variants referred to in the litera
commits in advance to a specific sequence of instances, thereby eliminating the instance-related uncertainty.
1.1 The Transductive Online Learning Model
The model of learning studied in this paper, due to [BKM97], is a zero-sum, finite, complete-information, sequential game involving two players, the learner and the adversary. Let  $n \in \mathbb{N}$ , let  $\mathcal{X}$  and  $\mathcal{Y}$  be sets, and let  $\mathcal{H} \subseteq \mathcal{Y}^{\mathcal{X}}$  be a collection of functions called the hypothesis class.
The game proceeds as follows (see Section 2 for further formal definitions). First, the adversary selects an arbitrary sequence of instances,  $x_{1}, \ldots, x_{n} \in \mathcal{X}$ . Then, for each  $t = 1, \ldots, n$ :
1. The learner selects a prediction,  $\hat{y}_t\in \mathcal{V}$
2. The adversary selects a label,  $y_{t}\in \mathcal{V}$
In each step  $t \in [n]$ , the adversary must select a label  $y_{t}$  such that the sequence  $(x_{1},y_{1}),\ldots ,(x_{t},y_{t})$  is realizable by  $\mathcal{H}$ , meaning that there exists  $h \in \mathcal{H}$  satisfying  $h(x_{i}) = y_{i}$  for all  $i \in [t]$ . The learner's objective is to minimize the quantity
$$
M (A, x, h) = | \{t \in [ n ]: \hat {y} _ {t} \neq h (x _ {t}) \} |,
$$
which is the number of mistakes when the learner plays strategy  $A$  and the adversary chooses sequence  $x \in \mathcal{X}^n$  and labels consistent with hypothesis  $h \in \mathcal{H}$ . We are interested in understanding the value of this game,
$$
M(\mathcal{H},n) = \inf_{A\in \mathcal{A}}\sup_{x\in \mathcal{X}^{n}}\sup_{h\in \mathcal{H}}M(A,x,h),
$$
where  $\mathcal{A}$  is the set of all learner strategies. Note that neither party can benefit from using randomness, so without loss of generality we consider only deterministic strategies.
1.2 A Motivating Example
Transductive predictions of the type studied in this paper appear in many real-world situations, essentially in any case where a party has a schedule or a to-do list known in advance of specific tasks that need to be completed in order, and there is some uncertainty as to the precise conditions that will arise in each task. As a concrete example, consider the logistics that the management of an airport faces when scheduling the work of passport-control officers.
Example 1.1. An airport knows in advance what flights are scheduled for each day. However, it does not know in advance exactly how many passengers will go through passport control each day (because tickets can be booked and cancelled in the last minute, and entire flights can be cancelled, delayed or rerouted, etc.). Each day can be 'regular' (the number of passengers is normal), or it can be 'busy' (more passengers than usual). Correspondingly, each day the airport must decide whether to schedule a 'regular shift' of passport-control officers, or an 'extended shift' that contains more officers.
If the airport assigns a standard shift for a busy day, then passengers experience long lines at passport control, and the airport suffers a loss of 1; if the airport assigns an extended shift for a regular day, then it wastes money on excess manpower, and it again experiences a loss of 1; If the airport assigns a regular shift to a regular day, or an extended shift to a busy day, then it experiences a loss of 0.
Hence, when the airport schedules its staff, it is essentially attempting to predict for each day whether it will be a regular day or a busy day, using information it knows well in advance about which flights are scheduled for each day. This is precisely a transductive online learning problem.
1.3 Our Contributions
I. Trichotomy. We show the following trichotomy. It shows that the rate at which  $M(\mathcal{H}, n)$  grows as a function of  $n$  is determined by the VC dimension and the Littlestone dimension (LD).
ture as 'transductive online learning'. For example, [SKS16] write of "a transductive setting [BKM97] in which the learner knows the arriving contexts a priori, or, less stringently, knows only the set, but not necessarily the actual sequence or multiplicity with which each context arrives." That is, in one setting, the learner knows the sequence  $(x_{1},\ldots ,x_{n})$  in advance, but in another setting the learner only knows the set  $\{x_{1},\dots,x_{n}\}$ . One could distinguish between these two settings by calling them 'sequence-transductive' and 'set-transductive', respectively. Seeing as the current paper deals exclusively with the sequence-transductive setting, we refer to it herein simply as the 'transductive' setting.
Theorem (Informal Version of Theorem 4.1). Every hypothesis class  $\mathcal{H} \subseteq \{0,1\}^{\mathcal{X}}$  satisfies precisely one of the following:
1.  $M(\mathcal{H},n) = n$  . This happens if  $\mathrm{VC}(\mathcal{H}) = \infty$
2.  $M(\mathcal{H},n) = \Theta (\log (n))$  . This happens if  $\mathsf{VC}(\mathcal{H}) <   \infty$  and  $\mathsf{LD}(\mathcal{H}) = \infty$
3.  $M(\mathcal{H},n) = \Theta (1)$ . This happens if  $\mathsf{LD}(\mathcal{H}) < \infty$ .
The  $\Theta(\cdot)$  notations in Items 2. and 3. hide a dependence on  $\mathsf{VC}(\mathcal{H})$ , and  $\mathsf{LD}(\mathcal{H})$ , respectively.
The proof uses bounds on the number of mistakes in terms of the threshold dimension (Section 3.2), among other tools.
II. Littlestone classes. The minimal constant upper bound in the  $\Theta(1)$  case of Theorem 4.1 is some value  $C(\mathcal{H})$  that depends on the class  $\mathcal{H}$ , but the precise mapping  $\mathcal{H} \mapsto C(\mathcal{H})$  is not known in general. [BKM97] showed that  $C(\mathcal{H}) = \Omega\left(\sqrt{\log(\mathrm{LD}(\mathcal{H}))}\right)$ . In Section 3 and Appendix A we improve upon their result as follows.
Theorem (Informal Version of Theorem 3.1). Let  $\mathcal{H} \subseteq \{0,1\}^{\mathcal{X}}$  such that  $\mathsf{LD}(\mathcal{H}) = d < \infty$ . Then  $M(\mathcal{H},n) = \Omega (\log (d))$ .
III. Multiclass setting. In Section 5, we generalize Theorem 4.1 to the multiclass setting with a finite label set  $\mathcal{V}$ , showing a trichotomy based on the Natarajan dimension. The proof uses a simple result from Ramsey theory, among other tools.
Additionally, we show that the DS dimension of [DS14] does not characterize multiclass transductive online learning.
IV. Agnostic setting. In the standard (non-transductive) agnostic online setting, [BPS09] showed that  $R_{\mathrm{online}}(\mathcal{H}, n)$ , the agnostic online regret for a hypothesis class  $\mathcal{H}$  for a sequence of length  $n$  satisfies
$$
\Omega \left(\sqrt {\operatorname {L D} (\mathcal {H}) \cdot n}\right) \leq R _ {\text {o n l i n e}} (\mathcal {H}, n) \leq O \left(\sqrt {\operatorname {L D} (\mathcal {H}) \cdot n \cdot \log n}\right). \tag {1}
$$
Later,  $[\mathrm{ABD}^+ 21]$  showed an improved bound of  $R_{\mathrm{online}}(\mathcal{H},n) = \Theta \Big(\sqrt{\mathrm{LD}(\mathcal{H})\cdot n}\Big)$ .
In Section 6 we show a result similar to Eq. (1), for the transductive agnostic online setting.
Theorem (Informl Version of Theorem 6.1). Let  $\mathcal{H} \subseteq \{0,1\}^{\mathcal{X}}$ , such that  $0 < \mathsf{VC}(\mathcal{H}) < \infty$ . Then the agnostic transductive regret for  $\mathcal{H}$  is
$$
\Omega \Big (\sqrt {\mathsf {V C} (\mathcal {H}) \cdot n} \Big) \leq R (\mathcal {H}, n) \leq O \Big (\sqrt {\mathsf {V C} (\mathcal {H}) \cdot n \cdot \log n} \Big).
$$
1.4 Related Works
The general idea of bounding the number of mistakes by learning algorithms in sequential prediction problems was introduced in the seminal work of Littlestone [Lit87]. That work introduced the online learning model, where the sequence of examples is revealed to the learner one example at a time. After each example  $x$  is revealed, the learner makes a prediction, after which the true target label  $y$  is revealed. The constraint, which makes learning even plausible, is that this sequence of  $(x,y)$  pairs should maintain the property that there is an (unknown) target concept in a given concept class  $\mathcal{H}$  which is correct on the entire sequence. Littlestone [Lit87] also identified the optimal predictor for this problem (called the SOA, for Standard Optimal Algorithm, and a general complexity measure which is precisely equal to the optimal bound on the number of mistakes: a quantity now referred to as the Littlestone dimension.
Later works discussed variations on this framework. In particular, as mentioned, the transductive model discussed in the present work was introduced in the work of [BKM97]. The idea (and terminology) of transductive learning was introduced by [VC74, Vap82, Kuh99], to capture scenarios
where learning may be easier due to knowing in advance which examples the learner will be tested on. [VC74, Vap82, Kuh99] study transductive learning in a model closer in spirit to the PAC framework, where some uniform random subset of examples have their labels revealed to the learner and it is tasked with predicting the labels of the remaining examples. In contrast, [BKM97] study transductive learning in a sequential prediction setting, analogous to the online learning framework of Littlestone. In this case, the sequence of examples  $x$  is revealed to the learner all at once, and only the target labels (the  $y$ 's) are revealed in an online fashion, with the label of each example revealed just after its prediction for that example in the given sequential order. Since a mistake bound in this setting is still required to hold for any sequence, for the purpose of analysis we may think of the sequence of  $x$ 's as being a worst case set of examples and ordering thereof, for a given learning algorithm. [BKM97] compare and contrast the optimal mistake bound for this setting to that of the original online model of [Lit87]. Denoting by  $d$  the Littlestone dimension of the concept class, it is clear that the optimal mistake bound would be no larger than  $d$ . However, they also argue that the optimal mistake bound in the transductive model is never smaller than  $\Omega(\sqrt{\log(d)})$  (as mentioned, we improve this to  $\log(d)$  in the present work). They further exhibit a family of concept classes of variable  $d$  for which the transductive mistake bound is strictly smaller by a factor of  $\frac{3}{2}$ . They additionally provide a general equivalent description of the optimal transductive mistake bound in terms of the maximum possible rank among a certain family of trees, each representing the game tree for the sequential game on a given sequence of examples  $x$ .
In addition to these two models of sequential prediction, the online learning framework has also been explored in other variations, including exploring the optimal mistake bound under a best-case order of the data sequence  $x$ , or even a self-directed adaptive order in which the learning algorithm selects the next point for prediction from the remaining  $x$ 's from the given sequence on each round. [BKM97, BEK95, GS94, BE98, Kuh99].
Unlike the online learning model of Littlestone, the transductive model additionally allows for nontrivial mistake bounds in terms of the sequence length  $n$  (the online model generally has  $\min \{d, n\}$  as the optimal mistake bound). In this case, it follows immediately from the Sauer-Shelah-Perles lemma and a Halving technique that the optimal transductive mistake bound is no larger than  $O(v \log(n))$  [KK05], where  $v$  is the VC dimension of the concept class [VC71, VC74].
Notation 2.1. Let  $\mathcal{X}$  be a set and  $n, k \in \mathbb{N}$ . For a sequence  $x = (x_{1},\ldots,x_{n}) \in \mathcal{X}^{n}$ , we write  $x_{\leq k}$  to denote the subsequence  $(x_{1},\ldots,x_{k})$ . If  $k \leq 0$  then  $x_{\leq k}$  denotes the empty sequence,  $\mathcal{X}^0$ .
Definition 2.2. Let  $k \in \mathbb{N}$ , let  $\mathcal{X}$  and  $\mathcal{Y}$  be sets, and let  $\mathcal{H} \subseteq \mathcal{Y}^{\mathcal{X}}$ . A sequence  $(x_{1},y_{1}),\ldots ,(x_{k},y_{k}) \in (\mathcal{X} \times \mathcal{Y})^{k}$  is realizable by  $\mathcal{H}$ , or  $\mathcal{H}$ -realizable, if  $\exists h \in \mathcal{H} \forall i \in [k] : h(x_{i}) = y_{i}$ .
Definition 2.3. Let  $\mathcal{X}$  be a set, let  $\mathcal{H} \subseteq \{0,1\}^{\mathcal{X}}$ , let  $d \in \mathbb{N}$ , and let  $X = \{x_{1},\ldots,x_{d}\} \subseteq \mathcal{X}$ . We say that  $\mathcal{H}$  shatters  $X$  if for every  $y \in \{0,1\}^{d}$  there exists  $h \in \mathcal{H}$  such that for all  $i \in [d]$ ,  $h(x_{i}) = y_{i}$ . The Vapnik-Chervonenkis (VC) dimension of  $\mathcal{H}$  is  $\mathsf{VC}(\mathcal{H}) = \sup \{|X| : X \subseteq \mathcal{X}$  finite  $\wedge$ $\mathcal{H}$  shatters  $X\}$ .
Definition 2.4 ([Lit87]). Let  $\mathcal{X}$  be a set and let  $d\in \mathbb{N}$ . A Littlestone tree of depth  $d$  with domain  $\mathcal{X}$  is a set
$$
T = \left\{x _ {u} \in \mathcal {X}: u \in \bigcup_ {s = 0} ^ {d} \{0, 1 \} ^ {s} \right\}. \tag {2}
$$
Let  $\mathcal{H} \subseteq \{0,1\}^{\mathcal{X}}$ . We say that  $\underline{\mathcal{H}}$  shatters a tree  $T$  as in Eq. (2) if for every  $u \in \{0,1\}^{d+1}$  there exists  $h_u \in \mathcal{H}$  such that
$$
\forall i \in [ d + 1 ]: h \left(x _ {u _ {\leq i - 1}}\right) = u _ {i}.
$$
The Littlestone dimension of  $\mathcal{H}$ , denoted  $\mathsf{LD}(\mathcal{H})$ , is the supremum over all  $d \in \mathbb{N}$  such that there exists a Littlestone tree of depth  $d$  with domain  $\mathcal{X}$  that is shattered by  $\mathcal{H}$ .
Theorem 2.5 ([Lit87]). Let  $\mathcal{X}$  be a set and let  $\mathcal{H} \subseteq \{0,1\}^{\mathcal{X}}$  such that  $d = \mathrm{LD}(\mathcal{H}) < \infty$ . Then there exists a strategy for the learner that guarantees that the learner will make at most  $d$  mistakes in the standard (non-transductive) online learning setting, regardless of the adversary's strategy and of number of instances to be labeled.
Figure 1: A shattered Littlestone tree of depth 2. The empty sequence is denoted by  $\lambda$ .
Theorem 2.6 (Sauer-Shelah-Perles; [She72, Sau72]). Let  $n, d \in \mathbb{N}$ , let  $\mathcal{X}$  be a set of cardinality  $n$ , and let  $\mathcal{H} \subseteq \{0,1\}^{\mathcal{X}}$  such that  $\mathsf{VC}(\mathcal{H}) = d$ . Then  $|\mathcal{H}| \leq \sum_{i=0}^{n} \binom{n}{i} \leq \left(\frac{en}{d}\right)^d$ .
3 Quantitative Bounds
3.1 Littlestone Dimension: A Tighter Lower Bound
The Littlestone dimension is an upper bound on the number of mistakes, namely
$$
\forall n \in \mathbb {N}: M (\mathcal {H}, n) \leq \mathrm {L D} (\mathcal {H}) \tag {3}
$$
for any class  $\mathcal{H}$ . This holds because  $\mathrm{LD}(\mathcal{H})$  is an upper bound on the number of mistakes for standard (non-transductive) online learning [Lit87], and the adversary in the transductive setting is strictly weaker.
The Littlestone dimension also supplies a lower bound. We give a quadratic improvement on the previous lower bound of [BKM97], as follows.
Theorem 3.1. Let  $\mathcal{X}$  be a set, let  $\mathcal{H} \subseteq \{0,1\}^{\mathcal{X}}$  such that  $d = \operatorname{LD}(\mathcal{H}) < \infty$ , and let  $n \in \mathbb{N}$ . Then
$$
M (\mathcal {H}, n) \geq \min  \left\{\lfloor \log (d) / 2 \rfloor , \lfloor \log \log (n) / 2 \rfloor \right\}.
$$
Proof idea for Theorem 3.1. Let  $T$  be a Littlestone tree of depth  $d$  that is shattered by  $\mathcal{H}$ , and let  $\mathcal{H}_1 \subseteq \mathcal{H}$  be a collection of  $2^{d+1}$  functions that witness the shattering. The adversary selects the sequence consisting of the nodes of  $T$  in breadth-first order. For each time step  $t \in [n]$ , let  $\mathcal{H}_t$  denote the version space, i.e., the subset of  $\mathcal{H}_1$  that is consistent with all previously-assigned labels. The adversary's adaptive labeling strategy at time  $t$  is as follows. If  $\mathcal{H}_t$  is very unbalanced, meaning that a large majority of functions in  $\mathcal{H}_t$  assign the same value to  $x_t$ , then the adversary chooses  $y_t$  to be that value. Otherwise, if  $\mathcal{H}_t$  is fairly balanced, the adversary forces a mistake (it can do so without violating  $\mathcal{H}$ -realizability). The pivotal observation is that: (1) under this strategy, the version space decreases in cardinality significantly more during steps where the adversary forces a mistake compared to steps where it did not force a mistake; (2) let  $x_t$  be the  $t$ -th node in the breadth-first order. It has distance  $\ell = \lfloor \log(t) \rfloor$  from the root of  $T$ . Because  $T$  is a binary tree, the subtree  $T'$  of  $T$  rooted at  $x_t$  is a tree of depth  $d - \ell$ . In particular, seeing as  $\mathcal{H}_t$  contains only functions necessary for shattering  $T'$ ,  $|\mathcal{H}_t| \leq 2^{d - \ell + 1}$ , so  $\mathcal{H}_t$  must decrease not too slowly with  $t$ . Combining (1) and (2) yields that the adversary must be able to force a mistake not too rarely. A careful quantitative analysis shows that the number of mistakes the adversary can force is at least logarithmic in  $d$ .
The full proof of Theorem 3.1 appears in Appendix A.
3.2 Threshold Dimension
We also show some bounds on the number of mistakes in terms of the threshold dimension.
Definition 3.2. Let  $\mathcal{X}$  be a set, let  $X = \{x_{1},\ldots ,x_{k}\} \subseteq \mathcal{X}$ , and let  $\mathcal{H}\subseteq \{0,1\}^{\mathcal{X}}$ . We say that  $X$  is threshold-shattered by  $\mathcal{H}$  if there exist  $h_1,\dots,h_k\in \mathcal{H}$  such that  $h_i(x_j) = \mathbb{1}(j\leq i)$  for all  $i,j\in [k]$ .
The threshold dimension of  $\mathcal{H}$ , denoted  $\mathsf{T D}(\mathcal{H})$ , is the supremum of the set of integers  $k$  for which there exists a threshold-shattered set of cardinality  $k$ .
The following connection between the threshold and Littlestone dimensions is well-known.
Theorem 3.3 ([She90, Hod97]). Let  $\mathcal{X}$  be a set, let  $\mathcal{H} \subseteq \{0,1\}^{\mathcal{X}}$ , and let  $d \in \mathbb{N}$ . Then:
1. If  $\mathsf{LD}(\mathcal{H})\geq d$  then  $\mathsf{T}\mathsf{D}(\mathcal{H})\geq \lfloor \log d\rfloor$
2. If  $\mathsf{T}\mathsf{D}(\mathcal{H})\geq d$  then  $\mathsf{LD}(\mathcal{H})\geq \lfloor \log d\rfloor$
Item 1 in Theorem 3.3 and Eq. (3) imply that
$$
\forall n \in \mathbb {N}: M (\mathcal {H}, n) \leq 2 ^ {\mathrm {T D} (\mathcal {H})}
$$
for any class  $\mathcal{H}$ . Similarly, Item 2 in Theorem 3.3 and Theorem 3.1 imply a mistake lower bound of  $\Omega (\log \log (\mathsf{T}\mathsf{D}(\mathcal{H})))$ . However, one can do exponentially better than that, as follows.
Claim 3.4. Let  $\mathcal{X}$  be a set, let  $\mathcal{H} \subseteq \{0,1\}^{\mathcal{X}}$  such that  $d = \mathsf{T}\mathsf{D}(\mathcal{H}) < \infty$ , and let  $n \in \mathbb{N}$ . Then
$$
M (\mathcal {H}, n) \geq \min  \left\{\lfloor \log (d) \rfloor , \lfloor \log (n) \rfloor \right\}.
$$
One of the ideas used in this proof appeared in an example called  $\sigma_{\mathrm{worst}}$  in Section 4.1 of [BKM97].
$$
q _ {1}: \quad x _ {\frac {N}{2}}
$$
$$
q _ {2}: \quad x _ {\frac {N}{4}} \quad x _ {\frac {3 N}{4}}
$$
$$
q _ {3}: \quad x _ {\frac {N}{8}} \quad x _ {\frac {3 N}{8}} \quad x _ {\frac {5 N}{8}} \quad x _ {\frac {7 N}{8}}
$$
Figure 2: Construction of the sequence  $q$  in the proof of Claim 3.4.  $q$  is a breadth-first enumeration of the depicted binary tree.
Proof of Claim 3.4. Let  $k = \min \left\{\lfloor \log (d)\rfloor ,\lfloor \log (n)\rfloor \right\}$  and let  $N = 2^{k}$ . Let  $X = \{x_{1},\ldots ,x_{N - 1}\} \subseteq \mathcal{X}$  be a set that is threshold-shattered by functions  $h_1,\dots ,h_{N - 1}\in \mathcal{H}$  and  $h_i(x_j) = \mathbb{1}(j\leq i)$  for all  $i,j\in [N - 1]$ . The strategy for the adversary is to present  $X$  in dyadic order, namely
$$
x _ {\frac {N}{2}}, x _ {\frac {N}{4}}, x _ {\frac {3 N}{4}}, x _ {\frac {N}{8}}, x _ {\frac {3 N}{8}}, x _ {\frac {5 N}{8}}, x _ {\frac {7 N}{8}}, \dots , x _ {\frac {(2 ^ {k} - 1) N}{2 ^ {k}}}.
$$
More explicitly, the adversary chooses the sequence  $q = q_{1} \circ q_{2} \circ \dots \circ q_{k}$ , where 'o' denotes sequence concatenation and
$$
q _ {i} = \left(x _ {\frac {1}{2 ^ {i}} N}, x _ {\frac {3}{2 ^ {i}} N}, x _ {\frac {5}{2 ^ {i}} N}, x _ {\frac {7}{2 ^ {i}} N}, \ldots , x _ {\frac {(2 ^ {i} - 1)}{2 ^ {i}} N}\right)
$$
for all  $i \in [k]$ . See Figure 2.