The Many Faces of Optimal Weak-to-Strong Learning
Authors: Mikael Møller Høgsgaard
Kasper Green Larsen
Markus Engelund Mathiasen
Venue: NeurIPS 2024 poster (2024)
Keywords: Learning Theory
Weak to Strong Learning
Boosting
Large Margin Classifiers
Generalization Bounds
Sample Complexity
Primary area: learning_theory
TL;DR: We propose a new, simpler, faster and optimal boosting algorithm in terms of sample complexity

Abstract: Boosting is an extremely successful idea, allowing one to combine multiple low accuracy classifiers into a much more accurate voting classifier. In this work, we present a new and surprisingly simple Boosting algorithm that obtains a provably optimal sample complexity. Sample optimal Boosting algorithms have only recently been developed, and our new algorithm has the fastest runtime among all such algorithms and is the simplest to describe: Partition your training data into 5 disjoint pieces of equal size, run AdaBoost on each, and combine the resulting classifiers via a majority vote. In addition to this theoretical contribution, we also perform the first empirical comparison of the proposed sample optimal Boosting algorithms. Our pilot empirical study suggests that our new algorithm might outperform previous algorithms on large data sets.

The Many Faces of Optimal Weak-to-Strong Learning
Mikael Møller Høgsgaard Department of Computer Science
Kasper Green Larsen
Department Computer Science Aarhus University larsenEcs au.dk
Aarhus University
hogsgaards@cs.. au.
Markus Engelund Mathiasen
Department of Computer Science Aarhus University markusm@cs. au. dk
Abstract
Boosting is extremely successful idea, allowing to combine multiple low accuracy classifiers into a much more accurate voting classifier. In this work, we present new and surprisingly simple Boosting algorithm that obtains a provably optimal sample complexity. Sample optimal Boosting algorithms have only recently been developed, and our new algorithm has the fastest runtime among all such algorithms and is the simplest to describe: Partition your training data into 5 disjoint pieces of equal size, run AdaBoost each, and combine the resulting classifiers via majority vote. In addition to this theoretical contribution, we also perform the first empirical comparison of the proposed sample optimal Boosting algorithms. Our pilot empirical study suggests that our new algorithm might outperform previous algorithms on large data sets.
Introduction
Boosting is one of the most powerful machine learning ideas, allowing one to improve the accuracy of simple base learning algorithm A. The main idea in Boosting, is iteratively invoke the base learning algorithm A on modified versions
training data set. Each invocation of A returns a classifier, and these classifiers finally combined via majority vote averaging. Variations of Boosting, including Gradient Boosting 100 16 77, are often among the best performing classifiers in practice, especially when data is tabular. Furthermore, when combined with decision regressors as the base learning algorithm, these algorithms are independent of scaling of data features and provides impressive out-of-the-boo performance. See the excellent survey [22] for further details.
The textbook Boosting algorithm for binary classification, AdaBoost [9, works by maintaining a weighing D2 (I), ......... Dj (m)) of a training set $ (I1, 311 VI)...., (Fms Bmn) with (z;, 34)
X' x 1, 1} for an input domain X' and labels (-1, 1,111 1}. In each Boosting iteration f, a classifier S, but with samples weighed according to D.. The weights are then updated such that samples (z;, ys) misclassified by by have a larger weight under D++1 and correctly classified samples have smaller weight. Finally, after a sufficient number of iterations T, AdaBoost combines the classifiers h1,-..,yy hor into a voting classifier f(xx) sign(E; (x)) taking weighted majority vote among the predictions made by the h2 Here the Or s are real-valued weights depending on the accuracy of h2 on D₁-
h4 X {-1,1} is trained to minimize the 0/1-loss
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Woak-to-St Learning. Boosting was originally introduced to address theoretical question asked by Kearns and Valiant [87]8]] on tecak-to-stron learning. A learning algorithm is called a weak learner if, for any distribution DD over X' x {-1,1}, when given some mo i.i.d. training samples S from D, it produces with probability least 1 do. a classiferrhypootheis hş E with erp(hs) S 1/2 % where (h) = Pr(a.))~D [xx)) o y) and H S X - (-1,1} 1] is predefined hypothesis set used by A. A weak learner thus produces, for any distribution D. a hypothesis that performs slightly better than random guessing when given enough samples from D. The parameter is called the advantage of the weak learner and we refer to the weak learner as a 7-weak learner. We think of by and my constants that may depend on H. but not D. A strong learner in contrast, is an algorithm that for any distribution D, and any parameters (E, 5) with 0 < E, < 1, when given m(e, 5) ii.d. samples S from D, produces with probability at least 1 ō a hypothesis hg X {-1.1]
with (hs)s) S 6. Here m(s, 5) is the sample complexity of the strong learner. A strong learner thus obtains arbitrarily high accuracy when given enough training samples. With these definitions in place, Kearns and Valiant asked whether it is always possible to produce a strong learner from a weak learner. This was indeed shown be the case [25], and AdaBoost is one among many examples of algorithms producing strong learner from weak learner.
Sample Complexity. Given that weak-to st-oon learning is always possible, natural question is "what is the best possible sample complexity m(E, 6) of weak-to-strong learning?". This is known to depend on the VC-dimension d of the hypothesis set H. used by the weak learner, as well as the advantage 7 of the weak learner. In particular, the best known analysis (27 of AdaBoost shows that it achieves sample complexity (A,a( 6) of
mAda () = (dln(t((e)))ndd(()()) + In(1/6)
728
Larsen and Ritzert 20 were the first to give an algorithm improving over AdaBoost. Their algorithm has sample complexity (Ln(e,,) 8) of
MLR(E, 8) = (, 1
They further complemented their algorithm with a lower bound proof showing that any weak-to-strong leamning algorithm must have sample complexity m(=, 8) of
m(s, 6) = Ω ( d +++++ ln(1/6) 728
The optimal sample complexity for weak-to-strong learning is thus fully understood from a theoretical point of view.
Other Performance Metrics. Sample complexity however not the only interesting performance metric of for practical purposes. is thus worthwhile to develop alternative optimal weak-to-strone learners and compare their empirical performance.
learner. Furthermore, O-))nnotation may hide constants that too large
The algorithm of Larsen and Ritzert for instance has a rather slow running time as it invokes the weak-learner a total of O(mlk4 8x-2 In m) = 0m. times on training set of m samples. This should be compared to AdaBoost that only invokes the weak learner 0(- 2 In m) times to achieve the sample complexity stated in (.]
An alternative sample optimal weak-to-strong learner was given by Larsen 19 proof that Bagging (5) is an optimal PAC learner in the realizable setting. Concretely, his work gives a weak-to-strong learner with an optimal sample complexity while only invoking the weak-learner a total of 0(>- In (m/m/) In m) times on training set of m samples.
corollary of
A natural question is whether the sample complexity of AdaBoost shown in can be improved to match the optimal sample complexity by a better analysis. Since AdaBoost only invokes its weak learner 0 In m) times samples, this would be even more efficient optimal weak- to-strong learner. Unfortunately, work by Høgsgaard et al. [15] shows that AdaBoost's sample complexity is sub-optimal by at least In(1/e) factor, i.e.
(1/=() 728
In(1/8)
mAda 8) =
(2)

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It thus remains an intriguing task to design weak-to-strong learners that have optimal sample complexity and yet match the runtime guarantees of AdaBoost. Furthermore, do the theoretical improvements translate to practice? Or are there large hidden constant factors in the O())-ootation? And how does vary among the different weak-to-strong learners?
1.1 Our Contributions
In this work, we first present new weak-to-strong learner with optimal sample complexity (at least in expectation). The algorithm, called MAJORITY-OF-5 and shown as Algorithm [ is extremely simple: Partition the training set into disjoint pieces of size m/5 and run AdaBoost on each to produce voting classifiers fi...., .......fs.. fs. Finally, combine them by taking a majority vote g(z) sign( (ii= (x)). This simple algorithm only invokes the weak learner O(y In m) times, asymptotically matching AdaBoost and improving over previous optimal weak-to-strong learners. Furthermore, since each invocation of AdaBoost is on training set of only m/5 samples, it is at least as fast as AdaBoost, even when considering constant factors. It is even trivial to parallelize the algorithm among up to 5 machines/threads.
Algorithm 1: MAJORITY-OR-5(5,,
Input:
Training set S (x1 (1),
(Imm Um). Weak learner W.
Result: Hypothesis g X' 1-1.1}.
Partition S into 5 disjoint pieces S Ss of size m/5.
2 for { ,....,5 do
Run AdaBoost on S. with W to obtain fe: X + {-1,1}.
4 9 4 *g+sign( (5, ft).
5 return g
The concrete guarantees we give for MAJORITY-OF-5 are as follows
Theorem 1. For any distribution D over X' x (-1, 1} and any "y-weak learner W using hypothesis set H of VC-dimension d, it holds for a training set $ ~ Dm that rumning MAJORITY-OF-5 on S to obtain hypothesis 9 satisfies
E(erp(g)) = 0
d
72m
In particular, Theorem Immpies that Egerp(g)) < E when given m = ((d/(??=)) samples. This is a slightly weaker guarantec than the alternative optimal weak-to-strong learners in the sense that do not provide high probability guarantees (i.c. with probability - 8). On the other hand, our algorithm is extremely simple and has running time comparable AdaBoost. Furthermore, the proof that AdaBoost is sub-optimal (see (2) shows that even the expected error of AdaBoost is sub-optimal by logarithmic factor. It is interesting that combining constant number of voting classifiers trained by AdaBoost makes it optimal when a single AdaBoost is provably sub-optimal. Let us also comment that the analysis of Algorithm [ is based recent work by Aden-Ali et al. [1 optimal PAC learning in the realizable setting, demonstrating new applications of their techniques. Furthermore, we believe the number is an artifact of our proof and we conjecture that can be replaced with by giving a better generalization bound for large margin voting classifiers. See Section for further details.
Empirical Comparison. Our second contribution is pilot empirical study, which gives the first empirical comparison between the alternative optimal weak-to-strong learners, both the algorithm of Larsen and Ritzert (20). the Bagging+Boosting based algorithm [19], our new MAJORITY-OF-. algorithm, as well as classic AdaBoost. We give the full details of the alternative algorithms in Section In our experiments, we compare their performance both on real-life data as well as the data distribution used by Høgsgaard et al. [15) in their proof that AdaBoost is sub-optimal as shown (2) Our pilot empirical study give indication that our new algorithm MAJORITY-OF-5 may outperform previous algorithms on large data sets, whereas Dagging+Boostie performs best on small data sets. See Section 4 for further details and the results of these experiments.
3

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2 Previous Optimal Weak-to-Strong Learners
In this section, we present the two previous optimal weak-to-strong learners. The first such algorithm. by Larsen and Ritzert (21), builds on a sub-sampling technique due to Hanneke [14] in his seminal work on optimal PAC learning in the realizable setting. This sub-sampling technique, named SUBSAMPLE, is shown as Algorithm
Algorithm 2: SUBSAMPLE(S, 7)
Input: Training set S, Stash T
Result: List of training sets L.
1 if [||] < then
Let L contain the single training S U..
3 return L
4 Partition S into 4 disjoint pieces So, S1, S2, S3 of size |S|/4 each.
5 Let L be empty list.
6 Append SUBS AMBLEEM) TU U 53) to L.
Append SUBSAMPLE(S,, TUS; S3) to L.
8 Append SUBSAMPLE(Ss, TUS, U2) to L.
9 return L
Given a training set S. SUBSAMPLE generates a list L of subsets S; C S. The list has size m m.79 when invoking SUBSAMPLEES, t) for training set S of size Larsen and Ritzert now give an optimal weak-to-strong learner using SUBSAMPLE as sub-routine as shown in Algorithm
Algorithm 3: LARSENRITZERT(S) W)
Input: Training set S (x1, V1) ..... (Im Mm). Weak learner W.
Result: Hypothesis g X'' + 111,1]
Invoke SUBSAMPLE(S, 0) obtain list L S Sk.
2 for f .....,K k do
Run AdaBoost on S₁ with W to obtain f:: 1-1,1}.
4 9 4 sign 3, ft).
5 return g
Finally, the algorithm by Larsen 19 based on Bagging (a.k.a. Bootstrap Aggregation) by Breiman (5) combines AdaBoost with sampling subsets of the training data with replacement. Unlike the algorithm above by Larsen and Ritzert, it requires target failure probability as input. The algorithm is shown as Algorithm
Algorithm 4: BAGGEDADABOOST(S.W..
Input: Training set (x1. ..... (I' Vm). Weak learner W. Failure probability 0< 8 1. Result: Hypothesis g: x
I for / 1,..., (l)(m/))) do
2
Let S, be a set of m independent samples with replacement from S.
Run AdaBoost on S, with W to obtain fc : X (1,
4 9 + 00s-igg(> (E, ft).
5 return g
Analysis of MAJORITY-OF-5
In this section, we give the proof of Theorem showing that our new algorithm MAJORITY- OF-5 has an optimal expected error. Before giving the formal details, we present the main ideas in our proof. Our analysis is at high level inspired by recent work of Aden-Ali et al. || for

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realizable PAC learning. The first ingredient we need is the notion of margins. For a voting classifier
f() = sign(Enn h 0izh(z)) with a, 2 0 for all h. consider the function f'(x) = EEEE aj (()(()) with a/, an. Ln ath That is. f is simply the voting classifier f without the sign(-) and normalized to have coefficients summing to 1. The margin of f on a sample (zz, c(x)) is then f(x) E [-11, 1]. The margin is if all hypotheses combined by f aggree and are correct. is 0 if half of the mass is on hypothesis that are correct and half of the mass is on the hypothesis that are wrong. We can thus think of the margin as a confidence of the voting classifier. Margins have been extensively studied in the context of boosting and were originally introduced to give theoretical justification for the impressive practical performance of AdaBoost (2]. In particular, there are strong generalization bounds for voting classifiers with large margins (2 6 11. Indeed, the best known sample complexity bound for AdaBoost, as stated in is derived by showing that AdaBoost produces a voting classifier with margins (2(7).
Returning to our outline of the proof of Theorem ! recall that the optimal error for weak-to-strong learning as a function of the number of samples m is ((d((?²m)). Now assume we can prove that for a set of m i.i.d. samples from distribution D, the expected maximum error under D of any voting classifier that has margins 22(7) on all the samples, is no more than (l (()d/()?m)) This fact follows from previous work on Rademacher complexity. Note that this is sub-optimal compared to our target error by polynomial factor since v 2 x for x between and 1. We want to argue that combining 5 instantiations of AdaBoost on disjoint training sets reduces this expected error to optimal ((d/(?²))).
For this argument, consider running AdaBoost on n = m/5 samples, For any x E X, consider the probability pa = Prs-D= (fs(()) of c(x)) where fs is the hypothesis produced by AdaBoost on S and c(x) is the correct label of ær Inspired by Aden-Ali et al. M. we now partition the input domain X into sets R,, such that R3 contains all for which Ps E (2 2.22-50 #+1]. The crucial observation is that if we consider & independently trained AdaBoosts, then the probability they all err on x is precisely pt- Since a majority vote among classifiers only fails when at least 3 of the involved classifiers fail, combining 5 AdaBoosts intuitively reduces the contribution to the expected error from points x E R2 to Prx~D[[ [X E R. What remains is thus to argue that Pr[X E R.] is small.
This last step is done by considering the distribution D;, which is D conditioned on receiving a sample from R2 The expected number of samples we see from R2 is m, Pr[X E Ralm. Furthermore, since AdaBoost obtains margins 2(7) on all its training data, it in particular obtains margins 12(7) on all its samples from D. This leads to an error probability of p: = ((vd/((22))) under DD But the definition of R2 implies p. 2- Hence vd/(?2m.) = (22-") + m, d(2")d/²²) + Pr[X
R.] = ((22)d/ d/(22m)). By summing over all R,, the final expected error is hence
00
22 31
32 =O(d/(²m)))
This completes the proof overview. Let us end by making a few remarks. First, it is worth noting that the generalization bound O (() seems much worse than the bound in (T) claimed for AdaBoost and other voting classifiers with large margins. Unfortunately, if we examine ( carefully and state as a function of m, we get e = O(d (dIIn(m))) (n(m/d) log-factors are not bounded by polynomial in d/ d/(r²m). In particular for m =Cd/-² with C > 0 a constant, any polynomial in d/ d/(?2m) must be constant. But In = In In(Cd/72) is not a constant independent of d and 7. Thus we have to use the generalization bound with v- that fortunately is within a polynomial factor of optimal for the full range of m. Let us also comment that if the lower bound for AdaBoost stated in (2) is tight also for m-margin voting classifiers, i.c. matched by an upper bound, then it suffices to take a majority of 3 AdaBoosts for optimal sample complexity.
. The problem is that the two
This concludes the description of the high level ideas in our proof.
3.1 Formal Analysis
We now give the formal details of the proof. We start by introducing some notation.
5

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Preliminaries. For hypothesis set H. we let A(H) denote the set of linear classifiers using hypothesis from that
A(H) - 1E(-1.)); E
E
hER
NEW
Note that we have termed these linear classifiers rather than voting classifier, to distinguish that we have yet applied sign(-) and insist on normalizing the coefficients so they sum to We define the sign function as being when the value is non-negative (so also 1 when x = 0) and -1 when negative.
We let c E {-1,1} be true labeling that we are trying to learn. For a distribution D over X. and we
we define the expected loss of f [-1, will use S E X²m to denote a point set of m i.i.d. samples from D i.e. S Dr (we see the point set vector so allow repetition of points). Define for any h N the majority of l linear classifiers f f& E [-1, as Maj(fi.... fx)(2) sign(()
erp(f) := EX-D (L{sign(f))XX) * c(X))
Let s be a set of points, i.. s E - and c(s) [-1.1)1t the labeling of the points that contains with that c(s); = c(s;). We define a 7-margin classifier algorithm f U
- (X' (-1, [-1, 1]* to be a mapping that takes input a point set with labels (s, ((s) E E (X (11, and outputs a function f(ss, (-)))()) from X' to the interval [-1, where f(s, ((s))())))) is such that for x E 8, 7. In the following we will fa denote f's c(s)) and write f A(H)
[s, (z))()))
for any (s, c(s)) E 1 (X' (-1,1})) we have that the output of the y-margin classifier algorithm
f(s, c(s) is in (HH).
Analysis. We now prove Theorem 1 which a direct consequence of the following Corollary
Corollary 2 states that running y-margin classifier algorithm on 5 disjoint training sets of size m
and forming the majority vote of the produced classifiers, has the optimal (dd((?²m) expected
error. Since AdaBoost, after (In(()))))))))) (mlnm²)))))))))) iterations, has (?()) margins on all points (26) [Section
5.4.1J. Corollary of AdaBoost. The proof of Corollary 2 follows the method used in ( where the authors show similar bound for PAC learning in the realizable setting. We now state Corollary 2
the claim in Theorem] Alternatively one could run AdaBoost (24] instead
Corollary 2. For any distribution D over X. hypothesis set H with VC-dimension i.i.d. point sets
St...., Ss, from D margin O < < and J' ((H) being a 7-margin classifier algorithm, we
have that
Se (er (Maj(Js Sss))) =
7
The result Corollary 2 is primarily a consequence of the following Lemma 3 which says that, in expectation, the probability that Js, J'g. Jss all misclassifying sample from 'D is (7d We now state Lemma 3 and give the proof of Corollary We postpone the proof of Lemma to later in this section.
Lemma 3. For any distribution D over X, hypothesis set H with VC-dimension d, i.i.d. point sets S1. S2, S3 from D margin O S 1 and A(H) being y-margin classifier algorithm we have that
(sign (ignnsss (xx))) - (14m
Es Sg,
PX
7
For the majority of fs....., Ss, , to fail on an x X. must the case that at least 3 of the trained y-margin classifiers Js, have sign(Js, (x)) o c(r). Using this combined with the S, being i.idd. we get that
Proof of Corvllary 2
Es SemD (erD (Maj(Js,.... [s:))]
Σ
Es₁ SygD [P (sgnns,, (X)) # x[]]]]
15h <h2
Px S.S
I (sign( x(x))
Σ
1.
15h
Now using Lemma and (2) = =1(), we get that the above O
as claimed.
7m

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We now move to prove Lemma 3 For this we need Lemma 4 Which we now state and give the proof in Appendix A
Lemma 4. Let
denote universal constant. For D distribution, R a subset of X such that P [R] = FX-D X E R] + O, hypothesis set H with VC-dimension d, S a point set of m i.dd points
from D, margin 0 7 1 and f E ((H) being "y-margin classifier algorithm we have that
ad Es RX [Xfs(X) + c(X))l] Es (erD, V |
where we use Dn to denote the conditional distribution on the subset R. That is, for any measurable function g. Ex~Da [g(X)] Ex~D [g(X)1(X RR) (R]//Px-5p [X E R).
Proof of Lemma 3 For E X' let P. = ES-DD [={fs(æ) * c(x)] and define for i = 1.... the sets Ri {x E X' Pe E . 2- Now using Tonelli, and that S₁, S2, S3 are i.i.d. with distribution D and that Ps S 2-++ for x E R₁ we get that
Sp PX [ (X) # c(x)]]]
=ExaD Sp
[1 [ss, (X) + d()))]
DO
=ExDD [p}]
Σ
Ex-D [xx x E PIX Rd] S 23
Σ 2 3 Ri).
thus if we can show that there exists universal constant d' such that PXWD (X Ro} < we get that
d'd2at 7
Sg PXND (fs, # c(x)]]] s 23 dd 2-'=
m
m2m
and we are done. Thus assume for contradiction that PX~D [X Rd] chosen large enough. Using Lemma we have that there exist a universal constant a > such that
d'd2a 3mm for c' > 1 to be
ad
EsaDm [erpa, (Js)] S
P 22,
By Tonelli, the definition of ps and that for x E Rí we have Ps E (2-1,2- 2-i+1] we get that
ES-D" [erD,, (fs)] E EX~DR; [ES-DM [L(s(() c())]]] BX~DA, [px] 2 2
Combining the above lower and upper bound on ES-D= [erDa] and PX-DD [X E R.j get that
2 7m
ad
ad
2'
S
2
P P
d' 7m
72m
c
which for c' sufficiently large strictly less than thus we reached a contradiction. Hence must be the case that IP PX~DD [X E R&]
<
c'aza: which concludes the proof of Lemma 7
Experiments
In this section, we present the results of our pilot empirical study between the different sample optimal weak-to-strong learners. We compare the algorithms on five different data sets. The first four are standard binary classification data sets and are the same data sets used in [12, whereas the last a synthetic binary classification data set developed from the lower bound [I5] showing that AdaBoost sub-optimal. For all real world data sets, we have shuffled the samples and randomly set aside 20% to use as test set. The weak learner we use for these is the scikit-learn Decision TreioooTeeelaaaa aasiilaassiii ier with max_ depth11.. This is default for the implementation of AdaBoost in scikit-learn, which the implementation used in our experiments. We describe the data sets in greater detail below.
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This data set represents measurements from particle detectors, and the labels tells whether they come from a process producing Higgs bosons or if they were a background process. The data set consists of 11 million labeled samples. However, we focus on the first 300,000 samples. Each sample consists of 28 features, where of these are derived from the other 21.
Higgs [29]:
In this data set, we try to distinguish electron neutrinos from muon neutrinos. The data set consists of 130,065 labeled samples. Each sample consists of 50 features.
Boone [33:
Forest Cover [ In this data set, we try to determine the forest cover type of 30 30 meter cells. The data set actually has 7 different forest cover types, so we have removed all samples of the 5 most uncommon make it into binary classification problem. This leaves us with 495,141 samples. Each sample consists of 54 features such as elevation, soil-type and more.
Diabetes (28) In this data set, we try to determine whether patient has diabetes or not from features such as BMI, insulin level, age and so on. This is the smallest real-world data set, consisting of only 768 samples. Each sample consists of 8 features.
Adversarial [5] This data set, as well as the weak learner, have been developed using the lower bound instance in [T]. Concretely, the data set consists of 1024 uniform random samples from the universe X' {1,.... 350). Every element of the input domain has the label 1, but all weak-to-stron learners are run simply by giving them access to a weak learner. The weak learner is adversarially designed. When it is queried with weighing of the training data set, it computes the set T containing the first 20 points from the input domain that receive zero mass under the query weighing. It then searches through set of hypotheses (chosen randomly) and returns the hypothesis with the worst performance on T, while respecting that it must have error more than 1/2 7 under the query weighing and having at least 1/2 + y error on T. Finally, the hypothesis set contains an additional special hypothesis by that is correct (returns 1) on all but the last 20 points of the input domain. This hypothesis is used to handle queries where none of the randomly chosen hypotheses have advantage 7. We refer the reader to [15] for further details and intuition on why this construction is hard for AdaBoost.
The data sets represent both large and small training sets. For each data set, we run simple AD- ABOOST (accuracy shown as a blue horizontal line in the plots), BAGGEDADABOOST (Algorithm { LARSENRITZERT (Algorithm)) 3) and our new MAJORITY-OF-X (for X varying from 3 to 29). In our experiments, we vary the number of AdaBoosts trained by each weak-to-strong learner from 3 to 29, instead of merely following the theoretical suggestions. Each of these voting classifiers then trained for 300 rounds on its respective input. This has been repeated 5 times with different random seeds, so the plots indicate the average accuracy across these runs. For Algorithm 3 that creates m/ sub-samples, we use the full set of sub-samples on the two small data sets Diabetes and Adversarial. For the three large data sets Higgs, Boone and Forest Cover, this creates a huge overhead in running time and we instead randomly sample without replacement from among the sub-samples resulting from the SUBSAMPLE procedure (Algorithm 2). This is the reason for the non-constant behavior of this algorithm in the corresponding experiments. For AAGGEDADABOOST we have chosen to sample 95% of the samples (with replacement) in our experiments. The results of the experiments on the three large data sets are shown in Fig. I
The results in Fig. D gives an initial suggestion that our new algorithm with disjoint training sets might have an advantage on large data sets. Quite surprisingly, we see that the two other optimal weak-to-strong learners perform no better, or even worse, than standard AdaBoost. Experiments on the small Diabetes data set, as well as the Adversarially designed data set, are shown in Fig.
The results in Fig. 2 suggest that our new algorithm may perform poorly on small training sets. This makes sense, as the training data for each weak learner is extremely small on these data sets. Instead, we find that the Bagging based variant outperforms classic AdaBoost. Since Bagging has a relative small overhead compared to simple AdaBoost, this suggests running both our new algorithm MAJORITY- OF-X and BAGGEDADA BOOST and using a validation set to pick the best classifier. We hope these first experiments may inspire future and more extensive empirical comparisons between the various weak to strong learners.
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0.635
ADABoOST
LARSENRITZERT
0.69-1
MAJORITY-OFF OF
BAGGEDADABOOST
Test Accuracy
0.603
0.602
0.691
0.690
3
11
15
21
29
Number Voting Classifiers
ADABOOST
0.774
0.923
LABSENHITZERT
MATORITY-OF-X
ADABooST
Accuracy
0.922
BAGOE
Test Accuracy
0.772
LARSENBITzRR
MATORITY-OP-X
0.921
0.770
BA
0.920
0.768
0.919
15
Number of Vating Classihers
Number Vating Classiheess
Figure 1: Top is Higgs, Left plot is Boone. Right plot is Forest Cover
0.80
ADABOOST
0.95
LARSENRITZERT
0.79
MAJOBITY-OF-I BAGCHEAAABBOSS
0.90
Test Accuracy
Test Accuracy
BoosT LABSENHITZERI
0.85
0.78
01500
MAJoRITy-o--
BAGGEDADABoOST
0.75
0.77
0.70
0.65
Number of Voting Classifiers
Number of Vating Classifiers
Figure 2: Left plot is Diabetes. Right plot is Adversarial

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Limitations
Our main results are proved under the theoretical assumption of i.idd training samples as well as access to weak-learner that always obtains an advantage of over random guessing. Since these might not be realistic assumptions, we also performed an empirical evaluation of our algorithm. Due to computational constraints, we have only been able to run experiments on 5 data sets. We have thus been careful not to over-emphasize the practical implications our results. In circumstances, we View the theoretical contributions the main novelty of this work.
Acknowledgment
This research is co-funded by the European Union (ERC, TUCLA, 101 125203) and Independent Research Fund Denmark (DFF) Sapere Aude Research Leader Grant No. 9064-00068B. Views and opinions expressed are however those the author(s) only and do not necessarily reflect those the European Union the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
References
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DOI:
htpp://chi.ng//0.2433//VVV11

---

A Proof of Lemma 4
We now give the proof of Lemma For this we need the following Corollary | that gives a high probability bound on the error for all classifiers in (HH) that have margins on a training set S.
Corollary 5. Let C > 1 denote a universal constant. For hypothesis set H with VC- YC-dimension d, distribution D, margin < ? S failure probability 0 < 8 1, and point set S D we have with probability at least 1 - over S, that any f E (H) such that f(z)e(x) 2 foror all x E S satisfies
2Cd 72m
2 (2/6)
erd(f) -
Corollary 5 follows for instance by modification of [26] [page 107-111] due to 3 and using the
stronger bound on the Rademacher Complexity of O O(vd/m) due [8 [See e.g. theorem 7.2 [3] ]. With Corollary in place we now give the proof of Lemma
Proof of Lemma
If R m S d/y² we have that
2 1,
2P [ m
and the claim holds since erp, (fs) is always less than 1. Thus we assume from now on that
P [ m d/7². We define the events Eg U RH = i} for 2 i > P [R] m/2 and E = U222tR/m/222 Eg = { n R| 2 IP [R] m/2]. We further define X, = 1[5, E R} for j = 1,.... m and notice that these are i.i.d. {0, }-randoom variables where X = Σ, X, has
expectation P [[] m and the event E, Xj 2 P [R] m/2 is contained in E. Thus by a Chernoff bound and that exp(-x) < 1/x x/ for x > 0, we get that
8
Ps E
1 - exp (-P(R) 2 P P m
We thus have Ps E
< 8/ (P [ m). Since we assumed that IP [R]] 2 d/>² 2 and using
0 f < V for 1, this further implies that
Ps S8
TT R m
We will soon show that Es (erD; Eg < 16 constant C² >
7= for m 2 i2 TP [7t m/2 for a universal
16CH
Now using these two relations combined with the law of total expectation on the partition E, Erjijj///..... Em and that CTD, I 1, we get that
Es (fs)
Σ
Es erD, (fs) E. Ps [.]
Es (erp, (fs) E Ps
R2
16Cd
16
P E
+8
PP R
7 PP [
16Cd
< 24
7P [R] m
as claimed in Lemma 4 with the universal constant 242 16C. Thus we have to show that
Es (Js)( < 16 *F non-negative random variable, have that
77 for m 2 1 [R] m/2. consider such an i. Since erp, (/p)
Es (fs) E.]
Iss ler (fs) 2 E.] dæ
12

---

We will thus upper bound this integral. Now conditioned on E,, we know that S contains i points that are samples according D and that fs on these examples has all margins at least y. Thus we have by Corollary that with probability at least å over S, it holds that
8Cd 8n (2/8)
erDx
max
724
where C 2 1 is universal constant. For ease of notation let r.
8Cd V
We notice that
7
Ti S
10C4 P since is assumed to be greater than IP [[t] m/2. Using this, we get that
Ps (fs) 2 x Ed] de S +
Ps (fr) 2 x] dx
(4)
Jr
S
16Cd
+
| Ps (er) 2 x] dr.
(5)
7PP[M]m
Thus if we can show that
Ps (fs) x] dx S 15 V d
get by combining this with Eq. 4 that
Es (fs) E.] < 16
I6cd 7
which would conclude the proof. Thus we now show 1,"
Ps [OrD, (fs) (fs) 2 2/ 15 V d
For this, we do the following non-trivial rewriting of a to make it resemble the second term in the max appearing in (3)
2
8ln
i-1
2exp
(-g2t
Now for any æ 2 Ti we have that
Ps jerpa (fs) 2 x E,] Ps (erD. (fs) 2 max(r;, x) Ed),
which combined with the rewriting of x and Eq. (3) with 5 = 2 exp first argument of the max in Eq. B3. we get that
and noticing r; is the
-22
Ps (crp)(fs) 2 x Ej} Fs (fss 2 max(r;,a)
E,] 2 exp
for any ri. Now using the density function of normal distribution with mean O and standard deviation is equal to (xp(-)(</))))/( (2T), and letting N(0, o) denote normal random variable with mean and standard deviation o, we get that
-x2
Ps (erD, (fs) 2 x Ed do <
2exp
dx
8
$ 22/8/ 8//28/2222)) 25 1
1
x
exp
dr
/8/(2))/2r
2
(8/(2i)
8
d
< 2 / P
N
0.
2n
8 (P [ m)
15
2i
21P [ m
where the second to last inequality follows from i d/72 2 1.
> P [R] m/2 and the last inequality by P R
>
13

---

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Reviews:
[{"review_id": "3tvtCvfWPT", "reviewer": "Reviewer_ynbn", "summary": "The authors present a new boosting algorithm: partition training data into 29 pieces of equal size, run AdaBoost on each, and output the majority vote over them. The authors prove that the sample complexity of MajorityVote29 is optimal and its running time is the same order as AdaBoost. Experimental results are also attached, which corroborate their theoretical findings.", "questions": "N/A", "limitations": "N/A", "rating": 7, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- Very strong and interesting result\n- Mathematically sound, based by my judgement\n- Good presentation, self-contained and well-structured", "weaknesses": "N/A", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "vSepSGuYe8", "reviewer": "Reviewer_KvEC", "summary": "This paper introduces a new Boosting algorithm, MAJORITY-OF-29, which achieves provably optimal sample complexity and is remarkably simple to implement. The algorithm partitions the training data into 29 disjoint subsets, applies AdaBoost to each subset, and combines the resulting classifiers through a majority vote. This approach not only matches the asymptotic performance of AdaBoost but also improves upon previous weak-to-strong learners in terms of simplicity and runtime efficiency.", "questions": "Please see the weakness.", "limitations": "Limitations are discussed.", "rating": 3, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 1, "strengths": "1. The paper introduces a novel method and provides detailed theoretical analysis.", "weaknesses": "2. Existing experiments fail to demonstrate the effectiveness of the proposed method, and there is a lack of analysis and discussion on current experimental results.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Regarding the question about experiments: Let us first re-iterate that our main focus in this work is on the theoretical results, which we believe are strong (the paper is also submitted with a primary area of \"Learning Theory\").  Perhaps we were not clear enough when claiming that our experimental results should be seen as \"indications\" (as phrased in Section 1 Empirical Comparison), and thus the experiments should be seen a small complement to our main focus the theoretical bound. We also try to allude to this in section 5 (Limitations). However it is seems like we have not made it clear enough that this is only a pilot empirical study as reviewer p9UC also points out the same things - and that the paper's main contribution is the theoretical result. We thank you for pointing this out and will make it clear in following iterations of the submission. \n\nWe were not certain about, if effectiveness in the comment \"Existing experiments fail to demonstrate the effectiveness of the proposed method\" refers to the reported Test Accuracy or the running time of the algorithms so we have given an answer to both in the following: \n\nTest accuracy: The experiments do not strongly indicate that the Majority-OF-29 (now Majority-OF-5) outperforms other optimal methods LarsenRitzert, BaggedAdaBoost. However, based on the theoretical results, it is also not expected that it outperforms them in terms of accuracy, as they are all asymptotically optimal. The benefit of our new algorithm is that it is simpler, easier to implement and more efficient in terms of computation.\n\nRuntime: For a runtime comparison, we have chosen not to directly include this. One reason for this, is that it will be heavily implementation dependent. In particular, if all invocations of AdaBoost are done in a single thread, then our algorithm will surely be faster in practice, as each data point is only included in precisely one AdaBoost invocation. For the other optimal methods, data points are included in many invocations. However, if properly multi-threaded, it is conceivable that other methods could be made as (or nearly as) efficient as our new algorithm.\n\nEven though we don't feel comfortable reporting the runtimes in the article, the time it takes to fit the given number of AdaBoosts in AdaBoost, Majority-of-X, LarsenRitzert, Bagging can be found in the zip file attached to the article in the folder \"results\", where for a given dataset and run, the time variable $\"fit$_$time\"$ describes the time it takes to fit the given number of AdaBoosts in a given a run.", "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": "JBrAWjWhFm", "reviewer": "Reviewer_p9UC", "summary": "This paper presents an efficient and simple weak-to-strong learner that has optimal in-expectation error. In weak-to-strong learning, we are given a dataset of $m$ points from a distribution, and a $\\gamma$-weak learner that returns hypotheses from a class of VC dimension $d$. AdaBoost, which is a textbook weak-to-strong learner, makes $O(\\ln(m)/\\gamma^2)$ total invokations to the weak learner, and the best-known analysis for it shows that it suffers an in-expectation error $O\\left(\\frac{d\\ln(m/d)\\ln(m)}{\\gamma^2 m}\\right)$. Larsen and Ritzert (2022) constructed a weak-to-strong learner, that has expected error $O(d/\\gamma^2 m)$. Furthermore, they showed that this is the optimal error that one can obtain from $m$ training examples and a $\\gamma$-weak learner. However, the weak-to-strong learner by Larsen and Ritzert (2022) makes $O(m^{0.8}/\\gamma^2)$ invokations to the weak learner --- which is exponentially worse than AdaBoost. Another bagging-based-boosting algorithm due to Larsen (2023), which also achieves the optimal expected error of $O(d/\\gamma^2m)$, makes only $O((\\ln m)^2/\\gamma^2)$ invokations to the weak-learner. This is still a log factor worse than AdaBoost. Could we then hope to obtain a tighter analysis of the error of AdaBoost, and show that it obtains the optimal error with only $O(\\ln(m)/\\gamma^2)$ invokations to the weak learner? Unfortunately, no. Høgsgaard et al. (2023) showed that AdaBoost necessarily suffers an expected error which is at least $\\Omega(d\\ln(m)/\\gamma^2 m)$.\n\nCan we then at least shoot for a different weak-to-strong learner that attains the optimal expected error of $O(d/\\gamma^2m)$, and also invokes the weak learner only $O(\\ln(m)/\\gamma^2)$ many times (which is the AdaBoost gold standard)? This paper answers the question in the affirmative, with a remarkably simple weak-to-strong learner that they call Majority-of-29. The algorithm is exceedingly simple to describe: Partition the training dataset into 29 disjoint sub-samples of size $m/29$ each. Run AdaBoost on each subsample, and return the majority vote over the AdaBoosts. Since each AdaBoost makes only $O(\\ln(m)/\\gamma^2)$ calls to the weak learner, and we run a constant (29) many AdaBoosts, the total number of calls to the weak learner is $O(\\ln(m)/\\gamma^2)$ as required. Further, using an analysis similar to the recent majority-of-3-ERMs algorithm of Aden-Ali et al. (2023), the authors are able to show that the expected error of Majority-of-29 is $O(d/\\gamma^2m)$. The analysis from that work does not extend in a trivial manner, and the authors are required to make appropriate technical modifications and enhancements. The number 29 emerges from the analysis --- the authors require showing a new generalization bound for margin-based classifiers (they show a generalization bound of the order $O((d/\\gamma^2m)^{\\alpha})$), for $\\alpha=1/14$, and this lets them obtain the result for Majority-of-$g(\\alpha)$, where $g(\\alpha)=2/\\alpha+1$. The authors conjecture that the analysis of the generalization bound could be improved, and a Majority-of-3 might well suffice for optimal error.\n\nFinally, the authors also do a (somewhat-limited) empirical comparison of the the performances of the three optimal weak-to-strong learners mentioned above (LarsenRitzert, Bagging-based-boosting, Majority-of-29) as well as AdaBoost. The authors find that for large datasets, Majority-of-29 outperforms the other optimal weak-to-strong learners. On the smaller datasets, the authors find that Bagging-based-boosting outperforms Majority-of-29.", "questions": "1) Do we have reason to believe that $O(\\ln(m)/\\gamma^2)$ calls to the weak learner is indeed the best gold standard we can hope for? To my understanding, the reason we need $O(\\ln(m)/\\gamma^2)$ calls to the weak learner in AdaBoost is because we want to use a margin-based generalization bound that expects a classifier to have at least $\\Omega(\\gamma)$ margin on every training sample---AdaBoost attains this guarantee only after $O(\\ln(m)/\\gamma^2)$ iterations. But could it perhaps be possible that there is a weak-to-strong learner out there that attains optimal error of $O(d/\\gamma^2m)$ with $o(\\ln(m)/\\gamma^2)$ calls to the weak learner?\n\n2) In the Experiments section, the x-axis in Figures 1 and 2 varies the number X of AdaBoosts trained on disjoint partitions in the Majority-of-X algorithm. But this is not a parameter in the other algorithms (BaggedAdaboost and LarsenRitzert). Hence, I would have expected to see a constant line for these other algorithms in the plots (like how the red and blue lines are constant in Figure 2). Why are there different numbers corresponding to different number of voting classifiers in BaggedAdaboost and LarsenRitzert in Figure 1 (and also for BaggedAdaboost in Figure 2)? Am I missing something?\n\nMinor/Typos: \\\nLine 133: It is 0 if half of hypotheses are  correct and half are wrong --- this is only true **in a weighted sense** right? \\\nLine 299: this suggests*", "limitations": "The authors have adequately addressed any limitations that I can foresee.", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The weak-to-strong learner that the authors propose is optimal, and also requires the fewest calls to the weak learner among all optimal weak-to-strong learners that we know. More importantly, it is exceedingly simple and elegant. It also empirically outperforms the other optimal weak-to-strong learners (at least in the experiments performed by the authors). It is also nice to see that the analysis technique from Aden-Ali et al. (2023) finds new applications. The paper is well-written, sets up the stage (along with relevant prior work) well in the first two sections, and provides a nice high-level summary of the formal analysis in Section 3.", "weaknesses": "While the theoretical contribution is substantial and undeniable, arguably, the experimental section is extremely limited (which is okay, and the authors admit this at the end, but this is still a limitation, especially if we want to draw conclusions about the empirical performance of the different weak-to-strong learners). The authors only perform experiments on 4 real-world datasets---there are admittedly many more out there, even just in the UCI repository. Could the authors at least elaborate on their rationale behind choosing the datasets that they did? (e.g., was it a random subset of 4? was it the first 4? was it the best 4 from 20 that they observed this trend on?) How might one believe that there is no cherry-picking of datasets involved? The authors make two conclusions from their experiments: 1) on larger datasets, Majority-of-29 outperforms both Bagging-based-boosting and LarsenRitzert. 2) on smaller datasets, Bagging-based-boosting outperforms Majority-of-29. Importantly, the former conclusion is drawn from results on just 3 datasets, and the latter is drawn from just 1! This can really make one skeptical about whether they should truly believe these conclusions. It is okay that this is just a pilot empirical study, but such claims call for significantly larger empirical validation. Also, please see the questions below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank you for taking the time to thoroughly assesing the article, asking interesting questions, and suggesting concrete improvements. \n\nExperiments:\nAs allotted to in the answer to reviewer KvEC, and as you correctly point out, we should have made it more clear that the experiments are very much a pilot. As you also point out, the main contribution of the paper is the theoretical result.\nRegarding the choice of the 4 real-word-datasets, we simply chose the same as used in \"Optimal Minimal Margin Maximization with Boosting\" by Gr\\o nlund et al. (ICML'19). We did not run on any other data sets and no results were left out, so we wouldn't say that there was any cherry picking done. \n\nLet us also comment on the reason for not conducting more experiments. We are in a theory research group and do not have access to other machines than our laptops. Hence conducting a large number of experiments was simply infeasible. As we also comment on for reviewer KvEC, if you all feel that the paper is stronger as a pure theory paper, we are okay with removing the experiments. In all circumstances, we will further downtime the significance of our experiments.\n\n\n(Question 1) Number of calls to weak learner: This is a very interesting question. Classic work by Freund (Y. Freund. Boosting a weak learning algorithm by majority. Information and Computation, 121(2):256– 285, 1995), and also a recent line of work on parallel boosting (Karbasi and Larsen ALT'24, Luy et al. SODA'24), shows that indeed one needs $\\Omega(\\gamma^{-2} \\log(1/\\varepsilon))$ calls to a weak learner to obtain error $\\varepsilon$. However, these lower bounds actually require $\\exp(d) \\geq \\gamma^{-1}$. If $d$ is assumed a constant, then work by Alon, Gohen, Hazan and Moran STOC'21 shows that there are algorithms using only $\\tilde{O}(\\gamma^{-1} \\log(1/\\varepsilon))$ calls.\n\n(Question 2): Regarding the question about BaggedAdaboost and LarsenRitzert not being one line: \n\nWe did attempt to explain this in the paragraph immediately following the description of the data sets. Our apologies if this was not clear enough. Since Bagging is well-defined (but not necessarily optimal) with an arbitrary number of bootstrap sub-samples, we simply run BaggedAdaboost with X many bootstrap samples (random sub-samples of the dataset) for varying X. For LarsenRitzert, as explained in that paragraph, the number of sub-samples needed grows unwieldy for large data sets. Thus we chose to instead sample X of the sub-samples defined by LarsenRitzert without replacement. This was due to computational constraints. Furthermore, we feel the plot is more informative when seeing how more and more combined AdaBoosts improve the accuracy.", "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:
We would like to thank all reviewers for their thoughtful reviews.

Let us add one general remark that we will leave to the reviewers whether to include in their evaluation of the submission or not. Recently we found a way to improve the result to the majority of 5 instead of majority of 29. The reason why this improvement is possible is due to us becoming aware of a stronger statement of corollary 5 which improves the previous  $O((d/(\gamma^2m)^{1/14})$ bound to $O((d/(\gamma^2m)^{1/2})$. For the intuition of why this leads to a majority of 5, we recall from the proof sketch that we needed to combine $2/\alpha+1$ many AdaBoosts where $\alpha$ is the number of the exponent that we obtain in corollary 5 ($O((d/(\gamma^2m))^{\alpha}$), which now can be set equal to $\alpha=1/2$ and thus we get that we only need to combine $2/\alpha+1=5$ many AdaBoosts to get the optimal in expectation sample complexity. Fortunately, out experiments also included Majority-of-5 so no significant changes to the paper are needed. The new proof only changes (and simplifies) the material in the appendix.

Also, as we wish to be assessed as a theory paper, so primarily on the theoretical contribution, and added the experiments merely as a very first pilot study, let us add that if the reviewers all feel the paper would be stronger without the experiments, we would like to remove the experiments.

Meta-review: If there is one thing that boosting teaches us, it is the value of simple and original ideas whose use relies on nice technical tricks. This paper is a good example which deserves to be seen by the NeurIPS audience.

Regarding experiments: those have generated substantial discussion, for a paper which is theoretical in nature. Looking at the paper, one can say that there has been substantial work in polishing the theory while the experiments comparatively look a bit "let down" (it only takes a look at Figs 1 and 2) and their presentation could have been substantially simplified. But the merits of the paper are principally in its theory. I concur with the idea that those experiments could be changed. A suggestion: reduce the 1.5 page to 0.5 page that would take place in a "Discussion" section before conclusion, along with the story around Corollary 5 (below) and the limitations of the work, currently a single section. Put all that is not necessary in Appendix (dataset descriptions, etc.), eventually find a summary plot / table, insist on the fact that the experiments are preliminary / pilot, tone down the narrative. Again, the theory is the message.

Regarding the replacement of Corollary 5, there are two strategies: the first would be to just replace Corollary 5 with its stronger version and change the rest. I do not support this because the change has not been reviewed. Another strategy would be to use the "Discussion" Section (above) to add a subsection on a potential change to Corollary 5, with a new Corollary properly stated and how it changed the 29 -> 5 split. Hopefully both Corollaries have the same backbone. Extensive proof in Appendix required and proofchecked.

With this, make the why "The answer is 29" clear from Section 1.1: write that 29 is an integer that allows for the appropriate convergence of a series handling the algorithm's error, discuss the 29 -> 5, etc. . This allows to install the formal decorum.
Decision: Accept (poster)