Title: The Many Faces of Optimal Weak-to-Strong Learning
Venue: NeurIPS 2024
Accepted: yes
Keywords: Learning Theory

Weak to Strong Learning

Boosting

Large Margin Classifiers

Generalization Bounds

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

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.

1 Introduction

Boosting is one of the most powerful machine learning ideas, allowing one to improve the accuracy of a simple base learning algorithm A . The main idea in Boosting, is to iteratively invoke the base learning algorithm A on modified versions of a training data set. Each invocation of A returns a classifier, and these classifiers are finally combined via a majority vote or averaging. Variations of Boosting, including Gradient Boosting [10, 16, 7], are often among the best performing classifiers in practice, especially when data is tabular. Furthermore, when combined with decision trees or regressors as the base learning algorithm, these algorithms are independent of scaling of data features and provides impressive out-of-the-box performance. See the excellent survey [22] for further details.

The textbook Boosting algorithm for binary classification, AdaBoost [9], works by maintaining a weighing D t = ( D t (1) , . . . , D t ( m )) of a training set S = ( x 1 , y 1 ) , . . . , ( x m , y m ) with ( x i , y i ) ∈ X × {1 , 1 } for an input domain X and labels {-1 , 1 } . In each Boosting iteration t , a classifier h t : X → {1 , 1 } is trained to minimize the 0 / 1 -loss on S , but with samples weighed according to D t . The weights are then updated such that samples ( x i , y i ) misclassified by h t have a larger weight under D t +1 and correctly classified samples have a smaller weight. Finally, after a sufficient number of iterations T , AdaBoost combines the classifiers h 1 , . . . , h T into a voting classifier f ( x ) = sign( ∑ t α t h t ( x )) taking a weighted majority vote among the predictions made by the h t 's. Here the α t 's are real-valued weights depending on the accuracy of h t on D t .

̸

Weak-to-Strong Learning. Boosting was originally introduced to address a theoretical question asked by Kearns and Valiant [17, 18] on weak-to-strong learning. A learning algorithm A is called a weak learner if, for any distribution D over X × {1 , 1 } , when given some m 0 i.i.d. training samples S from D , it produces with probability at least 1 -δ 0 , a classifier/hypothesis h S ∈ H with er D ( h S ) ≤ 1 / 2 -γ , where er D ( h ) = Pr ( x,y ) ∼D [ h ( x ) = y ] and H ⊆ X → {1 , 1 } is a 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 γ is called the advantage of the weak learner and we refer to the weak learner as a γ -weak learner. We think of δ 0 and m 0 as 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 ( ε, δ ) with 0 < ε, δ < 1 , when given m ( ε, δ ) i.i.d. samples S from D , produces with probability at least 1 -δ a hypothesis h S : X → {1 , 1 } with er D ( h S ) ≤ ε . Here m ( ε, δ ) 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 to be the case [25], and AdaBoost is one among many examples of algorithms producing a strong learner from a weak learner.

Sample Complexity. Given that weak-to-strong learning is always possible, a natural question is "what is the best possible sample complexity m ( ε, δ ) 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 γ of the weak learner. In particular, the best known analysis [27] of AdaBoost shows that it achieves a sample complexity m Ada ( ε, δ ) of

Larsen and Ritzert [20] were the first to give an algorithm improving over AdaBoost. Their algorithm has a sample complexity m LR ( ε, δ ) of

They further complemented their algorithm with a lower bound proof showing that any weak-to-strong learning algorithm must have a sample complexity m ( ε, δ ) of

The optimal sample complexity for weak-to-strong learning is thus fully understood from a theoretical point of view.

Other Performance Metrics. Sample complexity is however not the only interesting performance metric of a weak-to-strong learner. Furthermore, O ( · ) -notation may hide constants that are too large for practical purposes. It is thus worthwhile to develop alternative optimal weak-to-strong learners and compare their empirical performance.

The algorithm of Larsen and Ritzert for instance has a rather slow running time as it invokes the weak-learner a total of O ( m lg 4 3 γ -2 ln m ) = O ( m 0 . 8 γ -2 ) times on a training set of m samples. This should be compared to AdaBoost that only invokes the weak learner O ( γ -2 ln m ) times to achieve the sample complexity stated in (1).

An alternative sample optimal weak-to-strong learner was given by Larsen [19] as a corollary of a 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 O ( γ -2 ln( m/δ ) ln m ) times on a training set of m samples.

A natural question is whether the sample complexity of AdaBoost shown in (1) can be improved to match the optimal sample complexity by a better analysis. Since AdaBoost only invokes its weak learner O ( γ -2 ln m ) times on m samples, this would be an even more efficient optimal weakto-strong learner. Unfortunately, work by Høgsgaard et al. [15] shows that AdaBoost's sample complexity is sub-optimal by at least a ln(1 /ε ) factor, i.e.

It thus remains an intriguing task to design weak-to-strong learners that have an 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 ( · ) -notation? And how does it vary among the different weak-to-strong learners?

1.1 Our Contributions

In this work, we first present a new weak-to-strong learner with an optimal sample complexity (at least in expectation). The algorithm, called MAJORITY-OF-5 and shown as Algorithm 1, is extremely simple: Partition the training set into 5 disjoint pieces of size m/ 5 and run AdaBoost on each to produce voting classifiers f 1 , . . . , f 5 . Finally, combine them by taking a majority vote g ( x ) = sign( ∑ 5 t =1 f t ( x )) . This simple algorithm only invokes the weak learner O ( γ -2 ln m ) times, asymptotically matching AdaBoost and improving over previous optimal weak-to-strong learners. Furthermore, since each invocation of AdaBoost is on a 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.

The concrete guarantees we give for MAJORITY-OF-5 are as follows

Theorem 1. For any distribution D over X ×{1 , 1 } and any γ -weak learner W using a hypothesis set H of VC-dimension d , it holds for a training set S ∼ D m that running MAJORITY-OF-5 on S to obtain a hypothesis g satisfies

In particular, Theorem 1 implies that E [er D ( g )] ≤ ε when given m = Θ( d/ ( γ 2 ε )) samples. This is a slightly weaker guarantee than the alternative optimal weak-to-strong learners in the sense that we do not provide high probability guarantees (i.e. with probability 1 -δ ). On the other hand, our algorithm is extremely simple and has a running time comparable to AdaBoost. Furthermore, the proof that AdaBoost is sub-optimal (see (2)) shows that even the expected error of AdaBoost is sub-optimal by a logarithmic factor. It is interesting that combining a 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 1 is based on recent work by Aden-Ali et al. [1] on optimal PAC learning in the realizable setting, demonstrating new applications of their techniques. Furthermore, we believe the number 5 is an artifact of our proof and we conjecture that it can be replaced with 3 by giving a better generalization bound for large margin voting classifiers. See Section 3 for further details.

Empirical Comparison. Our second contribution is a 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-5 algorithm, as well as classic AdaBoost. We give the full details of the alternative algorithms in Section 2. 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 in (2). Our pilot empirical study give an indication that our new algorithm MAJORITY-OF-5 may outperform previous algorithms on large data sets, whereas Bagging+Boosting performs best on small data sets. See Section 4 for further details and the results of these experiments.

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 2.

Given a training set S , SUBSAMPLE generates a list L of subsets S i ⊂ S . The list L has size m lg 4 3 ≈ m 0 . 79 when invoking SUBSAMPLE( S, ∅ ) for a training set S of size m . Larsen and Ritzert now give an optimal weak-to-strong learner using SUBSAMPLE as a sub-routine as shown in Algorithm 3.

Algorithm 3: LARSENRITZERT( S, W )

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 a target failure probability δ as input. The algorithm is shown as Algorithm 4.

3 Analysis of MAJORITY-OF-5

In this section, we give the proof of Theorem 1, showing that our new algorithm MAJORITYOF-5 has an optimal expected error. Before giving the formal details, we present the main ideas in our proof. Our analysis is at a high level inspired by recent work of Aden-Ali et al. [1] for

realizable PAC learning. The first ingredient we need is the notion of margins. For a voting classifier f ( x ) = sign( ∑ h ∈H α h h ( x )) with α h ≥ 0 for all h , consider the function f ′ ( x ) = ∑ h ∈H α ′ h h ( x ) with α ′ h = α h / ∑ h α h . 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 ( x, c ( x )) is then c ( x ) f ′ ( x ) ∈ [ -1 , 1] . The margin is 1 if all hypotheses combined by f agree and are correct. It 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 (1), is derived by showing that AdaBoost produces a voting classifier with margins Ω( γ ) .

Returning to our outline of the proof of Theorem 1, recall that the optimal error for weak-to-strong learning as a function of the number of samples m is O ( d/ ( γ 2 m )) . Now assume we can prove that for a set of m i.i.d. samples from a distribution D , the expected maximum error under D of any voting classifier that has margins Ω( γ ) on all the samples, is no more than O ( √ d/ ( γ 2 m )) . This fact follows from previous work on Rademacher complexity. Note that this is sub-optimal compared to our target error by a polynomial factor since √ x ≥ x for x between 0 and 1 . We want to argue that combining 5 instantiations of AdaBoost on disjoint training sets reduces this expected error to optimal O ( d/ ( γ 2 m )) .

̸

For this argument, consider running AdaBoost on n = m/ 5 samples. For any x ∈ X , consider the probability p x = Pr S ∼D n [ f S ( x ) = c ( x )] where f S is the hypothesis produced by AdaBoost on S and c ( x ) is the correct label of x . Inspired by Aden-Ali et al. [1], we now partition the input domain X into sets R i , such that R i contains all x for which p x ∈ (2 -i , 2 -i +1 ] . The crucial observation is that if we consider k independently trained AdaBoosts, then the probability they all err on x is precisely p k x . Since a majority vote among 5 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 ∈ R i to Pr X ∼D [ X ∈ R i ]2 -3 i . What remains is thus to argue that Pr[ X ∈ R i ] is small.

This last step is done by considering the distribution D i , which is D conditioned on receiving a sample from R i . The expected number of samples we see from R i is m i = Pr[ X ∈ R i ] m . Furthermore, since AdaBoost obtains margins Ω( γ ) on all its training data, it in particular obtains margins Ω( γ ) on all its samples from D i . This leads to an error probability of p i = O ( √ d/ ( γ 2 m i )) under D i . But the definition of R i implies p i ≥ 2 -i . Hence √ d/ ( γ 2 m i ) = Ω(2 -i ) ⇒ m i = O (2 2 i d/γ 2 ) ⇒ Pr[ X ∈ R i ] = O (2 2 i d/ ( γ 2 m )) . By summing over all R i , the final expected error is hence

This completes the proof overview. Let us end by making a few remarks. First, it is worth noting that the generalization bound O ( √ d/ ( γ 2 m )) seems much worse than the bound in (1) claimed for AdaBoost and other voting classifiers with large margins. Unfortunately, if we examine (1) carefully and state ε as a function of m , we get ε = O ( d ln( m/d ) ln m/ ( γ 2 m )) . The problem is that the two log-factors are not bounded by a polynomial in d/ ( γ 2 m ) . In particular for m = Cd/γ 2 with C > 0 a constant, any polynomial in d/ ( γ 2 m ) must be constant. But ln m = ln( Cd/γ 2 ) is not a constant independent of d and γ . Thus we have to use the generalization bound with √ · 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 γ -margin voting classifiers, i.e. matched by an upper bound, then it suffices to take a majority of 3 AdaBoosts for optimal sample complexity.

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.

Preliminaries. For a hypothesis set H , we let ∆( H ) denote the set of linear classifiers using hypothesis from H that is

Note that we have termed these linear classifiers rather than voting classifier, to distinguish that we have not yet applied a sign( · ) and insist on normalizing the coefficients so they sum to 1 . We define the sign function as being 1 when the value is non-negative (so also 1 when x = 0 ) and -1 when negative.

̸

We let c ∈ {-1 , 1 } X be a true labeling that we are trying to learn. For a distribution D over X , we define the expected loss of f ∈ [ -1 , 1] X as er D ( f ) := E X ∼D [1 { sign( f )( X ) = c ( X ) } ] and we will use S ∈ X m to denote a point set of m i.i.d. samples from D i.e. S ∼ D m (we see the point set as a vector so we allow repetition of points). Define for any k ∈ N the majority of k linear classifiers f 1 , . . . , f k ∈ [ -1 , 1] X as Maj ( f 1 , . . . , f k )( x ) = sign( ∑ i sign( f i ( x ))) . Let s be a set of points, i.e. s ∈ ∪ ∞ i =1 X i and c ( s ) ∈ {-1 , 1 } | s | the labeling of the points that s contains with c , that is c ( s ) i = c ( s i ) . We define a γ -margin classifier algorithm f : ∪ ∞ i =1 ( X × {1 , 1 } ) i ⇒ [ -1 , 1] X to be a mapping that takes as input a point set with labels ( s, c ( s )) ∈ ∪ ∞ i =1 ( X × {1 , 1 } ) i and outputs a function f ( s, c ( s ))( · ) from X to the interval [ -1 , 1] , where f ( s, c ( s ))( · ) is such that for x ∈ s , f ( s, c ( s ))( x ) c ( x ) ≥ γ . In the following we will use f s to denote f ( s, c ( s )) and write f ∈ ∆( H ) if for any ( s, c ( s )) ∈ ∪ ∞ i =1 ( X × {1 , 1 } ) i we have that the output of the γ -margin classifier algorithm f ( s, c ( s )) is in ∆( H ) .

Analysis. We now prove Theorem 1, which is a direct consequence of the following Corollary 2. Corollary 2 states that running a γ -margin classifier algorithm on 5 disjoint training sets of size m and forming the majority vote of the produced 5 classifiers, has the optimal O ( d/ ( γ 2 m )) expected error. Since AdaBoost, after O (ln( m ) /γ 2 ) iterations, has Ω( γ ) margins on all points [26] [Section 5.4.1], Corollary 2 gives the claim in Theorem 1. Alternatively one could run AdaBoost ∗ v [24] instead of AdaBoost. The proof of Corollary 2 follows the method used in [1] where the authors show a similar bound for PAC learning in the realizable setting. We now state Corollary 2.

Corollary 2. For any distribution D over X , hypothesis set H with VC-dimension d , i.i.d. point sets S 1 , . . . , S 5 from D m , margin 0 < γ ≤ 1 and f ∈ ∆( H ) being a γ -margin classifier algorithm, we have that

The result in Corollary 2 is primarily a consequence of the following Lemma 3, which says that, in expectation, the probability that f S 1 , f S 2 , f S 3 all misclassifying a sample from D is O ( d/ ( γ 2 m )) . We now state Lemma 3 and give the proof of Corollary 2. We postpone the proof of Lemma 3 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 S 1 , S 2 , S 3 from D m , margin 0 < γ ≤ 1 and f ∈ ∆( H ) being a γ -margin classifier algorithm we have that

̸

̸

Proof of Corollary 2. For the majority of f S 1 , . . . , f S 5 to fail on an x ∈ X , it must be the case that at least 3 of the trained γ -margin classifiers f S i have sign( f S i ( x )) = c ( x ) . Using this combined with the S i 's being i.i.d. we get that

̸

̸

Now using Lemma 3 and ( 5 3 ) = O (1) , we get that the above is O ( d γ 2 m ) as claimed.

We now move on to prove Lemma 3. For this we need Lemma 4 which we now state and give the proof in Appendix A.

̸

Lemma 4. Let a > 1 denote a universal constant. For D a distribution, R a subset of X such that P [ R ] := P X ∼D [ X ∈ R ] = 0 , hypothesis set H with VC-dimension d, S a point set of m i.i.d. points from D , margin 0 < γ ≤ 1 and f ∈ ∆( H ) being a γ -margin classifier algorithm we have that

̸

where we use D R to denote the conditional distribution on the subset R . That is, for any measurable function g , E X ∼D R [ g ( X )] := E X ∼D [ g ( X )1 { X ∈ R } ] / P X ∼D [ X ∈ R ] .

̸

Proof of Lemma 3. For x ∈ X let p x = E S ∼D m [1 { f S ( x ) = c ( x ) } ] and define for i = 1 , . . . the sets R i = { x ∈ X : p x ∈ ( 2 -i , 2 -i +1 ] } . Now using Tonelli, and that S 1 , S 2 , S 3 are i.i.d. with distribution D m , and that p x ≤ 2 -i +1 for x ∈ R i we get that

̸

̸

thus if we can show that there exists a universal constant c ′ > 0 such that P X ∼D [ X ∈ R i ] ≤ c ′ d 2 2 i γ 2 m we get that

̸

and we are done. Thus assume for contradiction that P X ∼D [ X ∈ R i ] > c ′ d 2 2 i γ 2 m , for c ′ > 1 to be chosen large enough. Using Lemma 4 we have that there exist a universal constant a > 1 such that

By Tonelli, the definition of p x and that for x ∈ R i we have p x ∈ ( 2 -i , 2 -i +1 ] we get that

̸

Combining the above lower and upper bound on E S ∼D m [ er D R i ] and P X ∼D [ X ∈ R i ] > c ′ d 2 2 i γ 2 m we get that

which for c ′ sufficiently large is strictly less than 1 , thus we reached a contradiction. Hence it must be the case that P X ∼D [ X ∈ R i ] ≤ c ′ d 2 2 i γ 2 m , which concludes the proof of Lemma 3.

4 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 is a synthetic binary classification data set developed from the lower bound [15] showing that AdaBoost is 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 DecisionTreeClassifier with max_depth=1 . This is default for the implementation of AdaBoost in scikit-learn, which is the implementation used in our experiments. We describe the data sets in greater detail below.

Higgs [29]: 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 7 of these are derived from the other 21.

Boone [23]: 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.

Forest Cover [4]: In this data set, we try to determine the forest cover type of 30 x 30 meter cells. The data set actually has 7 different forest cover types, so we have removed all samples of the 5 most uncommon to make it into a 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 a 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 [15]: This data set, as well as the weak learner, have been developed using the lower bound instance in [15]. 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-strong learners are run simply by giving them access to a weak learner. The weak learner is adversarially designed. When it is queried with a 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 a set of hypotheses (chosen randomly) and returns the hypothesis with the worst performance on T , while respecting that it must have error no more than 1 / 2 -γ under the query weighing and having at least 1 / 2 + γ error on T . Finally, the hypothesis set contains an additional special hypothesis h 0 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 γ . 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 ADABOOST (accuracy shown as a blue horizontal line in the plots), BAGGEDADABOOST (Algorithm 4), 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 is 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 5 runs. For Algorithm 3 that creates m lg 4 3 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 BAGGEDADABOOST, 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. 1.

The results in Fig. 1 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. 2.

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 BAGGEDADABOOST 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.

0.695

ADABOOST

LARSENRITZERT

0.694

MAJORITY-OF-X

BAGGEDADABOOST

0.693

0.691

0.690

35

11

15

21

29

Number of Voting Classifiers

ADABOOST

0.774

0.923

LARSENRITZERT

MAJORITY-OF-X

ADABOOST

BAGGEDADABOOST

0.772

LARSENRITZERT

MAJORITY-OF-X

0.921

BAGGEDADABOOST

Test

0.770

0.920

0.768

0.919

35

11

15

21

29

35

11

15

21

29

Number of Voting Classifiers

Number of Voting Classifiers

0.80

ADABOOST

0.95

LARSENRITZERT

0.79

MAJORITY-OF-X

0.90

BAGGEDADABOOST

ADABOOST

0.85

LARSENRITZERT

0.78

Test

Test

0.80

MAJORITY-OF-X

BAGGEDADABOOST

0.75

0.77

0.70

0.76

0.65

35

11

15

21

29

35

11

15

21

29

Number of Voting Classifiers

Number of Voting Classifiers

5 Limitations

Our main results are proved under the theoretical assumption of i.i.d. training samples as well as access to a weak-learner that always obtains an advantage of γ 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 of our results. In all circumstances, we view the theoretical contributions as the main novelty of this work.

Acknowledgment

This research is co-funded by the European Union (ERC, TUCLA, 101125203) and Independent Research Fund Denmark (DFF) Sapere Aude Research Leader Grant No. 9064-00068B. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

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A Proof of Lemma 4

We now give the proof of Lemma 4. For this we need the following Corollary 5 that gives a high probability bound on the error for all classifiers in ∆( H ) that have γ margins on a training set S .

Corollary 5. Let C > 1 denote a universal constant. For hypothesis set H with VC-dimension d , distribution D , margin 0 < γ ≤ 1 , failure probability 0 < δ < 1 , and a point set S ∼ D m , we have with probability at least 1 -δ over S , that any f ∈ ∆( H ) such that f ( x ) c ( x ) ≥ γ for all x ∈ S satisfies

Corollary 5 follows for instance by a modification of [26] [page 107-111] due to [3] and using the stronger bound on the Rademacher Complexity of O ( √ d/m ) due to [8] [See e.g. theorem 7.2 [13] ]. With Corollary 5 in place we now give the proof of Lemma 4.

Proof of Lemma 4. If P [ R ] m ≤ d/γ 2 we have that

and the claim holds since er D R ( f S ) is always less than 1 . Thus we assume from now on that P [ R ] m > d/γ 2 . We now define the events E i := {|{ S ∩ R }| = i } for m ≥ i ≥ P [ R ] m/ 2 and E = ∪ i ≥ P [ R ] m/ 2 E i = {| S ∩ R | ≥ P [ R ] m/ 2 } . We further define X j = 1 { S j ∈ R } for j = 1 , . . . , m and notice that these are i.i.d. { 0 , 1 } -random variables where X = ∑ j X j has expectation P [ R ] m and the event ∑ j X j ≥ P [ R ] m/ 2 is contained in E . Thus by a Chernoff bound and that exp( -x ) ≤ 1 /x for x > 0 , we get that

We thus have P S [ ¯ E ] ≤ 8 / ( P [ R ] m ) . Since we assumed that P [ R ] m ≥ d/γ 2 ≥ 1 and using 0 ≤ x ≤ √ x for x ≤ 1 , this further implies that

We will soon show that E S [er D R ( f S ) | E i ] ≤ 16 · √ 16 Cd γ 2 P [ R ] m for m ≥ i ≥ P [ R ] m/ 2 for a universal constant C ≥ 1 . Now using these two relations combined with the law of total expectation on the partition ¯ E,E P [ R ] m/ 2 , . . . , E m and that er D R ≤ 1 , we get that

as claimed in Lemma 4 with the universal constant a = 24 2 · 16 C . Thus we have to show that E S [er D R ( f S ) | E i ] ≤ 16 · √ 16 Cd γ 2 P [ R ] m for m ≥ i ≥ P [ R ] m/ 2 . So consider such an i . Since er D R ( f S ) is a non-negative random variable, we have that

We will thus upper bound this integral. Now conditioned on E i , we know that S contains i points that are samples according to D R and that f S on these examples has all margins at least γ . Thus we have by Corollary 5 that with probability at least 1 -δ over S , it holds that

where C ≥ 1 is a universal constant. For ease of notation let r i = √ 8 Cd γ 2 i . We notice that r i ≤ √ 16 Cd γ 2 P [ R ] m since i is assumed to be greater than P [ R ] m/ 2 . Using this, we get that

we get by combining this with Eq. (4) that

which would conclude the proof. Thus we now show ∫ ∞ r i P S [er D R ( f S ) ≥ x ] dx ≤ 15 · √ d γ 2 P [ R ] m . For this, we do the following non-trivial rewriting of x to make it resemble the second term in the max appearing in (3)

Now for any x ≥ r i we have that

which combined with the rewriting of x and Eq. (3) with δ = 2exp ( -x 2 i 8 ) and noticing r i is the first argument of the max in Eq. (3), we get that

for any x ≥ r i . Now using the density function of a normal distribution with mean 0 and standard deviation σ is equal to exp( -1 2 ( x/σ ) 2 ) / ( σ √ 2 π ) , and letting N (0 , σ ) denote a normal random variable with mean 0 and standard deviation σ , we get that

where the second to last inequality follows from i ≥ P [ R ] m/ 2 and the last inequality by P [ R ] m ≥ d/γ 2 ≥ 1 .

Thus if we can show that

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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\nThe 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.\n\nWEAKNESSES:\nWhile 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.\n\nQUESTIONS:\n1) 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*\n\nLIMITATIONS:\nThe authors have adequately addressed any limitations that I can foresee.", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-06-19"}

comments: 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.

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.

QUESTIONS:
Please see the weakness.

LIMITATIONS:
Limitations are discussed.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 2
DATE: 2024-07-12

comments: 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.

STRENGTHS:
- Very strong and interesting result
- Mathematically sound, based by my judgement
- Good presentation, self-contained and well-structured

WEAKNESSES:
N/A

QUESTIONS:
N/A

LIMITATIONS:
N/A
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2024-07-12