When are ensembles really effective?

Ensembling has a long history in statistical data analysis, with many impactful applications. However, in many modern machine learning settings, the benefits of ensembling are less ubiquitous and less obvious. We study, both theoretically and empirically, the fundamental question of when ensembling yields significant performance improvements in classification tasks. Theoretically, we prove new results relating the \emph{ensemble improvement rate} (a measure of how much ensembling decreases the error rate versus a single model, on a relative scale) to the \emph{disagreement-error ratio}. We show that ensembling improves performance significantly whenever the disagreement rate is large relative to the average error rate; and that, conversely, one classifier is often enough whenever the disagreement rate is low relative to the average error rate. On the way to proving these results, we derive, under a mild condition called \emph{competence}, improved upper and lower bounds on the average test error rate of the majority vote classifier. To complement this theory, we study ensembling empirically in a variety of settings, verifying the predictions made by our theory, and identifying practical scenarios where ensembling does and does not result in large performance improvements. Perhaps most notably, we demonstrate a distinct difference in behavior between interpolating models (popular in current practice) and non-interpolating models (such as tree-based methods, where ensembling is popular), demonstrating that ensembling helps considerably more in the latter case than in the former.

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Peer review

Reviewer E93x7/10 · confidence 4/52023-06-26

Summary

The authors offer a novel theoretical analysis of the majority vote classifier. In the process they define the "ensemble improvement rate" and give lower and upper bounds on it in terms of another entity they call the "disagreement-error ratio", under a novel assumption (competence) that they show empirically is mild. They also show how other works can be phrased in terms of lower/upper bounds on the ensemble improvement rate.

Strengths

As they say in the paper, without any assumptions all that can be said is that EIR (the ensemble improvement rate) is >=-1. In MLIS20 they make additional assumptions and yield better results, but the current work improves upon the work in MLIS20 by a factor of 2. The main novelty and strength of the paper is in the definition of "competence", a seemingly mild assumption that allows to get much better theoretical results. (Theorem 1: competent ensembles never heart, i.e. EIR>=0; and Theorem 2 giving explicit bounds of EIR under the assumption of competence.) Another strength of the paper is that they are not satisfied that "competence" feels like a mild assumption, they show empirical proof that this is often the case. The mathematical definitions are elegant, and the paper is easy to read.

Weaknesses

It is not clear how much the analysis can translate to regression tasks, perhaps some words to address this would be helpful. It may also be nice to say some words about whether the bounds become trivial as K goes to infinity.

Questions

This is a bit nit-picky, but I find the $W_{\rho}$ notation confusing: $E_{h\sim\rho}(1_{(h(X)\neq Y)})$ should really be $E_{h\sim \rho}(1_{(h(X)\neq Y)}|X, Y)$. On the other hand, others may find the notation "$E_{h\sim\rho}(1_{(h(X)\neq Y)}|X, Y)$" confusing, despite being exactly correct. (Conditional expectation on random variables yields a random variable.) Perhaps the solution is to keep $W_{\rho}$ as $W_{\rho}(X,Y)$ (meaning not to neglect $X$ and $Y$ in the notation)? Or maybe explain a little more that $P(W_{\rho} \in [t, 1/2))$ is over the distribution of $(X,Y)$? I am also a little confused about the relationship between competence and a similar notion where you change the order of expectation: replace $P_{(X,Y)\sim D}(E_{h\sim \rho}(1_{(h(X)\neq Y)}|X, Y) \text{is in some interval})$ with $P_{h\sim \rho}(E_{(X,Y)\sim D}(1_{(h(X)\neq Y)}|h) \text{is in some interval})$. Is this more or less intuitive? Can anything be said about ensembles with this alternate definition of competency? How does this work compare with the seminal "Super Learner" paper (Van der Laan, M. J., Polley, E. C., and Hubbard, A. E.) that is commonly cited in theoretical ensemble research? They compare ensemble performance to the best performing base learner, but they don't give guarantees for when it is better. Would they be able to benefit from a regression-equivalent of "competence"?

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

4 excellent

Presentation

4 excellent

Contribution

4 excellent

Limitations

None that I can see.

Reviewer un8p9/10 · confidence 5/52023-07-05

Summary

The paper "When are ensembles really effective?" discusses the question asked in the title: When are ensembles really effective? To do so, the authors introduce novel theoretical measures called the ensemble improvement rate and the disagreement-error ratio. The ensemble improvement rate measures the relative rate an ensemble improves over a single learner, whereas the disagreement-error ratio measures the expected disagreement between members of the ensemble scaled by the expected performance of each model. The authors relate both terms to each other through a series of inequalities, i.e. provide an upper and lower bound for the ensemble improvement rate wrt. to the disagreement-error ratio. Moreover, the authors introduce the concept of competent ensembles which more closely resemble ensembles we are seeing in practice: While we might construct ensembles with arbitrarily bad members, we usually see ensembles that, on average, have well-performing members. Last, the authors conclude the paper with an empirical investigation that shows that a) ensembles are indeed competent in practice and b) that the disagreement-error ratio is valuable in predicting if an ensemble might improve performance compared to a single model.

Strengths

- The paper is well-written and easy to follow. The authors introduce remarks where necessary and re-visit existing results from the literature if it adds valuable information to the paper. - The authors introduce new theoretical concepts, argue in favor of them, and show their practical value in experiments. In addition, the authors also validate any assumptions they are making in their empirical investigation. Hence, the authors offer a very complete overview of their ideas, studying them from a theoretical and a practical point of view. This should be the gold standard for all papers introducing novel concepts. - The introduced ensemble improvement rate and disagreement-error ratio seem to work well in practice and hence are valuable. Second, I think the idea of competent ensembles can be potentially impactful for future studies about ensembles, as it brings the theory about ensembles closer to what we see in practice

Weaknesses

- In some form, the paper presents many things we already know in a new form. For example, the disagreement-error ratio can be seen as some form of diversity that has been studied for a long time in the literature. Similarly, the concept of weak learnability and competent ensembles seem to be closely related to each other (see my question below). While the authors did a decent job relating their work to existing work, a deeper discussion here would be interstring - There is a minor typo in the paper: Section 3 uses "K" as the number of classes, whereas in Section 4 "C" is used for the number of classes

Questions

- In the introduction, you present the following scenario: "In particular, consider the following practical scenario: a practitioner has trained a single (perhaps large and expensive) model, and would like to know whether they can expect significant gains in performance from training additional models and ensembling them" -- How can you answer this question with EIR and DER? As far as I can see, EIR and DER require already trained ensembles and explain their performance and not predict the performance of a yet untrained ensemble? In this sense, it is similar to the bias-variance decomposition that explains the performance of a model/ensemble post-hoc (i.e. after training) and, therefore might be used as a theoretical guidance tool but not necessarily as a prediction of what model/ensemble performs better or worse in a given scenario. Did I miss something here? - Did you explore the relationship between the disagreement-error ratio and the bias-variance decomposition, and if so, what is it? I assume that given a suitable disagreement rate D, one could come up with a term that is close to the variance (i.e., diversity), but the normalization via the expected error rate this novel here.

Rating

9: Very Strong Accept: Technically flawless paper with groundbreaking impact on at least one area of AI/ML and excellent impact on multiple areas of AI/ML, with flawless evaluation, resources, and reproducibility, and no unaddressed ethical considerations.

Confidence

5: You are absolutely certain about your assessment. You are very familiar with the related work and checked the math/other details carefully.

Soundness

3 good

Presentation

4 excellent

Contribution

4 excellent

Limitations

The authors empirically validate their assumption of competent ensembles that might limit the applicability of the presented theory. Moreover, they point out the unique case of Random Forests that arguably justifies a study in its own right. I agree with the authors that there do not seem to be any negative societal impacts from the theory presented in this paper.

Reviewer sRJm7/10 · confidence 4/52023-07-05

Summary

The paper asks the following question: when is ensembling beneficial? The benefit is measured by the ensemble improvement rate (EIR), defined as the difference of average and majority voting risk, divided by the average risk. It is shown that 'competent ensembles never hurt', in the sense that under mild assumptions, EIR \geq 0 (Th. 1). A more precise result (Th. 2) provides quantitative bounds relating EIR and another quantity called the disagreement-error ratio (DER). DER is simply the expected disagreement between our ensembles, divided by the expected loss. A consequence of this result is that improvement is moderate when disagreement is small, whenever it can be large whenever disagreement is large. The paper concludes with some empirical evaluation.

Strengths

The key question of the paper is of utmost importance to the machine learning community. The answers provided by the present work, although simple, are quite interesting. What is really promising is that competence can be evaluated in practice (Section 4.2), giving an actionable way to check whether it is profitable to do ensembling in a specific concrete case.

Weaknesses

The main weakness in my opinion is the restriction to the classification setting, and, further, to the use of the 0-1 loss.

Questions

Do you suspect that Th. 2 is tight? Could examples similar to those developed for Th. 1 achieve tightness in the string of inequalities?

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

see weaknesses

Reviewer GNtk4/10 · confidence 2/52023-07-06

Summary

Analysis the ensemble improvement rate vs. the disagreement-error ratio

Strengths

A solid analytical analysi.

Weaknesses

It is not clear for me what is the usefulness of obtained results. The scope of the experiments is limited.

Questions

What is the usefulness of this study? How to apply them in practice?

Rating

4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.

Confidence

2: You are willing to defend your assessment, but it is quite likely that you did not understand the central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

2 fair

Presentation

3 good

Contribution

2 fair

Limitations

Not applicable

Reviewer G3j57/10 · confidence 4/52023-07-28

Summary

This study explores when ensembling leads to significant performance enhancements in classification tasks. Theoretical findings reveal that ensembling improves performance considerably when the disagreement rate among classifiers is large compared to the average error rate. The study also establishes improved upper and lower bounds on the average test error rate of the majority vote classifier under a condition called competence. In addition to the theoretical analysis, the researchers conduct empirical experiments in various scenarios to validate their theory. The study highlights a distinction in the behaviour of interpolating models and non-interpolating models, showing that ensembling provides more significant benefits in the latter case.

Strengths

- The work addresses a relevant and, still, open problem: when are ensembles really effective? The presented theoretical framework is specially designed to answer this question. Although it builds on previous theoretical analysis, it introduces a new concept, “competent ensembles”, which give rises to more insightful conclusions. - The empirical evaluation of this work is quite strong. Besides validating the proposed theoretical framework, authors also provide insightful analysis about the effectiveness of ensembles. The analysis of what happens at the interpolation regime is of high importance for current practices of machine learning.

Weaknesses

- Assumption 1 is central in the theoretical analysis of this work. Although authors make an effort in describing assumption 1 as a condition guaranteeing that ensembles are better than individual classifiers, this assumption is very technical and it is not well described. - Although this is not the main aim of this paper, the provided theoretical framework provides little to no guidance in how to build better ensembles.

Questions

Could you derive Thm 2 under the assumption that EIR>1? Because, in this case, your assumption should be EIR>1.

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

4 excellent

Presentation

4 excellent

Contribution

3 good

Limitations

Authors do not fully discussed the limitations of their work.

Reviewer G3j52023-08-14

Dear authors, In relation to my question. Sorry, it was a typo. My question is: Could you derive Thm 2 under the assumption that EIR>0? Because, in this case, your assumption should be EIR>0. I would like to see your response.

Authorsrebuttal2023-08-16

Thank you for clarifying, this is a good question. It turns out that assuming $EIR > 0$ is _not_ sufficient to prove Theorem 2 (that is, competence is not necessary for $EIR \geq 0$). To see this, we can provide a numerical example for when $EIR > 0$ but the ensemble is not competent, and the lower bound in Theorem 2 does not hold. We summarize the setup of this example in the following table: | Set $A \subset \mathcal{X}\times \mathcal{Y}$, $P_{X,Y}(A) = 0.6$ | Set $B\subset \mathcal{X}\times \mathcal{Y}$, $P_{X,Y}(B)=0.4$ | |------------------------|---------------------| | For all $(x,y)\in A$, $P_h(h(x) = y) = 0.7$ | For all $(x,y)\in B$, $P_h(h(x)=y) = 0.4$ | where here we assume we are in the binary $K=2$ case. For this setup, one can compute $L(h_{MV}) = 0.4$ (note $h_{MV}(x) \neq y \Leftrightarrow (x,y)\in B$), $E[L(h)] = 0.42$, and $E[D(h,h')] = 0.444$. In this case, $EIR = 0.02/0.42 > 0$, but the lower bound is $DER - 1 = 0.444/0.42 - 1 = 0.024/0.42 > 0.02/0.42 = EIR$, and so the lower bound in Theorem 2 does not hold.

Reviewer E93x2023-08-17

I thank the authors for the interesting comments. I retain my score, and recommend acceptance. About "Super Learner", I think the authors are mistaken: stacked generalizations are not a "specific method for ensembling", but rather a vast generalization of almost all ensembling techniques used in practice. One would be hard pressed to find an ensembling method that does not fall into that category. (Many classic ensembling methods such as simple mean/median etc. are simply the case where the stacker is constant.) I'd still urge the authors to make the connection and explain the differences in the final draft.

Authorsrebuttal2023-08-22

Thank you again for your positive feedback. It seems we may have misinterpreted the results of the “Super Learner” paper; we will do a more thorough review of this work and include a discussion in an updated draft of the paper.

Reviewer un8p2023-08-22

Thank you for the clarifications, which solidified my original scoring :-)

Reviewer sRJm2023-08-22

Post-rebuttal

Thank you for the added clarification regarding the loss functions and the comment on the tightness of Th. 2.

Program Chairsdecision2023-09-21

Decision

Accept (poster)

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