NeurIPS 2020

Towards Convergence Rate Analysis of Random Forests for Classification

Review 1

Summary and Contributions: The paper discusses classification with random forests and gives convergence rates to the Bayes error for various methods to construct the individual trees: pure random trees with random splits of the leaf-cells, random trees with midpoint splits, and a simplified version of Breimans random forest, where splitting terminates on cells which only contain example points of equal labels. In the latter case the rate matches the minimax rates for plug-in classifiers up to a logarithmic factor. This is further improved under an assumption on the data distribution. All rates are in expectation.

Strengths: I did not study the entire appendix, but I checked the proofs in the main part of the paper and of several supporting lemmata in the appendix. They seemed correct (but see caveat below). I liked the estimate on the height of a random tree. I am not an expert on random forests, but the reviewing duties motivated a brief survey of the literature, where I did not find comparable results on classification. As random forests are widely used in practice, their theoretical study seems quite relevant.

Weaknesses: There are several ambiguities and imprecisions. I very much disliked the argument in the proof of Theorem 1 where the unit cube in R^d is treated as a discrete set by summing over its elements, although I am convinced that the argument can be made correct without damage to the result. This kind of hand-waving would be more acceptable if there was a rigorous proof in the appendix. The assumption on the distribution in Theorem 4 is in terms of the algorithm itself and remains quite obscure. Some elucidation of the assumption would be in order. There are a number of small grammatical errors, mostly missing articles or near typos like "3D objection recognition". The paper would benefit from a proof-reading in this respect.(selecting "a dimension at random, uniformly over dimensions of the longest side length" presumably means random selection of any dimension along which the side length of the cell is maximal). ---------------------------update-------------------------------------------------- The authors have addressed these concerns and promise to make corresponing improvements. In the hope that they keep this promise I improve my score to 7.

Correctness: The claims seem correct, but please see the previous comments.

Clarity: Apart from the many small grammatical errors, the paper is well written and various methods of tree construction are well explained.

Relation to Prior Work: This is done in the introduction and in the section on related work on page 6. As mentioned above I am not an expert on random forests, and the discussion in the paper may not be completely exhaustive.

Reproducibility: Yes

Additional Feedback: Please check the grammar in another reading. Otherwise see the comments above.

Review 2

Summary and Contributions: This work studies the finite sample convergence of random forests when the probability of class one is Lipschitz. Rates are derived of the form n^{-1/cd}, which scale extremely poorly in dimension but are somewhat intuitive. When the data is sufficiently structured the scaling is improved to the parametric rate.

Strengths: See above.

Weaknesses: No experiments, which could be helpful to verify the derived rates/identify if they are pessimistic.

Correctness: The claims seem technically correct.

Clarity: Clearly written. Minor points: First sentence of the conclusion and introduction are nearly identical, which is somewhat jarring. Line 181: Intuitive -> Intuitively Line 182: makes quite -> makes it quite Lines 56-57: missing "the" several places. Line 57: estimation -> estimate Line 35: when do random -> when random Line 34: along variable -> along with variable

Relation to Prior Work: Yes.

Reproducibility: Yes

Additional Feedback:

Review 3

Summary and Contributions: This article deals with the generalization performance of random forests. The authors establish convergence rates (non-asymptotic bounds) for different algorithms used for pattern classification (computation of dichotomies). They introduce a simplified variant of Breiman's original random forest (Algorithm 1). For this classifier, under mild hypotheses, the convergence rate is optimal up to a logarithmic factor.

Strengths: The theory of random forests has primarily been developed for regression, not pattern classification. Thus, this paper contributes to bridging the gap between theory and practice. It is noticeable that the formula of Theorem 4 does not depend on the dimension d of the description space.

Weaknesses: There is no argument suggesting that those theorems could extend nicely to the multi-class case. I doubt that it could be the case.

Correctness: I checked large parts of the computations without identifying any flaw.

Clarity: The paper is clearly written. However, the scope of the results obtained could be discussed in more detail.

Relation to Prior Work: The reader unfamiliar with the domain will find it difficult to figure out what the state of the art is (for pattern classification).

Reproducibility: Yes

Additional Feedback:

Review 4

Summary and Contributions: The paper provides finite-sample convergence rates for two simplified variants of random forests: - Breiman's pure random forests, where k nodes are split at random dimensions and at (uniformly) random positions or at midpoint. The latter giving faster convergence rates than the former. - A new variant where nodes are split at midpoint along one of their longest dimensions picked at random and where node splitting is stopped as soon as all examples in a node are of the same class. Better convergence rates are also derived for this latter method by making further assumption about the problem.

Strengths: - While convergence of simplified RF models has been studied in regression, this paper proposes the first convergence rate analyses in the context of classification. Results are non trivial and they also nicely follows intuition, in the sense that the faster methods are the expected ones. - The second simplified method that is studied is closer to the original random forests than simplified RF methods usually analysed in the literature. - Convergence rates come with interesting new side results, such as a new link between the convergence of forests and trees and a better estimation of the height of random trees.

Weaknesses: - The studied algorithms remain quite far from real random forests (no bootstrap sampling, split choices are fully independent of the data, trees are pruned, etc.) - As in other results in the literature, convergence rates for forests are by-product of convergence rate of individual trees (using Lemma 1). The results therefore do not really show the benefit of using forests instead of trees in terms of convergence rate. This should be discussed in the paper I think. - Overall, the contribution is purely theoretical. No real conclusion is drawn from the theoretical results that would help better understand standard RF or suggest modification to these methods. - The structure of the paper could be improved, as well as the discussion of related works (see below). - The paper is very technical with 14 pages of mathematical proofs in the supplement and 2.5 additional pages of proofs of Theorem 1 in the main text. I think this kind of very technical contribution would be more appropriate for a journal submission than for a conference (given the limited time allotted for reviewing). ---------- update after the author's response ---------- I thank the authors for their response. They mostly confirm the limitations that I highlighted in my review, which do not prevent acceptance. Concerning A3, the theoretical results indeed allow to compare simplified RF models but they do not really motivate changes to the standard RF algorithm (which uses neither random splits, nor midpoint splits for example).

Correctness: The results are plausible to me. However, I didn't fully check the proofs in the supplement.

Clarity: The paper is clearly written. One problem that I see is however in the definition of the random variables Y_i and U_i between lines 254 and 261. These variables are related to a specific dimension j. They should thus be indexed by j in addition to i. Note that Lemma 3 does not need them. These variables are only needed for the proof of Lemma 8 in Appendix C that considers a fixed dimension j. There are several redundancies in the text, with several statements repeated several times between the introduction, related works and conclusions. I think the paper could also be better structured. All main results are gathered in Section 4 in a very unstructured way. I think this section could be subdivided into several sections corresponding to the different simplified RF models and the related works could be also addressed in a separate section.

Relation to Prior Work: The discussion of related works could be improved. Between lines 198 and 207, the authors discuss several random forests extensions that I think are unrelated to the present contribution. It would have been interesting to discuss instead in more details convergence rates derived in the literature for simplified RF models in regression (e.g., in [8] or [25]).

Reproducibility: Yes

Additional Feedback: Some typos: - Line 181: Intuitive => Intuitively - Lines 236-237: "the node of containing x" => "the node containing x" - Line 222 and also line 298: "it is interesting to exploit the convergence rates": Do you mean "explore" or "analyse"?