Gérard Biau
email: gerard.biau@upmc.fr

Analysis of a Random Forests Model

Keywords: Random forests, randomization, sparsity, dimension reduction, consistency, rate of convergence. 2010 Mathematics Subject Classification: 62G05, 62G20

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1 Introduction

Random forests

In a series of papers and technical reports, Breiman [START_REF] Breiman | Bagging predictors[END_REF][START_REF] Breiman | Some infinity theory for predictor ensembles[END_REF][START_REF] Breiman | Random forests[END_REF][START_REF] Breiman | Consistency for a simple model of random forests[END_REF] demonstrated that substantial gains in classification and regression accuracy can be achieved by using ensembles of trees, where each tree in the ensemble is grown in accordance with a random parameter. Final predictions are obtained by aggregating over the ensemble. As the base constituents of the ensemble are tree-structured predictors, and since each of these trees is constructed using an injection of randomness, these procedures are called "random forests".

Breiman's ideas were decisively influenced by the early work of Amit and Geman [START_REF] Amit | Shape quantization and recognition with randomized trees[END_REF] on geometric feature selection, the random subspace method of Ho [START_REF] Ho | The random subspace method for constructing decision forests[END_REF] and the random split selection approach of Dietterich [START_REF] Dietterich | An experimental comparison of three methods for constructing ensembles of decision trees: Bagging, boosting, and randomization[END_REF]. As highlighted by various empirical studies (see [START_REF] Breiman | Random forests[END_REF][START_REF] Svetnik | Random forest: A classification and regression tool for compound classification and QSAR modeling[END_REF][START_REF] Diaz-Uriarte | Gene selection and classification of microarray data using random forest[END_REF][START_REF] Genuer | Random Forests: Some methodological insights[END_REF][START_REF] Genuer | Variable selection using random forests[END_REF] for instance), random forests have emerged as serious competitors to state-of-the-art methods such as boosting (Freund [22]) and support vector machines (Shawe-Taylor and Cristianini [START_REF] Shawe-Taylor | Kernel Methods for Pattern Analysis[END_REF]). They are fast and easy to implement, produce highly accurate predictions and can handle a very large number of input variables without overfitting. In fact, they are considered to be one of the most accurate general-purpose learning techniques available. The survey by Genuer et al. [START_REF] Genuer | Random Forests: Some methodological insights[END_REF] may provide the reader with practical guidelines and a good starting point for understanding the method. In Breiman's approach, each tree in the collection is formed by first selecting at random, at each node, a small group of input coordinates (also called features or variables hereafter) to split on and, secondly, by calculating the best split based on these features in the training set. The tree is grown using CART methodology (Breiman et al. [START_REF] Breiman | Classification and Regression Trees[END_REF]) to maximum size, without pruning. This subspace randomization scheme is blended with bagging ( [START_REF] Breiman | Bagging predictors[END_REF][START_REF] Bühlmann | Analyzing bagging[END_REF][START_REF] Buja | Observations on bagging[END_REF][START_REF] Biau | On the rate of convergence of the bagged nearest neighbor estimate[END_REF]) to resample, with replacement, the training data set each time a new individual tree is grown.

Although the mechanism appears simple, it involves many different driving forces which make it difficult to analyse. In fact, its mathematical properties remain to date largely unknown and, up to now, most theoretical studies have concentrated on isolated parts or stylized versions of the algorithm. Interesting attempts in this direction are by Lin and Jeon [START_REF] Lin | Random forests and adaptive nearest neighbors[END_REF], who establish a connection between random forests and adaptive nearest neighbor methods (see also [START_REF] Biau | On the layered nearest neighbour estimate, the bagged nearest neighbour estimate and the random forest method in regression and classification[END_REF] for further results); Meinshausen [START_REF] Meinshausen | Quantile regression forests[END_REF], who studies the consistency of random forests in the context of conditional quantile prediction; and Devroye et al. [START_REF] Biau | Consistency of random forests and other averaging classifiers[END_REF], who offer consistency theorems for various simplified versions of random forests and other randomized ensemble predictors. Nevertheless, the statistical mechanism of "true" random forests is not yet fully understood and is still under active investigation.

In the present paper, we go one step further into random forests by working out and solidifying the properties of a model suggested by Breiman in [START_REF] Breiman | Consistency for a simple model of random forests[END_REF]. Though this model is still simple compared to the "true" algorithm, it is nevertheless closer to reality than any other scheme we are aware of. The short draft [START_REF] Breiman | Consistency for a simple model of random forests[END_REF] is essentially based on intuition and mathematical heuristics, some of them are questionable and make the document difficult to read and understand. However, the ideas presented by Breiman are worth clarifying and developing, and they will serve as a starting point for our study.

Before we formalize the model, some definitions are in order. Throughout the document, we suppose that we are given a training sample D n = {(X 1 , Y 1 ), . . . , (X n , Y n )} of i.i.d. [0, 1] d × R-valued random variables (d ≥ 2) with the same distribution as an independent generic pair (X, Y ) satisfying EY 2 < ∞. The space [0, 1] d is equipped with the standard Euclidean metric. For fixed x ∈ [0, 1] d , our goal is to estimate the regression function r(x) = E[Y |X = x] using the data D n . In this respect, we say that a regression function estimate r n is consistent if E[r n (X) -r(X)] 2 → 0 as n → ∞. The main message of this paper is that Breiman's procedure is consistent and adapts to sparsity, in the sense that its rate of convergence depends only on the number of strong features and not on how many noise variables are present.

The model

Formally, a random forest is a predictor consisting of a collection of randomized base regression trees {r n (x, Θ m , D n ), m ≥ 1}, where Θ 1 , Θ 2 , . . . are i.i.d. outputs of a randomizing variable Θ. These random trees are combined to form the aggregated regression estimate rn (X,

D n ) = E Θ [r n (X, Θ, D n )] ,
where E Θ denotes expectation with respect to the random parameter, conditionally on X and the data set D n . In the following, to lighten notation a little, we will omit the dependency of the estimates in the sample, and write for example rn (X) instead of rn (X, D n ). Note that, in practice, the above expectation is evaluated by Monte Carlo, i.e., by generating M (usually large) random trees, and taking the average of the individual outcomes (this procedure is justified by the law of large numbers, see the appendix in Breiman [START_REF] Breiman | Random forests[END_REF]). The randomizing variable Θ is used to determine how the successive cuts are performed when building the individual trees, such as selection of the coordinate to split and position of the split.

In the model we have in mind, the variable Θ is assumed to be independent of X and the training sample D n . This excludes in particular any bootstrapping or resampling step in the training set. This also rules out any data-dependent strategy to build the trees, such as searching for optimal splits by optimizing some criterion on the actual observations. However, we allow Θ to be based on a second sample, independent of, but distributed as, D n . This important issue will be thoroughly discussed in Section 3.

With these warnings in mind, we will assume that each individual random tree is constructed in the following way. All nodes of the tree are associated with rectangular cells such that at each step of the construction of the tree, the collection of cells associated with the leaves of the tree (i.e., external nodes) forms a partition of [0, 1] d . The root of the tree is [0, 1] d itself. The following procedure is then repeated log 2 k n times, where log 2 is the base-2 logarithm, . the ceiling function and k n ≥ 2 a deterministic parameter, fixed beforehand by the user, and possibly depending on n.

1. At each node, a coordinate of X = (X (1) , . . . , X (d) ) is selected, with the j-th feature having a probability p nj ∈ (0, 1) of being selected.

2. At each node, once the coordinate is selected, the split is at the midpoint of the chosen side.

Each randomized tree r n (X, Θ) outputs the average over all Y i for which the corresponding vectors X i fall in the same cell of the random partition as X. In other words, letting A n (X, Θ) be the rectangular cell of the random partition containing X,

r n (X, Θ) = n i=1 Y i 1 [X i ∈An(X,Θ)] n i=1 1 [X i ∈An(X,Θ)] 1 En(X,Θ) ,
where the event E n (X, Θ) is defined by

E n (X, Θ) = n i=1 1 [X i ∈An(X,Θ)] = 0 .
(Thus, by convention, the estimate is set to 0 on empty cells.) Taking finally expectation with respect to the parameter Θ, the random forests regression estimate takes the form

rn (X) = E Θ [r n (X, Θ)] = E Θ n i=1 Y i 1 [X i ∈An(X,Θ)] n i=1 1 [X i ∈An(X,Θ)]
1 En(X,Θ) .

Let us now make some general remarks about this random forests model. First of all, we note that, by construction, each individual tree has exactly 2 log 2 kn (≈ k n ) terminal nodes, and each leaf has Lebesgue measure 2 -log 2 kn (≈ 1/k n ). Thus, if X has uniform distribution on [0, 1] d , there will be on average about n/k n observations per terminal node. In particular, the choice k n = n induces a very small number of cases in the final leaves, in accordance with the idea that the single trees should not be pruned.

Next, we see that, during the construction of the tree, at each node, each candidate coordinate X (j) may be chosen with probability p nj ∈ (0, 1). This implies in particular d j=1 p nj = 1. Although we do not precise for the moment the way these probabilities are generated, we stress that they may be induced by a second sample. This includes the situation where, at each node, randomness is introduced by selecting at random (with or without replacement) a small group of input features to split on, and choosing to cut the cell along the coordinate-inside this group-which most decreases some empirical criterion evaluated on the extra sample. This scheme is close to what the original random forests algorithm does, the essential difference being that the latter algorithm uses the actual data set to calculate the best splits. This point will be properly discussed in Section 3.

Finally, the requirement that the splits are always achieved at the middle of the cell sides is mainly technical, and it could eventually be replaced by a more involved random mechanism-based on the second sample-, at the price of a much more complicated analysis.

The document is organized as follows. In Section 2, we prove that the random forests regression estimate rn is consistent and discuss its rate of convergence. As a striking result, we show under a sparsity framework that the rate of convergence depends only on the number of active (or strong) variables and not on the dimension of the ambient space. This feature is particularly desirable in high-dimensional regression, when the number of variables can be much larger than the sample size, and may explain why random forests are able to handle a very large number of input variables without overfitting. Section 3 is devoted to a discussion, and a small simulation study is presented in Section 4. For the sake of clarity, proofs are postponed to Section 5.

Asymptotic analysis

Throughout the document, we denote by N n (X, Θ) the number of data points falling in the same cell as X, i.e.,

N n (X, Θ) = n i=1 1 [X i ∈An(X,Θ)] .
We start the analysis with the following simple theorem, which shows that the random forests estimate rn is consistent.

Theorem 2.1 Assume that the distribution of X has support on [0, 1] d . Then the random forests estimate rn is consistent whenever p nj log k n → ∞ for all j = 1, . . . , d and k n /n → 0 as n → ∞.

Theorem 2.1 mainly serves as an illustration of how the consistency problem of random forests predictors may be attacked. It encompasses, in particular, the situation where, at each node, the coordinate to split is chosen uniformly at random over the d candidates. In this "purely random" model, p nj = 1/d, independently of n and j, and consistency is ensured as long as k n → ∞ and k n /n → 0. This is however a radically simplified version of the random forests used in practice, which does not explain the good performance of the algorithm. To achieve this goal, a more in-depth analysis is needed.

There is empirical evidence that many signals in high-dimensional spaces admit a sparse representation. As an example, wavelet coefficients of images often exhibit exponential decay, and a relatively small subset of all wavelet coefficients allows for a good approximation of the original image. Such signals have few non-zero coefficients and can therefore be described as sparse in the signal domain (see for instance [START_REF] Bruckstein | From sparse solutions of systems of equations to sparse modeling of signals and images[END_REF]). Similarly, recent advances in high-throughput technologies-such as array comparative genomic hybridization-indicate that, despite the huge dimensionality of problems, only a small number of genes may play a role in determining the outcome and be required to create good predictors ( [START_REF] Van't Veer | Gene expression profiling predicts clinical outcome of breast cancer[END_REF] for instance). Sparse estimation is playing an increasingly important role in the statistics and machine learning communities, and several methods have recently been developed in both fields, which rely upon the notion of sparsity (e.g. penalty methods like the Lasso and Dantzig selector, see [START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF][START_REF] Candès | The Dantzig selector: Statistical estimation when p is much larger than n[END_REF][START_REF] Bunea | Sparsity oracle inequalities for the Lasso[END_REF][START_REF] Bickel | Simultaneous analysis of Lasso and Dantzig selector[END_REF] and the references therein).

Following this idea, we will assume in our setting that the target regression function r(X) = E[Y |X], which is initially a function of X = (X (1) , . . . , X (d) ), depends in fact only on a nonempty subset S (for Strong) of the d features. In other words, letting X S = (X j : j ∈ S) and S = Card S, we have

r(X) = E[Y |X S ] or equivalently, for any x ∈ [0, 1] d , r(x) = r (x S ) µ-a.s., (2.1)
where µ is the distribution of X and r : [0, 1] S → R is the section of r corresponding to S. To avoid trivialities, we will assume throughout that S is nonempty, with S ≥ 2. The variables in the set W = {1, . . . , d} -S (for Weak) have thus no influence on the response and could be safely removed.

In the dimension reduction scenario we have in mind, the ambient dimension d can be very large, much larger than the sample size n, but we believe that the representation is sparse, i.e., that very few coordinates of r are non-zero, with indices corresponding to the set S. Note however that representation (2.1) does not forbid the somehow undesirable case where S = d. As such, the value S characterizes the sparsity of the model: The smaller S, the sparser r.

Within this sparsity framework, it is intuitively clear that the coordinatesampling probabilities should ideally satisfy the constraints p nj = 1/S for j ∈ S (and, consequently, p nj = 0 otherwise). However, this is a too strong requirement, which has no chance to be satisfied in practice, except maybe in some special situations where we know beforehand which variables are important and which are not. Thus, to stick to reality, we will rather require in the following that p nj = (1/S)(1+ξ nj ) for j ∈ S (and p nj = ξ nj otherwise), where p nj ∈ (0, 1) and each ξ nj tends to 0 as n tends to infinity. We will see in Section 3 how to design a randomization mechanism to obtain such probabilities, on the basis of a second sample independent of the training set D n . At this point, it is important to note that the dimensions d and S are held constant throughout the document. In particular, these dimensions are not functions of the sample size n, as it may be the case in other asymptotic studies.

We have now enough material for a deeper understanding of the random forests algorithm. To lighten notation a little, we will write

W ni (X, Θ) = 1 [X i ∈An(X,Θ)] N n (X, Θ) 1 En(X,Θ) ,
so that the estimate takes the form

rn (X) = n i=1 E Θ [W ni (X, Θ)] Y i .
Let us start with the variance/bias decomposition

E [r n (X) -r(X)] 2 = E [r n (X) -rn (X)] 2 + E [r n (X) -r(X)] 2 , (2.2)
where we set

rn (X) = n i=1 E Θ [W ni (X, Θ)] r(X i ).
The two terms of (2.2) will be examined separately, in Proposition 2.1 and Proposition 2.2, respectively. Throughout, the symbol V denotes variance.

Proposition 2.1 Assume that X is uniformly distributed on [0, 1] d and, for all

x ∈ R d , σ 2 (x) = V[Y | X = x] ≤ σ 2
for some positive constant σ 2 . Then, if

p nj = (1/S)(1 + ξ nj ) for j ∈ S, E [r n (X) -rn (X)] 2 ≤ Cσ 2 S 2 S -1 S/2d (1 + ξ n ) k n n(log k n ) S/2d , where C = 288 π π log 2 16 S/2d
.

The sequence (ξ n ) depends on the sequences {(ξ nj ) : j ∈ S} only and tends to 0 as n tends to infinity.

Remark 1 A close inspection of the end of the proof of Proposition 2.1 reveals that

1 + ξ n = j∈S (1 + ξ nj ) -1 1 - ξ nj S -1 -1 1/2d
.

In particular, if a < p nj < b for some constants a, b ∈ (0, 1), then

1 + ξ n ≤ S -1 S 2 a(1 -b) S/2d
. The main message of Proposition 2.1 is that the variance of the forests estimate is O(k n /(n(log k n ) S/2d )). This result is interesting by itself since it shows the effect of aggregation on the variance of the forest. To understand this remark, recall that individual (random or not) trees are proved to be consistent by letting the number of cases in each terminal node become large (see [START_REF] Devroye | A Probabilistic Theory of Pattern Recognition[END_REF]Chapter 20]), with a typical variance of the order k n /n. Thus, for such trees, the choice k n = n (i.e., about one observation on average in each terminal node) is clearly not suitable and leads to serious overfitting and variance explosion. On the other hand, the variance of the forest is of the order k n /(n(log k n ) S/2d ). Therefore, letting k n = n, the variance is of the order 1/(log n) S/2d , a quantity which still goes to 0 as n grows! Proof of Proposition 2.1 reveals that this log term is a by-product of the Θ-averaging process, which appears by taking into consideration the correlation between trees. We believe that it provides an interesting perspective on why random forests are still able to do a good job, despite the fact that individual trees are not pruned.

Note finally that the requirement that X is uniformly distributed on the hypercube could be safely replaced by the assumption that X has a density with respect to the Lebesgue measure on [0, 1] d and the density is bounded from above and from below. The case where the density of X is not bounded from below necessitates a specific analysis, which we believe is beyond the scope of the present paper. We refer the reader to [START_REF] Biau | On the layered nearest neighbour estimate, the bagged nearest neighbour estimate and the random forest method in regression and classification[END_REF] for results in this direction (see also Remark 5 in Section 5).

Let us now turn to the analysis of the bias term in equality (2.2). Recall that r denotes the section of r corresponding to S.

Proposition 2.2 Assume that X is uniformly distributed on [0, 1] d and r is L-Lipschitz on [0, 1] S . Then, if p nj = (1/S)(1 + ξ nj ) for j ∈ S, E [r n (X) -r(X)] 2 ≤ 2SL 2 k 0.75 S log 2 (1+γn) n + sup x∈[0,1] d r 2 (x) e -n/2kn
, where γ n = min j∈S ξ nj tends to 0 as n tends to infinity.

This result essentially shows that the rate at which the bias decreases to 0 depends on the number of strong variables, not on d. In particular, the quantity k n -(0.75/(S log 2))(1+γn) should be compared with the ordinary partitioning estimate bias, which is of the order k n -2/d under the smoothness conditions of Proposition 2.2 (see for instance [START_REF] Györfi | A Distribution-Free Theory of Nonparametric Regression[END_REF]). In this respect, it is easy to see

that k n -(0.75/(S log 2))(1+γn) = o(k n -2/d
) as soon as S ≤ 0.54d ( . is the integer part function). In other words, when the number of active variables is less than (roughly) half of the ambient dimension, the bias of the random forests regression estimate decreases to 0 much faster than the usual rate. The restriction S ≤ 0.54d is not severe, since in all practical situations we have in mind, d is usually very large with respect to S (this is, for instance, typically the case in modern genome biology problems, where d may be of the order of billion, and in any case much larger than the actual number of active features). Note at last that, contrary to Proposition 2.1, the term e -n/2kn prevents the extreme choice k n = n (about one observation on average in each terminal node). Indeed, an inspection of the proof of Proposition 2.2 reveals that this term accounts for the probability that N n (X, Θ) is precisely 0, i.e., A n (X, Θ) is empty.

Recalling the elementary inequality ze -nz ≤ e -1 /n for z ∈ [0, 1], we may finally join Proposition 2.1 and Proposition 2.2 and state our main theorem.

Theorem 2.2 Assume that X is uniformly distributed on [0, 1] d , r is L- Lipschitz on [0, 1] S and, for all x ∈ R d , σ 2 (x) = V[Y | X = x] ≤ σ 2 for some positive constant σ 2 . Then, if p nj = (1/S)(1+ξ nj ) for j ∈ S, letting γ n = min j∈S ξ nj , we have E [r n (X) -r(X)] 2 ≤ Ξ n k n n + 2SL 2 k 0.75 S log 2 (1+γn) n
, where

Ξ n = Cσ 2 S 2 S -1 S/2d (1 + ξ n ) + 2e -1 sup x∈[0,1] d r 2 (x) and C = 288 π π log 2 16 S/2d
.

The sequence (ξ n ) depends on the sequences {(ξ nj ) : j ∈ S} only and tends to 0 as n tends to infinity.

As we will see in Section 3, it may be safely assumed that the randomization process allows for ξ nj log n → 0 as n → ∞, for all j ∈ S. Thus, under this condition, Theorem 2.2 shows that with the optimal choice

k n ∝ n 1/(1+ 0.75 S log 2 ) ,
we get

E [r n (X) -r(X)] 2 = O n -0.75 S log 2+0.75
. This result can be made more precise. Denote by F S the class of (L, σ 2 )smooth distributions (X, Y ) such that X has uniform distribution on [0, 1] d , the regression function r is Lipschitz with constant L on [0, 1] S and, for all

x ∈ R d , σ 2 (x) = V[Y | X = x] ≤ σ 2 . Corollary 2.1 Let Ξ = Cσ 2 S 2 S -1 S/2d + 2e -1 sup x∈[0,1] d r 2 (x) and C = 288 π π log 2 16 S/2d
.

Then, if p nj = (1/S)(1 + ξ nj ) for j ∈ S, with ξ nj log n → 0 as n → ∞, for the choice

k n ∝ L 2 Ξ 1/(1+ 0.75 S log 2 ) n 1/(1+ 0.75 S log 2 ) , we have lim sup n→∞ sup (X,Y )∈F S E [r n (X) -r(X)] 2 ΞL 2S log 2 0.75 0.75 S log 2+0.75 n -0.75 S log 2+0.75 ≤ Λ,
where Λ is a positive constant independent of r, L and σ 2 . This result reveals the fact that the L 2 -rate of convergence of rn (X) to r(X) depends only on the number S of strong variables, and not on the ambient dimension d. The main message of Corollary 2.1 is that if we are able to properly tune the probability sequences (p nj ) n≥1 and make them sufficiently fast to track the informative features, then the rate of convergence of the random forests estimate will be of the order n -0.75 S log 2+0.75 . This rate is strictly faster than the usual rate n -2/(d+2) as soon as S ≤ 0.54d . To understand this point, just recall that the rate n -2/(d+2) is minimax optimal for the class F d (see for example Ibragimov and Khasminskii [START_REF] Ibragimov | On nonparametric estimation of regression[END_REF][START_REF] Ibragimov | Statistical Estimation: Asymptotic Theory[END_REF][START_REF] Ibragimov | On the bounds for quality of nonparametric regression function estimation[END_REF]), seen as a collection of regression functions over [0, 1] d , not [0, 1] S . However, in our setting, the intrinsic dimension of the regression problem is S, not d, and the random forests estimate cleverly adapts to the sparsity of the problem. As an illustration, Figure 1 shows the plot of the function S → 0.75/(S log 2 + 0.75) for S ranging from 2 to d = 100.

It is noteworthy that the rate of convergence of the ξ nj to 0 (and, consequently, the rate at which the probabilities p nj approach 1/S for j ∈ S) will eventually depend on the ambient dimension d through the ratio S/d. The same is true for the Lipschitz constant L and the factor sup x∈[0,1] d r 2 (x) which both appear in Corollary 2.1. To figure out this remark, remember first that the support of r is contained in R S , so that the later supremum (respectively, the Lipschitz constant) is in fact a supremum (respectively, a Lipschitz constant) over R S , not over R d . Next, denote by C p (s) the collection of functions η : [0, 1] p → [0, 1] for which each derivative of order s satisfies a Lipschitz condition. It is well known that the ε-entropy log 2 (N ε ) of C p (s) is Φ(ε -p/(s+1) ) as ε ↓ 0 (Kolmogorov and Tihomirov [START_REF] Kolmogorov | ε-entropy and ε-capacity of sets in functional spaces[END_REF]), where

a n = Φ(b n ) means that a n = O(b n ) and b n = O(a n ).
Here we have an interesting interpretation of the dimension reduction phenomenon: Working with Lipschitz functions on R S (that is, s = 0) is roughly equivalent to working with functions on R d for which all [(d/S) -1]-th order derivatives are Lipschitz! For example, if S = 1 and d = 25, (d/S) -1 = 24 and, as there are 25 24 such partial derivatives in R 25 , we note immediately the potential benefit of recovering the "true" dimension S.

Remark 2

The reduced-dimensional rate n -0.75 S log 2+0.75 is strictly larger than the S-dimensional optimal rate n -2/(S+2) , which is also shown in Figure 1 for S ranging from 2 to 100. We do not know whether the latter rate can be achieved by the algorithm.

Remark 3

The optimal parameter k n of Corollary 2.1 depends on the unknown distribution of (X, Y ), especially on the smoothness of the regression function and the effective dimension S. To correct this situation, adaptive (i.e., data-dependent) choices of k n , such as data-splitting or cross-validation, should preserve the rate of convergence of the estimate. Another route we may follow is to analyse the effect of bootstrapping the sample before growing the individual trees (i.e., bagging). It is our belief that this procedure should also preserve the rate of convergence, even for overfitted trees (k n ≈ n), in the spirit of [START_REF] Biau | On the rate of convergence of the bagged nearest neighbor estimate[END_REF]. However, such a study is beyond the scope of the present paper.

Remark 4 For further references, it is interesting to note that Proposition 2.1 (variance term) is a consequence of aggregation, whereas Proposition 2.2 (bias term) is a consequence of randomization.

It is also stimulating to keep in mind the following analysis, which has been suggested to us by a referee. Suppose, to simplify, that Y = r(X) (no-noise regression) and that n i=1 W ni (X, Θ) = 1 a.s. In this case, the variance term is 0 and we have

rn (X) = rn (X) = n i=1 E Θ [W ni (Θ, X)] Y i . Set Z n = (Y, Y 1 , . . . , Y n ). Then E [r n (X) -r(X)] 2 = E [r n (X) -Y ] 2 = E E (r n (X) -Y ) 2 | Z n = E E (r n (X) -E[r n (X) | Z n ]) 2 | Z n + E [E[r n (X) | Z n ] -Y ] 2 .
The conditional expectation in the first of the two terms above may be rewritten under the form

E [Cov (E Θ [r n (X, Θ)] , E Θ [r n (X, Θ )] | Z n )]
, where Θ is distributed as, and independent of, Θ. Attention shows that this last term is indeed equal to

E [E Θ,Θ Cov (r n (X, Θ), r n (X, Θ ) | Z n )]
The key observation is that if trees have strong predictive power, then they can be unconditionally strongly correlated while being conditionally weakly correlated. This opens an interesting line of research for the statistical analysis of the bias term, in connection with Amit [START_REF] Amit | 2D Object Detection and Recognition: Models, Algorithms, and Networks[END_REF] and Blanchard [START_REF] Blanchard | Different paradigms for choosing sequential reweighting algorithms[END_REF] conditional covariance-analysis ideas.

Discussion

The results which have been obtained in Section 2 rely on appropriate behavior of the probability sequences (p nj ) n≥1 , j = 1, . . . , d. We recall that these sequences should be in (0, 1) and obey the constraints p nj = (1/S)(1 + ξ nj ) for j ∈ S (and p nj = ξ nj otherwise), where the (ξ nj ) n≥1 tend to 0 as n tends to infinity. In other words, at each step of the construction of the individual trees, the random procedure should track and preferentially cut the strong coordinates. In this more informal section, we briefly discuss a random mechanism for inducing such probability sequences. Suppose, to start with an imaginary scenario, that we already know which coordinates are strong, and which are not. In this ideal case, the random selection procedure described in the introduction may be easily made more precise as follows. A positive integer M n -possibly depending on n-is fixed beforehand and the following splitting scheme is iteratively repeated at each node of the tree:

1. Select at random, with replacement, M n candidate coordinates to split on.

2. If the selection is all weak, then choose one at random to split on. If there is more than one strong variable elected, choose one at random and cut.

Within this framework, it is easy to see that each coordinate in S will be cut with the "ideal" probability

p n = 1 S 1 -1 - S d Mn .
Though this is an idealized model, it already gives some information about the choice of the parameter M n , which, in accordance with the results of Section 2 (Corollary 2.1), should satisfy

1 - S d Mn log n → 0 as n → ∞.
This is true as soon as

M n → ∞ and M n log n → ∞ as n → ∞.
This result is consistent with the general empirical finding that M n (called mtry in the R package RandomForests) does not need to be very large (see, for example, Breiman [START_REF] Breiman | Random forests[END_REF]), but not with the widespread belief that M n should not depend on n. Note also that if the M n features are chosen at random without replacement, then things are even more simple since, in this case, p n = 1/S for all n large enough.

In practice, we have only a vague idea about the size and content of the set S. However, to circumvent this problem, we may use the observations of an independent second set D n (say, of the same size as D n ) in order to mimic the ideal split probability p n . To illustrate this mechanism, suppose-to keep things simple-that the model is linear, i.e., Y = j∈S a j X (j) + ε, where X = (X (1) , . . . , X (d) ) is uniformly distributed over [0, 1] d , the a j are non-zero real numbers, and ε is a zero-mean random noise, which is assumed to be independent of X and with finite variance. Note that, in accordance with our sparsity assumption, r(X) = j∈S a j X (j) depends on X S only.

Assume now that we have done some splitting and arrived at a current set of terminal nodes. Consider any of these nodes, say A = d j=1 A j , fix a coordinate j ∈ {1, . . . , d}, and look at the weighted conditional variance V[Y |X (j) ∈ A j ] P(X (j) ∈ A j ). It is a simple exercise to prove that if X is uniform and j ∈ S, then the split on the j-th side which most decreases the weighted conditional variance is at the midpoint of the node, with a variance decrease equal to a 2 j /16 > 0. On the other hand, if j ∈ W, the decrease of the variance is always 0, whatever the location of the split.

On the practical side, the conditional variances are of course unknown, but they may be estimated by replacing the theoretical quantities by their respective sample estimates (as in the CART procedure, see Breiman et al. [START_REF] Breiman | Random forests[END_REF]Chapter 8] for a thorough discussion) evaluated on the second sample D n . This suggests the following procedure, at each node of the tree:

1. Select at random, with replacement, M n candidate coordinates to split on.

2. For each of the M n elected coordinates, calculate the best split, i.e., the split which most decreases the within-node sum of squares on the second sample D n .

3. Select one variable at random among the coordinates which output the best within-node sum of squares decreases, and cut.

This procedure is indeed close to what the random forests algorithm does.

The essential difference is that we suppose to have at hand a second sample D n , whereas the original algorithm performs the search of the optimal cuts on the original observations D n . This point is important, since the use of an extra sample preserves the independence of Θ (the random mechanism) and D n (the training sample). We do not know whether our results are still true if Θ depends on D n (as in the CART algorithm), but the analysis does not appear to be simple. Note also that, at step 3, a threshold (or a test procedure, as suggested in Amaratunga et al. [START_REF] Amaratunga | Enriched random forests[END_REF]) could be used to choose among the most significant variables, whereas the actual algorithm just selects the best one. In fact, depending on the context and the actual cut selection procedure, the informative probabilities p nj (j ∈ S) may obey the constraints p nj → p j as n → ∞ (thus, p j is not necessarily equal to 1/S), where the p j are positive and satisfy j∈S p j = 1. This should not affect the results of the article.

This empirical randomization scheme leads to complicate probabilities of cuts which, this time, vary at each node of each tree and are not easily amenable to analysis. Nevertheless, observing that the average number of cases per terminal node is about n/k n , it may be inferred by the law of large numbers that each variable in S will be cut with probability

p nj ≈ 1 S 1 -1 - S d Mn (1 + ζ nj ),
where ζ nj is of the order O(k n /n), a quantity which anyway goes fast to 0 as n tends to infinity. Put differently, for j ∈ S,

p nj ≈ 1 S (1 + ξ nj ) ,
where ξ nj goes to 0 and satisfies the constraint ξ nj log n → 0 as n tends to infinity, provided k n log n/n → 0, M n → ∞ and M n / log n → ∞. This is coherent with the requirements of Corollary 2.1. We realize however that this is a rough approach, and that more theoretical work is needed here to fully understand the mechanisms involved in CART and Breiman's original randomization process.

It is also noteworthy that random forests use the so-called out-of-bag samples (i.e., the bootstrapped data which are not used to fit the trees) to construct a variable importance criterion, which measures the prediction strength of each feature (see, e.g., Genuer et al. [START_REF] Genuer | Variable selection using random forests[END_REF]). As far as we are aware, there is to date no systematic mathematical study of this criterion. It is our belief that such a study would greatly benefit from the sparsity point of view developed in the present paper, but is unfortunately much beyond its scope. Lastly, it would also be interesting to work out and extend our results to the context of unsupervised learning of trees. A good route to follow with this respect is given by the strategies outlined in Amit and Geman [3, Section 5.5].

A small simulation study

Even though the first vocation of the present paper is theoretical, we offer in this short section some experimental results on synthetic data. Our aim is not to provide a thorough practical study of the random forests method, but rather to illustrate the main ideas of the article. As for now, we let U([0, 1] d ) (respectively, N (0, 1)) be the uniform distribution over [0, 1] d (respectively, the standard Gaussian distribution). Specifically, three models were tested:

1.

[Sinus] For x ∈ [0, 1] d , the regression function takes the form r(x) = 10 sin(10πx (1) ).

We let Y = r(X) + ε and X ∼ U([0, 1] d ) (d ≥ 1), with ε ∼ N (0, 1).

[Friedman #1]

This is a model proposed in Friedman [START_REF] Friedman | Multivariate adaptive regression splines[END_REF]. Here, r(x) = 10 sin(πx (1) x (2) ) + 20(x (3) -.05) 2 + 10x (4) + 5x (5) and Y = r(X) + ε, where X ∼ U([0, 1] d ) (d ≥ 5) and ε ∼ N (0, 1).

[Tree]

In this example, we let Y = r(X) + ε, where X ∼ U([0, 1] d ) (d ≥ 5), ε ∼ N (0, 1) and the function r has itself a tree structure. This tree-type function, which is shown in Figure 2, involves only five variables.

We note that, although the ambient dimension d may be large, the effective dimension of model 1 is S = 1, whereas model 2 and model 3 have S = 5.

In other words, S = {1} for model 1, whereas S = {1, . . . , 5} for model 2 and model 3. Observe also that, in our context, the model Tree should be considered as a "no-bias" model, on which the random forests algorithm is expected to perform well.

In a first series of experiments, we let d = 100 and, for each of the three models and different values of the sample size n, we generated a learning set of size n and fitted a forest (10 000 trees) with mtry = d. For j = 1, . . . , d, the ratio (number of times the j-th coordinate is split)/(total number of splits over the forest) was evaluated, and the whole experiment was repeated 100 times. Figure 3, Figure 4 and Figure 5 report the resulting boxplots for each of the first twenty variables and different values of n. These figures clearly enlighten the fact that, as n grows, the probability of cuts does concentrate on the informative variables only and support the assumption that ξ nj → 0 as n → ∞ for each j ∈ S.

Next, in a second series of experiments, for each model, for different values of d and for sample sizes n ranging from 10 to 1000, we generated a learning set of size n, a test set of size 50 000 and evaluated the mean squared error (MSE) of the random forests (RF) method via the Monte Carlo approximation MSE ≈ 1 50 000

50 000 j=1 [RF(test data #j) -r(test data #j)] 2 .
All results were averaged over 100 data sets. The random forests algorithm was performed with the parameter mtry automatically tuned by the R package RandomForests, 1000 random trees and the minimum node size set to 5 (which is the default value for regression). Besides, in order to compare the "true" algorithm with the approximate model discussed in the present document, an alternative method was also tested. This auxiliary algorithm has characteristics which are identical to the original ones (same mtry, same q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5

n = 10

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5

n = 25

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5

n = 50

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5

n = 100

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5

n = 200

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5 number of random trees), with the notable difference that now the maximum number of nodes is fixed beforehand. For the sake of coherence, since the minimum node size is set to 5 in the RandomForests package, the number of terminal nodes in the custom algorithm was calibrated to n/5 . It must be stressed that the essential difference between the standard random forests algorithm and the alternative one is that the number of cases in the final leaves is fixed in the former, whereas the latter assumes a fixed number of terminal nodes. In particular, in both algorithms, cuts are performed using the actual sample, just as CART does. To keep things simple, no data-splitting procedure has been incorporated in the modified version.

Figure 6, Figure 7 and Figure 8 illustrate the evolution of the MSE value with respect to n and d, for each model and the two tested procedures. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5

n = 10

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5

n = 25

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5

n = 50

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5 7 9 11 [START_REF] Breiman | Classification and Regression Trees[END_REF]

n = 100

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5 7 9 11 [START_REF] Breiman | Classification and Regression Trees[END_REF]

n = 500

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5 7 9 11 [START_REF] Breiman | Classification and Regression Trees[END_REF] First, we note that the overall performance of the alternative method is very similar to the one of the original algorithm. This confirms our idea that the model discussed in the present paper is a good approximation of the authentic Breiman's forests. Next, we see that for a sufficiently large n, the capabilities of the forests are nearly independent of d, in accordance with the idea that the (asymptotic) rate of convergence of the method should only depend on the "true" dimensionality S (Theorem 2.2). Finally, as expected, it is noteworthy that both algorithms perform well on the third model, which has been precisely designed for a tree-structured predictor.

q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5

n = 10

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5

n = 25

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5

n = 50

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5 7 9 11 [START_REF] Breiman | Classification and Regression Trees[END_REF]

n = 100

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5 7 9 11 [START_REF] Breiman | Classification and Regression Trees[END_REF]

n = 200

Variable index Proportion of cuts q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 3 5 7 9 11 [START_REF] Breiman | Classification and Regression Trees[END_REF]

Proofs

Throughout this section, we will make repeated use of the following two facts.

Fact 5.1 Let K nj (X, Θ) be the number of times the terminal node A n (X, Θ) is split on the j-th coordinate (j = 1, . . . , d). Then, conditionally on X, K nj (X, Θ) has binomial distribution with parameters log 2 k n and p nj (by independence of X and Θ). Moreover, by construction, Recall that we denote by N n (X, Θ) the number of data points falling in the same cell as X, i.e.,

N n (X, Θ) = n i=1 1 [X i ∈An(X,Θ)] .
Let λ be the Lebesgue measure on [0, 1] d .

Fact 5.2 By construction,

λ (A n (X, Θ)) = 2 -log 2 kn .
In particular, if X is uniformly distributed on [0, 1] d , then the distribution of N n (X, Θ) conditionally on X and Θ is binomial with parameters n and 2 -log 2 kn (by independence of the random variables X, X 1 , . . . , X n , Θ).

Remark 5 If X is not uniformly distributed but has a probability density f on [0, 1] d , then, conditionally on X and Θ, N n (X, Θ) is binomial with parameters n and P(X 1 ∈ A n (X, Θ) | X, Θ). If f is bounded from above and from below, this probability is of the order λ (A n (X, Θ)) = 2 -log 2 kn , and the whole approach can be carried out without difficulty. On the other hand, for more general densities, the binomial probability depends on X, and this makes the analysis significantly harder.

Proof of Theorem 2.1

Observe first that, by Jensen's inequality,

E [r n (X) -r(X)] 2 = E [E Θ [r n (X, Θ) -r(X)]] 2 ≤ E [r n (X, Θ) -r(X)] 2 .
A slight adaptation of Theorem 4.2 in Györfi et al. [START_REF] Györfi | A Distribution-Free Theory of Nonparametric Regression[END_REF] shows that rn is consistent if both diam(A n (X, Θ)) → 0 in probability and N n (X, Θ) → ∞ in probability.

Let us first prove that N n (X, Θ) → ∞ in probability. To see this, consider the random tree partition defined by Θ, which has by construction exactly 2 log 2 kn rectangular cells, say A 1 , . . . , A 2 log 2 kn . Let N 1 , . . . , N 2 log 2 kn denote the number of observations among X, X 1 , . . . , X n falling in these 2 log 2 kn cells, and let C = {X, X 1 , . . . , X n } denote the set of positions of these n + 1 points. Since these points are independent and identically distributed, fixing the set C and Θ, the conditional probability that X falls in the -th cell equals N /(n + 1). Thus, for every fixed M ≥ 0,

P (N n (X, Θ) < M ) = E [P (N n (X, Θ) < M | C, Θ)] = E   =1,...,2 log 2 kn :N <M N n + 1   ≤ M 2 log 2 kn n + 1 ≤ 2M k n n + 1 ,
which converges to 0 by our assumption on k n .

It remains to show that diam(A n (X, Θ)) → 0 in probability. To this aim, let V nj (X, Θ) be the size of the j-th dimension of the rectangle containing X. Clearly, it suffices to show that V nj (X, Θ) → 0 in probability for all j = 1, . . . , d. To this end, note that

V nj (X, Θ) D = 2 -K nj (X,Θ) ,
where, conditionally on X, K nj (X, Θ) has a binomial B( log 2 k n , p nj ) distribution, representing the number of times the box containing X is split along the j-th coordinate (Fact 5.1). Thus

E [V nj (X, Θ)] = E 2 -K nj (X,Θ) = E E 2 -K nj (X,Θ) | X = (1 -p nj /2) log 2 kn ,
which tends to 0 as p nj log k n → ∞.

Proof of Proposition 2.1

Recall that rn (X) =

n i=1 E Θ [W ni (X, Θ)] Y i , where W ni (X, Θ) = 1 [X i ∈An(X,Θ)] N n (X, Θ) 1 En(X,Θ)
and

E n = [N n (X, Θ) = 0] . Similarly, rn (X) = n i=1 E Θ [W ni (X, Θ)] r(X i ).
We have

E [r n (X) -rn (X)] 2 = E n i=1 E Θ [W ni (X, Θ)] (Y i -r(X i )) 2 = E n i=1 E 2 Θ [W ni (X, Θ)] (Y i -r(X i )) 2 (the cross terms are 0 since E[Y i |X i ] = r(X i )) = E n i=1 E 2 Θ [W ni (X, Θ)] σ 2 (X i ) ≤ σ 2 E n i=1 E 2 Θ [W ni (X, Θ)] = nσ 2 E E 2 Θ [W n1 (X, Θ)] ,
where we used a symmetry argument in the last equality. Observe now that

E 2 Θ [W n1 (X, Θ)] = E Θ [W n1 (X, Θ)] E Θ [W n1 (X, Θ )]
(where Θ is distributed as, and independent of, Θ)

= E Θ,Θ [W n1 (X, Θ)W n1 (X, Θ )] = E Θ,Θ 1 [X 1 ∈An(X,Θ)] 1 [X 1 ∈An(X,Θ )] N n (X, Θ)N n (X, Θ ) 1 En(X,Θ) 1 En(X,Θ ) = E Θ,Θ 1 [X 1 ∈An(X,Θ)∩An(X,Θ )] N n (X, Θ)N n (X, Θ ) 1 En(X,Θ) 1 En(X,Θ ) .
Consequently,

E [r n (X) -rn (X)] 2 ≤ nσ 2 E 1 [X 1 ∈An(X,Θ)∩An(X,Θ )] N n (X, Θ)N n (X, Θ ) 1 En(X,Θ) 1 En(X,Θ ) . Therefore E [r n (X) -rn (X)] 2 ≤ nσ 2 E 1 [X 1 ∈An(X,Θ)∩An(X,Θ )] 1 + n i=2 1 [X i ∈An(X,Θ)] 1 + n i=2 1 [X i ∈An(X,Θ )] = nσ 2 E E 1 [X 1 ∈An(X,Θ)∩An(X,Θ )] 1 + n i=2 1 [X i ∈An(X,Θ)] × 1 1 + n i=2 1 [X i ∈An(X,Θ )] | X, X 1 , Θ, Θ = nσ 2 E 1 [X 1 ∈An(X,Θ)∩An(X,Θ )] E 1 1 + n i=2 1 [X i ∈An(X,Θ)] × 1 1 + n i=2 1 [X i ∈An(X,Θ )] | X, X 1 , Θ, Θ = nσ 2 E 1 [X 1 ∈An(X,Θ)∩An(X,Θ )] E 1 1 + n i=2 1 [X i ∈An(X,Θ)] × 1 1 + n i=2 1 [X i ∈An(X,Θ )] | X, Θ, Θ
by the independence of the random variables X, X 1 , . . . , X n , Θ, Θ . Using the Cauchy-Schwarz inequality, the above conditional expectation can be upper bounded by

E 1/2 1 1 + n i=2 1 [X i ∈An(X,Θ)] 2 | X, Θ × E 1/2 1 1 + n i=2 1 [X i ∈An(X,Θ )] 2 | X, Θ ≤ 3 × 2 2 log 2 kn
n 2 (by Fact 5.2 and technical Lemma 5.1)

≤ 12k 2 n n 2 . It follows that E [r n (X) -rn (X)] 2 ≤ 12σ 2 k 2 n n E 1 [X 1 ∈An(X,Θ)∩An(X,Θ )] = 12σ 2 k 2 n n E E X 1 1 [X 1 ∈An(X,Θ)∩An(X,Θ )] = 12σ 2 k 2 n n E [P X 1 (X 1 ∈ A n (X, Θ) ∩ A n (X, Θ ))] . (5.1)
Next, using the fact that X 1 is uniformly distributed over [0, 1] d , we may write

P X 1 (X 1 ∈ A n (X, Θ) ∩ A n (X, Θ )) = λ (A n (X, Θ) ∩ A n (X, Θ )) = d j=1 λ (A nj (X, Θ) ∩ A nj (X, Θ )) , where A n (X, Θ) = d j=1 A nj (X, Θ) and A n (X, Θ ) = d j=1
A nj (X, Θ ).

On the other hand, we know (Fact 5.1) that, for all j = 1, . . . , d, λ (A nj (X, Θ))

D = 2 -K nj (X,Θ) ,
where, conditionally on X, K nj (X, Θ) has a binomial B( log 2 k n , p nj ) distribution and, similarly,

λ (A nj (X, Θ )) D = 2 -K nj (X,Θ ) ,
where, conditionally on X, K nj (X, Θ ) is binomial B( log 2 k n , p nj ) and independent of K nj (X, Θ). In the rest of the proof, to lighten notation, we write K nj and K nj instead of K nj (X, Θ) and K nj (X, Θ ), respectively. Clearly,

λ (A nj (X, Θ) ∩ A nj (X, Θ ))≤2 -max(K nj ,K nj ) = 2 -K nj 2 -(K nj -K nj ) +
and, consequently,

d j=1 λ (A nj (X, Θ) ∩ A nj (X, Θ )) ≤2 -log 2 kn d j=1 2 -(K nj -K nj ) +
(since, by Fact 5.1,

d j=1 K nj = log 2 k n ).
Plugging this inequality into (5.1) and applying Hölder's inequality, we obtain

E [r n (X) -rn (X)] 2 ≤ 12σ 2 k n n E d j=1 2 -(K nj -K nj ) + = 12σ 2 k n n E E d j=1 2 -(K nj -K nj ) + | X ≤ 12σ 2 k n n E d j=1 E 1/d 2 -d(K nj -K nj ) + | X .
Each term in the product may be bounded by technical Proposition 5.1, and this leads to

E [r n (X) -rn (X)] 2 ≤ 288σ 2 k n πn d j=1 min 1, π 16 log 2 k n p nj (1 -p nj ) 1/2d ≤ 288σ 2 k n πn d j=1 min 1, π log 2 16(log k n )p nj (1 -p nj ) 1/2d .
Using the assumption on the form of the p nj , we finally conclude that

E [r n (X) -rn (X)] 2 ≤ Cσ 2 S 2 S -1 S/2d (1 + ξ n ) k n n(log k n ) S/
= j∈S (1 + ξ nj ) -1 1 - ξ nj S -1 -1 1/2d
.

Clearly, the sequence (ξ n ), which depends on the {(ξ nj ) : j ∈ S} only, tends to 0 as n tends to infinity.

Proof of Proposition 2.2

We start with the decomposition

E [r n (X) -r(X)] 2 = E n i=1 E Θ [W ni (X, Θ)] (r(X i ) -r(X)) + n i=1 E Θ [W ni (X, Θ)] -1 r(X) 2 =E E Θ n i=1 W ni (X, Θ) (r(X i ) -r(X)) + n i=1 W ni (X, Θ) -1 r(X) 2 ≤ E n i=1 W ni (X, Θ) (r(X i ) -r(X)) + n i=1 W ni (X, Θ) -1 r(X) 2 ,
where, in the last step, we used Jensen's inequality. Consequently,

E [r n (X) -r(X)] 2 ≤ E n i=1 W ni (X, Θ) (r(X i ) -r(X)) 2 + E r(X) 1 E c n (X,Θ) 2 ≤ E n i=1 W ni (X, Θ) (r(X i ) -r(X)) 2 + sup x∈[0,1] d r 2 (x) P (E c n (X, Θ)) . (5.2)
Let us examine the first term on the right-hand side of (5.2). Observe that, 29 by the Cauchy-Schwarz inequality,

E n i=1 W ni (X, Θ) (r(X i ) -r(X)) 2 ≤ E n i=1 W ni (X, Θ) W ni (X, Θ) |r(X i ) -r(X)| 2 ≤ E n i=1 W ni (X, Θ) n i=1 W ni (X, Θ) (r(X i ) -r(X)) 2 ≤ E n i=1 W ni (X, Θ) (r(X i ) -r(X)) 2
(since the weights are subprobability weights).

Thus, denoting by X S the norm of X evaluated over the components in S, we obtain

E n i=1 W ni (X, Θ) (r(X i ) -r(X)) 2 ≤ E n i=1 W ni (X, Θ) (r (X iS ) -r (X S )) 2 ≤ L 2 n i=1 E W ni (X, Θ) X i -X 2 S = nL 2 E W n1 (X, Θ) X 1 -X 2 S (by symmetry). But E W n1 (X, Θ) X 1 -X 2 S = E X 1 -X 2 S 1 [X 1 ∈An(X,Θ)] N n (X, Θ) 1 En(X,Θ) = E X 1 -X 2 S 1 [X 1 ∈An(X,Θ)] 1 + n i=2 1 [X i ∈An(X,Θ)] = E E X 1 -X 2 S 1 [X 1 ∈An(X,Θ)] 1 + n i=2 1 [X i ∈An(X,Θ)] | X, X 1 , Θ .
Thus,

E W n1 (X, Θ) X 1 -X 2 S = E X 1 -X 2 S 1 [X 1 ∈An(X,Θ)] E 1 1 + n i=2 1 [X i ∈An(X,Θ)] | X, X 1 , Θ = E X 1 -X 2 S 1 [X 1 ∈An(X,Θ)] E 1 1 + n i=2 1 [X i ∈An(X,Θ)] | X, Θ
(by the independence of the random variables X, X 1 , . . . , X n , Θ).

By Fact 5.2 and technical Lemma 5.1,

E 1 1 + n i=2 1 [X i ∈An(X,Θ)] | X, Θ ≤ 2 log 2 kn n ≤ 2k n n .
Consequently,

E n i=1 W ni (X, Θ) (r(X i ) -r(X)) 2 ≤ 2L 2 k n E X 1 -X 2 S 1 [X 1 ∈An(X,Θ)] . Letting A n (X, Θ) = d j=1
A nj (X, Θ), we obtain

E n i=1 W ni (X, Θ) (r(X i ) -r(X)) 2 ≤ 2L 2 k n j∈S E |X (j) 1 -X (j) | 2 1 [X 1 ∈An(X,Θ)] = 2L 2 k n j∈S E ρ j (X, X 1 , Θ)E X (j) 1 |X (j) 1 -X (j) | 2 1 [X (j) 1 ∈A nj (X,Θ)]
where, in the last equality, we set ρ j (X, X 1 , Θ) = t=1,...,d,t =j

1 [X (t)
1 ∈Ant(X,Θ)] .

Therefore, using the fact that X 1 is uniformly distributed over [0, 1] d ,

E n i=1 W ni (X, Θ) (r(X i ) -r(X)) 2 ≤ 2L 2 k n j∈S E ρ j (X, X 1 , Θ)λ 3 (A nj (X, Θ)) .
Observing that λ (A nj (X, Θ)) × E [X (t) 1 : t=1,...,d,t =j] [ρ j (X, X 1 , Θ)] = λ (A n (X, Θ)) = 2 -log 2 kn (Fact 5.2), we are led to

E n i=1 W ni (X, Θ) (r(X i ) -r(X)) 2 ≤ 2L 2 j∈S E λ 2 (A nj (X, Θ)) = 2L 2 j∈S E 2 -2K nj (X,Θ) = 2L 2 j∈S E E 2 -2K nj (X,Θ) | X ,
where, conditionally on X, K nj (X, Θ) has a binomial B( log 2 k n , p nj ) distribution (Fact 5.1). Consequently,

E n i=1
W ni (X, Θ) (r(X i ) -r(X))

Some technical results

The following result is an extension of Lemma 4.1 in Györfi et al. [START_REF] Györfi | A Distribution-Free Theory of Nonparametric Regression[END_REF]. Its proof is given here for the sake of completeness. (ii) For all j ∈ N,

P(Z 1 -Z 2 = j) = 1 2πi Γ ϕ(z) z j+1 dz,
where Γ is the positively oriented unit circle.

(iii) For all d ≥ 1,

E 2 -d(Z 1 -Z 2 ) + ≤ 24 π 1 0 exp -4N p(1 -p)t 2 dt.
Proof of Lemma 5.2 Statement (i) is clear and (ii) is an immediate consequence of Cauchy's integral formula (Rudin [34]). To prove statement (iii), write The conclusion follows by observing that C d ≤ 24/π for all d ≥ 1.

Evaluating the integral in statement (iii) of Lemma 5.2 leads to the following proposition:

Proposition 5.1 Let Z 1 and Z 2 be two independent binomial B(N, p) random variables, with p ∈ (0, 1). Then, for all d ≥ 1,

E 2 -d(Z 1 -Z 2 ) + ≤ 24 π min 1, π 16N p(1 -p)
.

Figure 1 :

 1 Figure 1: Solid line: Plot of the function S → 0.75/(S log 2 + 0.75) for S ranging from 2 to d = 100. Dotted line: Plot of the minimax rate power S → 2/(S + 2). The horizontal line shows the value of the d-dimensional rate power 2/(d + 2) ≈ 0.0196.

Figure 2 :

 2 Figure 2: The tree used as regression function in the model Tree.

Figure 3 :

 3 Figure 3: Boxplots of the empirical probabilities of cuts for model Sinus (S = {1}).

Figure 4 :

 4 Figure 4: Boxplots of the empirical probabilities of cuts for model Friedman #1 (S = {1, . . . , 5}).

Figure 5 :

 5 Figure 5: Boxplots of the empirical probabilities of cuts for model Tree (S = {1, . . . , 5}).

KFigure 6 :

 6 Figure 6: Evolution of the MSE for model Sinus (S = 1).

22

 22

Figure 7 :

 7 Figure 7: Evolution of the MSE for model Friedman #1 (S = 5).

Figure 8 :

 8 Figure 8: Evolution of the MSE for model Tree (S = 5).

2 ≤ 2L 2 j∈S( 1 -, 1 [ 1 [= 1 - 2 -r 2

 22111122 0.75p nj ) log 2 kn with γ n = min j∈S ξ nj .To finish the proof, it remains to bound the second term on the right-hand side of (5.2), which is easier. Just note thatP (E c n (X, Θ)) = P n i=1 X i ∈An(X,Θ)] X i ∈An(X,Θ)] = 0 | X, Θ log 2 kn n (by Fact 5.2) ≤ e -n/2kn .Putting all the pieces together, we finally conclude that E [r n (X) -r(X)] (x) e -n/2kn , as desired.

Lemma 5 . 1

 51 Let Z be a binomial B(N, p) random variable, with p ∈ (0, 1]. Then

2 .

 2 Proof of Lemma 5.1 To prove statement (i), we write

Lemma 5 . 2

 52 Let Z 1 and Z 2 be two independent binomial B(N, p) random variables. Set, for all z ∈ C , ϕ(z) = E[z Z 1 -Z 2 ]. Then

( i )

 i For all z ∈ C , ϕ(z) = p(1 -p)(z + z -1 ) + 1 -2p(1 -p) N .

E 2 - 2 - 2 -j=0 2 -[ 1 +[ 1 +[ 1 + 2 π 0 [ 1 + 2 π 0 1 - 2 π/ 2 0 1 -

 222211120121221 d(Z 1 -Z 2 ) + = N j=0 dj P ((Z 1 -Z 2 ) + = j) dj P (Z 1 -Z 2 = j) ≤ ∞ dj P (Z 1 -Z 2 = j) iθ ) 1 -2 -d e -iθ dθ (by setting z = e iθ , θ ∈ [-π, π]) 2p(1 -p)(cos θ -1)] N e iθ 2 d e iθ -1 dθ (by statement (i)).Noting thate iθ 2 d e iθ -1 = 2 d -e iθ 2 2d -2 d+1 cos θ + 1 ,we obtainE 2 -d(Z 1 -Z 2 ) 2p(1 -p)(cos θ -1)] N 2 d -cos θ 2 2d -2 d+1 cos θ + 1 dθ. The bound 2 d -cos θ 2 2d -2 d+1 cos θ + 1 ≤ 2 d + 1 (2 d -1) 2 leads to E 2 -d(Z 1 -Z 2 ) + ≤ 2 d-1 (2 d + 1) π(2 d -1) 2p(1 -p)(cos θ -1)] N dθ = 2 d (2 d + 1) π(2 d -1) 2p(1 -p)(cos θ -1)] N dθ = 2 d (2 d + 1) π(2 d -1) 4p(1 -p) sin 2 (θ/2) N dθ (cos θ -1 = -2 sin 2 (θ/2)) = 2 d+1 (2 d + 1) π(2 d -1) 4p(1 -p) sin 2 θ N dθ.Using the elementary inequality (1-z) N ≤ e -N z for z ∈ [0, 1] and the change of variable t = tan(θ/2), we finally obtain E 2 -d(Z 1 -Z 2 ) p(1 -p)t 2 dt, with C d = 2 d+2 (2 d + 1) π(2 d -1) 2 .

Research carried out within the INRIA project "CLASSIC" hosted by Ecole Normale Supérieure and CNRS.

Acknowledgments. I greatly thank the Action Editor and two referees for valuable comments and insightful suggestions, which lead to a substantial improvement of the paper. I would also like to thank my colleague Jean-Patrick Baudry for his precious help on the simulation section.

 [START_REF] Amaratunga | Enriched random forests[END_REF]Research partially supported by the French National Research Agency under grant ANR-09-BLAN-0051-02 "CLARA".