# Adaptive Concentration of Regression Trees, with Application to Random Forests

@article{Wager2015AdaptiveCO, title={Adaptive Concentration of Regression Trees, with Application to Random Forests}, author={Stefan Wager and Guenther Walther}, journal={arXiv: Statistics Theory}, year={2015} }

We study the convergence of the predictive surface of regression trees and forests. To support our analysis we introduce a notion of adaptive concentration for regression trees. This approach breaks tree training into a model selection phase in which we pick the tree splits, followed by a model fitting phase where we find the best regression model consistent with these splits. We then show that the fitted regression tree concentrates around the optimal predictor with the same splits: as d and n…

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## References

SHOWING 1-10 OF 45 REFERENCES

### Random Forests

- Computer ScienceMachine Learning
- 2004

Internal estimates monitor error, strength, and correlation and these are used to show the response to increasing the number of features used in the forest, and are also applicable to regression.

### Consistency of Random Forests

- Computer Science
- 2015

A step forward in forest exploration is taken by proving a consistency result for Breiman's original algorithm in the context of additive regression models, and sheds an interesting light on how random forests can nicely adapt to sparsity.

### Tree-structured regression and the differentiation of integrals

- Mathematics
- 2007

This paper provides answers to questions regarding the almost sure limiting behavior of rooted, binary tree-structured rules for regression. Examples show that questions raised by Gordon and Olshen…

### Analysis of purely random forests bias

- Computer Science, MathematicsArXiv
- 2014

Under some regularity assumptions on the regression function, it is shown that the bias of an infinite forest decreases at a faster rate (with respect to the size of each tree) than a single tree, and infinite forests attain a strictly better risk rate than single trees.

### Histogram regression estimation using data-dependent partitions

- Computer Science, Mathematics
- 1996

The consistency of histograms regression estimates based on cubic partitions with data-dependent offsets, k-thresholding in one dimension and empirically optimal nearest-neighbor clustering schemes are established.

### Analysis of a Random Forests Model

- Computer ScienceJ. Mach. Learn. Res.
- 2012

An in-depth analysis of a random forests model suggested by Breiman (2004), which is very close to the original algorithm, and shows in particular that the 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.

### Random Forests and Adaptive Nearest Neighbors

- Computer Science
- 2006

It is shown that random forests with adaptive splitting schemes assign weights to k-PNNs in a desirable way: for the estimation at a given target point, these random forests assign voting weights to the k- PNNs of the target point according to the local importance of different input variables.

### Quantile Regression Forests

- Computer Science, MathematicsJ. Mach. Learn. Res.
- 2006

It is shown here that random forests provide information about the full conditional distribution of the response variable, not only about the conditional mean, in order to be competitive in terms of predictive power.

### Impact of subsampling and pruning on random forests

- Computer Science
- 2016

It is shown that fully developed sub-sampled forests and pruned (without subsampling) forests have similar performances, as long as respective parameters are well chosen.