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Random forests are a popular class of algorithms used for regression and classification. The algorithm introduced by Breiman in 2001 and many of its variants are ensembles of randomized decision trees built from axis-aligned partitions of the feature space. One such variant, called Mondrian forests, was proposed to handle the online setting and is the first class of random forests for which minimax rates were obtained in arbitrary dimension. However, the restriction to axis-aligned splits fails to capture dependencies between features, and random forests that use oblique splits have shown improved empirical performance for many tasks. In this work, we show that a large class of random forests with general split directions also achieve minimax rates in arbitrary dimension. This class includes STIT forests, a generalization of Mondrian forests to arbitrary split directions, as well as random forests derived from Poisson hyperplane tessellations. These are the first results showing that random forest variants with oblique splits can obtain minimax optimality in arbitrary dimension. Our proof technique relies on the novel application of the theory of stationary random tessellations in stochastic geometry to statistical learning theory.
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Randomization as Regularization: A Degrees of Freedom Explanation for Random Forest Success
Random forests remain among the most popular off-the-shelf supervised machine learning tools with a well-established track record of predictive accuracy in both regression and classification settings. Despite their empirical success as well as a bevy of recent work investigating their statistical properties, a full and satisfying explanation for their success has yet to be put forth. Here we aim to take a step forward in this direction by demonstrating that the additional randomness injected into individual trees serves as a form of implicit regularization, making random forests an ideal model in low signal-to-noise ratio (SNR) settings. Specifically, from a model-complexity perspective, we show that the mtry parameter in random forests serves much the same purpose as the shrinkage penalty in explicitly regularized regression procedures like lasso and ridge regression. To highlight this point, we design a randomized linear-model-based forward selection procedure intended as an analogue to tree-based random forests and demonstrate its surprisingly strong empirical performance. Numerous demonstrations on both real and synthetic data are provided.
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- Award ID(s):
- 2015400
- PAR ID:
- 10226126
- Date Published:
- Journal Name:
- Journal of machine learning research
- Volume:
- 21
- Issue:
- 171
- ISSN:
- 1533-7928
- Page Range / eLocation ID:
- 1-36
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- The DOI is not currently available.
