A cautionary tale on fitting decision trees to data from additive models: generalization lower bounds

Yan Shuo Tan, Abhineet Agarwal, Bin Yu
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:9663-9685, 2022.

Abstract

Decision trees are important both as interpretable models amenable to high-stakes decision-making, and as building blocks of ensemble methods such as random forests and gradient boosting. Their statistical properties, however, are not well understood. The most cited prior works have focused on deriving pointwise consistency guarantees for CART in a classical nonparametric regression setting. We take a different approach, and advocate studying the generalization performance of decision trees with respect to different generative regression models. This allows us to elicit their inductive bias, that is, the assumptions the algorithms make (or do not make) to generalize to new data, thereby guiding practitioners on when and how to apply these methods. In this paper, we focus on sparse additive generative models, which have both low statistical complexity and some nonparametric flexibility. We prove a sharp squared error generalization lower bound for a large class of decision tree algorithms fitted to sparse additive models with $C^1$ component functions. This bound is surprisingly much worse than the minimax rate for estimating such sparse additive models. The inefficiency is due not to greediness, but to the loss in power for detecting global structure when we average responses solely over each leaf, an observation that suggests opportunities to improve tree-based algorithms, for example, by hierarchical shrinkage. To prove these bounds, we develop new technical machinery, establishing a novel connection between decision tree estimation and rate-distortion theory, a sub-field of information theory.

Cite this Paper

BibTeX
@InProceedings{pmlr-v151-shuo-tan22a, title = { A cautionary tale on fitting decision trees to data from additive models: generalization lower bounds }, author = {Shuo Tan, Yan and Agarwal, Abhineet and Yu, Bin}, booktitle = {Proceedings of The 25th International Conference on Artificial Intelligence and Statistics}, pages = {9663--9685}, year = {2022}, editor = {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel}, volume = {151}, series = {Proceedings of Machine Learning Research}, month = {28--30 Mar}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v151/shuo-tan22a/shuo-tan22a.pdf}, url = {https://proceedings.mlr.press/v151/shuo-tan22a.html}, abstract = { Decision trees are important both as interpretable models amenable to high-stakes decision-making, and as building blocks of ensemble methods such as random forests and gradient boosting. Their statistical properties, however, are not well understood. The most cited prior works have focused on deriving pointwise consistency guarantees for CART in a classical nonparametric regression setting. We take a different approach, and advocate studying the generalization performance of decision trees with respect to different generative regression models. This allows us to elicit their inductive bias, that is, the assumptions the algorithms make (or do not make) to generalize to new data, thereby guiding practitioners on when and how to apply these methods. In this paper, we focus on sparse additive generative models, which have both low statistical complexity and some nonparametric flexibility. We prove a sharp squared error generalization lower bound for a large class of decision tree algorithms fitted to sparse additive models with $C^1$ component functions. This bound is surprisingly much worse than the minimax rate for estimating such sparse additive models. The inefficiency is due not to greediness, but to the loss in power for detecting global structure when we average responses solely over each leaf, an observation that suggests opportunities to improve tree-based algorithms, for example, by hierarchical shrinkage. To prove these bounds, we develop new technical machinery, establishing a novel connection between decision tree estimation and rate-distortion theory, a sub-field of information theory. } }
Endnote
%0 Conference Paper %T A cautionary tale on fitting decision trees to data from additive models: generalization lower bounds %A Yan Shuo Tan %A Abhineet Agarwal %A Bin Yu %B Proceedings of The 25th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2022 %E Gustau Camps-Valls %E Francisco J. R. Ruiz %E Isabel Valera %F pmlr-v151-shuo-tan22a %I PMLR %P 9663--9685 %U https://proceedings.mlr.press/v151/shuo-tan22a.html %V 151 %X Decision trees are important both as interpretable models amenable to high-stakes decision-making, and as building blocks of ensemble methods such as random forests and gradient boosting. Their statistical properties, however, are not well understood. The most cited prior works have focused on deriving pointwise consistency guarantees for CART in a classical nonparametric regression setting. We take a different approach, and advocate studying the generalization performance of decision trees with respect to different generative regression models. This allows us to elicit their inductive bias, that is, the assumptions the algorithms make (or do not make) to generalize to new data, thereby guiding practitioners on when and how to apply these methods. In this paper, we focus on sparse additive generative models, which have both low statistical complexity and some nonparametric flexibility. We prove a sharp squared error generalization lower bound for a large class of decision tree algorithms fitted to sparse additive models with $C^1$ component functions. This bound is surprisingly much worse than the minimax rate for estimating such sparse additive models. The inefficiency is due not to greediness, but to the loss in power for detecting global structure when we average responses solely over each leaf, an observation that suggests opportunities to improve tree-based algorithms, for example, by hierarchical shrinkage. To prove these bounds, we develop new technical machinery, establishing a novel connection between decision tree estimation and rate-distortion theory, a sub-field of information theory.
APA
Shuo Tan, Y., Agarwal, A. & Yu, B.. (2022). A cautionary tale on fitting decision trees to data from additive models: generalization lower bounds . Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 151:9663-9685 Available from https://proceedings.mlr.press/v151/shuo-tan22a.html.

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