skip to main content

Title: A cautionary tale on fitting decision trees to data from additive models: generalization lower bounds
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 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 more » 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. « less
Authors:
; ;
Award ID(s):
2023505
Publication Date:
NSF-PAR ID:
10343667
Journal Name:
Proeedings of the International Workshop on Artificial Intelligence and Statistics
ISSN:
1525-531X
Sponsoring Org:
National Science Foundation
More Like this
  1. Modern machine learning has achieved impressive prediction performance, but often sacrifices interpretability, a critical consideration in many problems. Here, we propose Fast Interpretable Greedy-Tree Sums (FIGS), an algorithm for fitting concise rule-based models. Specifically, FIGS generalizes the CART algorithm to simultaneously grow a flexible number of trees in a summation. The total number of splits across all the trees can be restricted by a pre-specified threshold, thereby keeping both the size and number of its trees under control. When both are small, the fitted tree-sum can be easily visualized and written out by hand, making it highly interpretable. A partially oracle theoretical result hints at the potential for FIGS to overcome a key weakness of single-tree models by disentangling additive components of generative additive models, thereby reducing redundancy from repeated splits on the same feature. Furthermore, given oracle access to optimal tree structures, we obtain l2 generalization bounds for such generative models in the case of C component functions, matching known minimax rates in some cases. Extensive experiments across a wide array of real-world datasets show that FIGS achieves state-of-the-art prediction performance (among all popular rule-based methods) when restricted to just a few splits (e.g. less than 20). We findmore »empirically that FIGS is able to avoid repeated splits, and often provides more concise decision rules than fitted decision trees, without sacrificing predictive performance. All code and models are released in a full-fledged package on Github.« less
  2. Nonparametric regression on complex domains has been a challenging task as most existing methods, such as ensemble models based on binary decision trees, are not designed to account for intrinsic geometries and domain boundaries. This article proposes a Bayesian additive regression spanning trees (BAST) model for nonparametric regression on manifolds, with an emphasis on complex constrained domains or irregularly shaped spaces embedded in Euclidean spaces. Our model is built upon a random spanning tree manifold partition model as each weak learner, which is capable of capturing any irregularly shaped spatially contiguous partitions while respecting intrinsic geometries and domain boundary constraints. Utilizing many nice properties of spanning tree structures, we design an efficient Bayesian inference algorithm. Equipped with a soft prediction scheme, BAST is demonstrated to significantly outperform other competing methods in simulation experiments and in an application to the chlorophyll data in Aral Sea, due to its strong local adaptivity to different levels of smoothness.
  3. Sparse decision tree optimization has been one of the most fundamental problems in AI since its inception and is a challenge at the core of interpretable machine learning. Sparse decision tree optimization is computationally hard, and despite steady effort since the 1960's, breakthroughs have been made on the problem only within the past few years, primarily on the problem of finding optimal sparse decision trees. However, current state-of-the-art algorithms often require impractical amounts of computation time and memory to find optimal or near-optimal trees for some real-world datasets, particularly those having several continuous-valued features. Given that the search spaces of these decision tree optimization problems are massive, can we practically hope to find a sparse decision tree that competes in accuracy with a black box machine learning model? We address this problem via smart guessing strategies that can be applied to any optimal branch-and-bound-based decision tree algorithm. The guesses come from knowledge gleaned from black box models. We show that by using these guesses, we can reduce the run time by multiple orders of magnitude while providing bounds on how far the resulting trees can deviate from the black box's accuracy and expressive power. Our approach enables guesses about howmore »to bin continuous features, the size of the tree, and lower bounds on the error for the optimal decision tree. Our experiments show that in many cases we can rapidly construct sparse decision trees that match the accuracy of black box models. To summarize: when you are having trouble optimizing, just guess.« less
  4. Tensors are becoming prevalent in modern applications such as medical imaging and digital marketing. In this paper, we propose a sparse tensor additive regression (STAR) that models a scalar response as a flexible nonparametric function of tensor covariates. The proposed model effectively exploits the sparse and low-rank structures in the tensor additive regression. We formulate the parameter estimation as a non-convex optimization problem, and propose an efficient penalized alternating minimization algorithm. We establish a non-asymptotic error bound for the estimator obtained from each iteration of the proposed algorithm, which reveals an interplay between the optimization error and the statistical rate of convergence. We demonstrate the efficacy of STAR through extensive comparative simulation studies, and an application to the click-through-rate prediction in online advertising.
  5. Binary classification is a fundamental machine learning task defined as correctly assigning new objects to one of two groups based on a set of training objects. Driven by the practical importance of binary classification, numerous machine learning techniques have been developed and refined over the last three decades. Among the most popular techniques are artificial neural networks, decision trees, ensemble methods, logistic regression, and support vector machines. We present here machine learning and pattern recognition algorithms that, unlike the commonly used techniques, are based on combinatorial optimization and make use of information on pairwise relations between the objects of the data set, whether training objects or not. These algorithms solve the respective problems optimally and efficiently, in contrast to the primarily heuristic approaches currently used for intractable problem models in pattern recognition and machine learning. The algorithms described solve efficiently the classification problem as a network flow problem on a graph. The technical tools used in the algorithm are the parametric cut procedure and a process called sparse computation that computes only the pairwise similarities that are “relevant.” Sparse computation enables the scalability of any algorithm that uses pairwise similarities. We present evidence on the effectiveness of the approaches, measuredmore »in terms of accuracy and running time, in pattern recognition, image segmentation, and general data mining.« less