Gradient boosting is widely popular due to its flexibility and predictive accuracy. However, statistical inference and uncertainty quantification for gradient boosting remain challenging and under-explored. We propose a unified framework for statistical inference in gradient boosting regression. Our framework integrates dropout or parallel training with a recently proposed regularization procedure called Boulevard that allows for a central limit theorem (CLT) for boosting. With these enhancements, we surprisingly find that \textit{increasing} the dropout rate and the number of trees grown in parallel at each iteration substantially enhances signal recovery and overall performance. Our resulting algorithms enjoy similar CLTs, which we use to construct built-in confidence intervals, prediction intervals, and rigorous hypothesis tests for assessing variable importance in only $O(nd^2)$ time with the Nyström method. Numerical experiments verify the asymptotic normality and demonstrate that our algorithms perform well, do not require early stopping, interpolate between regularized boosting and random forests, and confirm the validity of their built-in statistical inference procedures.

Feature importance scores are ubiquitous tools for understanding the predictions of machine learning models. However, many popular attribution methods suffer from high instability due to random sampling. Leveraging novel ideas from hypothesis testing, we devise techniques that ensure the most important features are correct with high-probability guarantees. These are capable of assessing both the set of $K$ top-ranked features as well as the order of its elements. Given local or global importance scores, we demonstrate how to retrospectively verify the stability of the highest ranks. We then introduce two efficient sampling algorithms that identify the $K$ most important features, perhaps in order, with probability at least $1-\alpha$. The theoretical justification for these procedures is validated empirically on SHAP and LIME.

Variable importance is one of the most widely used measures for interpreting machine learning with significant interest from both statistics and machine learning communities. Recently, increasing attention has been directed toward uncertainty quantification in these metrics. Current approaches largely rely on one-step procedures, which, while asymptotically efficient, can present higher sensitivity and instability in finite sample settings. To address these limitations, we propose a novel method by employing the targeted learning (TL) framework, designed to enhance robustness in inference for variable importance metrics. Our approach is particularly suited for conditional permutation variable importance. We show that it (i) retains the asymptotic efficiency of traditional methods, (ii) maintains comparable computational complexity, and (iii) delivers improved accuracy, especially in finite sample contexts. We further support these findings with numerical experiments that illustrate the practical advantages of our method and validate the theoretical results.

Abstract Model distillation has been a popular method for producing interpretable machine learning. It uses an interpretable “student” model to mimic the predictions made by the black box “teacher” model. However, when the student model is sensitive to the variability of the data sets used for training even when keeping the teacher fixed, the corresponded interpretation is not reliable. Existing strategies stabilize model distillation by checking whether a large enough sample of pseudo-data is generated to reliably reproduce student models, but methods to do so have so far been developed separately for each specific class of student model. In this paper, we develop a generic approach for stable model distillation based on central limit theorem for the estimated fidelity of the student to the teacher. We start with a collection of candidate student models and search for candidates that reasonably agree with the teacher. Then we construct a multiple testing framework to select a sample size such that the consistent student model would be selected under different pseudo samples. We demonstrate the application of our proposed approach on three commonly used intelligible models: decision trees, falling rule lists and symbolic regression. Finally, we conduct simulation experiments on Mammographic Mass and Breast Cancer datasets and illustrate the testing procedure throughout a theoretical analysis with Markov process. The code is publicly available at https://github.com/yunzhe-zhou/GenericDistillation .

Abstract Varying coefficient models are a flexible extension of generic parametric models whose coefficients are functions of a set of effect-modifying covariates instead of fitted constants. They are capable of achieving higher model complexity while preserving the structure of the underlying parametric models, hence generating interpretable predictions. In this paper we study the use of gradient boosted decision trees as those coefficient-deciding functions in varying coefficient models with linearly structured outputs. In contrast to the traditional choices of splines or kernel smoothers, boosted trees are more flexible since they require no structural assumptions in the effect modifier space. We introduce our proposed method from the perspective of a localized version of gradient descent, prove its theoretical consistency under mild assumptions commonly adapted by decision tree research, and empirically demonstrate that the proposed tree boosted varying coefficient models achieve high performance qualified by their training speed, prediction accuracy and intelligibility as compared to several benchmark algorithms.

This paper examines a novel gradient boosting framework for regression. We regularize gradient boosted trees by introducing subsampling and employ a modified shrinkage algorithm so that at every boosting stage the estimate is given by an average of trees. The resulting algorithm, titled "Boulevard'", is shown to converge as the number of trees grows. This construction allows us to demonstrate a central limit theorem for this limit, providing a characterization of uncertainty for predictions. A simulation study and real world examples provide support for both the predictive accuracy of the model and its limiting behavior.

An increasing number of machine learning models have been deployed in domains with high stakes such as finance and healthcare. Despite their superior performances, many models are black boxes in nature which are hard to explain. There are growing efforts for researchers to develop methods to interpret these black-box models. Post hoc explanations based on perturbations, such as LIME [39], are widely used approaches to interpret a machine learning model after it has been built. This class of methods has been shown to exhibit large instability, posing serious challenges to the effectiveness of the method itself and harming user trust. In this paper, we propose S-LIME, which utilizes a hypothesis testing framework based on central limit theorem for determining the number of perturbation points needed to guarantee stability of the resulting explanation. Experiments on both simulated and real world data sets are provided to demonstrate the effectiveness of our method.

As machine learning procedures become an increasingly popular modeling option among applied researchers, there has been a corresponding interest in developing valid tools for understanding their statistical properties and uncertainty. Tree-based ensembles like random forests remain one such popular option for which several important theoretical advances have been made in recent years by drawing upon a connection between their natural subsampled structure and the classical theory of $U$-statistics. Unfortunately, the procedures for estimating predictive variance resulting from these studies are plagued by severe bias and extreme computational overhead. Here, we argue that the root of these problems lies in the use of subsampling without replacement and that with-replacement subsamples, resulting in $V$-statistics, substantially alleviates these problems. We develop a general framework for analyzing the asymptotic behavior of $V$-statistics, demonstrating asymptotic normality under precise regularity conditions and establishing previously unreported connections to $U$-statistics. Importantly, these findings allow us to produce a natural and efficient means of estimating the variance of a conditional expectation, a problem of wide interest across multiple scientific domains that also lies at the heart of uncertainty quantification for supervised learning ensembles.

We propose a modification that corrects for split-improvement variable importance measures in Random Forests and other tree-based methods. These methods have been shown to be biased towards increasing the importance of features with more potential splits. We show that by appropriately incorporating split-improvement as measured on out of sample data, this bias can be corrected yielding better summaries and screening tools.

Models which estimate main effects of individual variables alongside interaction effects have an identifiability challenge: effects can be freely moved between main effects and interaction effects without changing the model prediction. This is a critical problem for interpretability because it permits “contradictory" models to represent the same function. To solve this problem, we propose pure interaction effects: variance in the outcome which cannot be represented by any subset of features. This definition has an equivalence with the Functional ANOVA decomposition. To compute this decomposition, we present a fast, exact algorithm that transforms any piecewise-constant function (such as a tree-based model) into a purified, canonical representation. We apply this algorithm to Generalized Additive Models with interactions trained on several datasets and show large disparity, including contradictions, between the apparent and the purified effects. These results underscore the need to specify data distributions and ensure identifiability before interpreting model parameters.

Black-box risk scoring models permeate our lives, yet are typically proprietary or opaque. We propose Distill-and-Compare, an approach to audit such models without probing the black-box model API or pre-defining features to audit. To gain insight into black-box models, we treat them as teachers, training transparent student models to mimic the risk scores assigned by the black-box models. We compare the mimic model trained with distillation to a second, un-distilled transparent model trained on ground truth outcomes, and use differences between the two models to gain insight into the black-box model. We demonstrate the approach on four data sets: COMPAS, Stop-and-Frisk, Chicago Police, and Lending Club. We also propose a statistical test to determine if a data set is missing key features used to train the black-box model. Our test finds that the ProPublica data is likely missing key feature(s) used in COMPAS.

This work develops formal statistical inference procedures for predictions generated by supervised learning ensembles. Ensemble methods based on bootstrapping, such as bagging and random forests, have improved the predictive accuracy of individual trees, but fail to provide a framework in which distributional results can be easily determined. Instead of aggregating full bootstrap samples, we consider predicting by averaging over trees built on subsamples of the training set and demonstrate that the resulting estimator takes the form of a U-statistic. As such, predictions for individual feature vectors are asymptotically normal, allowing for confidence intervals to accompany predictions. In practice, a subset of subsamples is used for computational speed; here our estimators take the form of incomplete U-statistics and equivalent results are derived. We further demonstrate that this setup provides a framework for testing the significance of features. Moreover, the internal estimation method we develop allows us to estimate the variance parameters and perform these inference procedures at no additional computational cost. Simulations and illustrations on a real data set are provided.

Standard generalized additive models (GAMs) usually model the dependent variable as a sum of univariate models. Although previous studies have shown that standard GAMs can be interpreted by users, their accuracy is significantly less than more complex models that permit interactions.

There has historically been very little concern with extrapolation in Machine Learning, yet extrapolation can be critical to diagnose. Predictor functions are almost always learned on a set of highly correlated data comprising a very small segment of predictor space. Moreover, flexible predictors, by their very nature, are not controlled at points of extrapolation. This becomes a problem for diagnostic tools that require evaluation on a product distribution. It is also an issue when we are trying to optimize a response over some variable in the input space. Finally, it can be a problem in non-static systems in which the underlying predictor distribution gradually drifts with time or when typographical errors misrecord the values of some predictors.We present a diagnosis for extrapolation as a statistical test for a point originating from the data distribution as opposed to a null hypothesis uniform distribution. This allows us to employ general classification methods for estimating such a test statistic. Further, we observe that CART can be modified to accept an exact distribution as an argument, providing a better classification tool which becomes our extrapolation-detection procedure. We explore some of the advantages of this approach and present examples of its practical application.

Many automated learning procedures lack interpretability, operating effectively as a black box: providing a prediction tool but no explanation of the underlying dynamics that drive it. A common approach to interpretation is to plot the dependence of a learned function on one or two predictors. We present a method that seeks not to display the behavior of a function, but to evaluate the importance of nonadditive interactions within any set of variables. Should the function be close to a sum of low dimensional components, these components can be viewed and even modeled parametrically. Alternatively, the work here provides an indication of where intrinsically high-dimensional behavior takes place. The calculations used in this paper correspond closely with the functional ANOVA decomposition; a well-developed construction in Statistics. In particular, the proposed score of interaction importance measures the loss associated with the projection of the prediction function onto a space of additive models. The algorithm runs in linear time and we present displays of the output as a graphical model of the function for interpretation purposes.