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.
Componentwise boosting (CWB), also known as model-based boosting, is a variant of gradient boosting that builds on additive models as base learners to ensure interpretability. CWB is thus often used in research areas where models are employed as tools to explain relationships in data. One downside of CWB is its computational complexity in terms of memory and runtime. In this paper, we propose two techniques to overcome these issues without losing the properties of CWB: feature discretization of numerical features and incorporating Nesterov momentum into functional gradient descent. As the latter can be prone to early overfitting, we also propose a hybrid approach that prevents a possibly diverging gradient descent routine while ensuring faster convergence. We perform extensive benchmarks on multiple simulated and real-world data sets to demonstrate the improvements in runtime and memory consumption while maintaining state-of-the-art estimation and prediction performance.
The "shape" of a planar curve and/or landmark configuration is considered its equivalence class under translation, rotation and scaling, its "form" its equivalence class under translation and rotation while scale is preserved. We extend generalized additive regression to models for such shapes/forms as responses respecting the resulting quotient geometry by employing the squared geodesic distance as loss function and a geodesic response function to map the additive predictor to the shape/form space. For fitting the model, we propose a Riemannian $L_2$-Boosting algorithm well suited for a potentially large number of possibly parameter-intensive model terms, which also yields automated model selection. We provide novel intuitively interpretable visualizations for (even non-linear) covariate effects in the shape/form space via suitable tensor-product factorization. The usefulness of the proposed framework is illustrated in an analysis of 1) astragalus shapes of wild and domesticated sheep and 2) cell forms generated in a biophysical model, as well as 3) in a realistic simulation study with response shapes and forms motivated from a dataset on bottle outlines.
In this paper, we consider the problem of predicting survey response rates using a family of flexible and interpretable nonparametric models. The study is motivated by the US Census Bureau's well-known ROAM application, which uses a linear regression model trained on the US Census Planning Database data to identify hard-to-survey areas. A crowdsourcing competition (Erdman and Bates, 2016) organized more than ten years ago revealed that machine learning methods based on ensembles of regression trees led to the best performance in predicting survey response rates; however, the corresponding models could not be adopted for the intended application due to their black-box nature. We consider nonparametric additive models with a small number of main and pairwise interaction effects using $\ell_0$-based penalization. From a methodological viewpoint, we study our estimator's computational and statistical aspects and discuss variants incorporating strong hierarchical interactions. Our algorithms (open-sourced on GitHub) extend the computational frontiers of existing algorithms for sparse additive models to be able to handle datasets relevant to the application we consider. We discuss and interpret findings from our model on the US Census Planning Database. In addition to being useful from an interpretability standpoint, our models lead to predictions comparable to popular black-box machine learning methods based on gradient boosting and feedforward neural networks - suggesting that it is possible to have models that have the best of both worlds: good model accuracy and interpretability.
This study examines the impact of nighttime light intensity on child health outcomes in Bangladesh. We use nighttime light intensity as a proxy measure of urbanization and argue that the higher intensity of nighttime light, the higher is the degree of urbanization, which positively affects child health outcomes. In econometric estimation, we employ a methodology that combines parametric and non-parametric approaches using the Gradient Boosting Machine (GBM), K-Nearest Neighbors (KNN), and Bootstrap Aggregating that originate from machine learning algorithms. Based on our benchmark estimates, findings show that one standard deviation increase of nighttime light intensity is associated with a 1.515 rise of Z-score of weight for age after controlling for several control variables. The maximum increase of weight for height and height for age score range from 5.35 to 7.18 units. To further understand our benchmark estimates, generalized additive models also provide a robust positive relationship between nighttime light intensity and children's health outcomes. Finally, we develop an economic model that supports the empirical findings of this study that the marginal effect of urbanization on children's nutritional outcomes is strictly positive.
We present a winning method of the IEEE DataPort Competition on Day-Ahead Electricity Demand Forecasting: Post-COVID Paradigm. The day-ahead load forecasting approach is based on online forecast combination of multiple point prediction models. It contains four steps: i) data cleaning and preprocessing, ii) a holiday adjustment procedure, iii) training of individual forecasting models, iv) forecast combination by smoothed Bernstein Online Aggregation (BOA). The approach is flexible and can quickly adopt to new energy system situations as they occurred during and after COVID-19 shutdowns. The pool of individual prediction models ranges from rather simple time series models to sophisticated models like generalized additive models (GAMs) and high-dimensional linear models estimated by lasso. They incorporate autoregressive, calendar and weather effects efficiently. All steps contain novel concepts that contribute to the excellent forecasting performance of the proposed method. This holds particularly for the holiday adjustment procedure and the fully adaptive smoothed BOA approach.
Model compression is generally performed by using quantization, low-rank approximation or pruning, for which various algorithms have been researched in recent years. One fundamental question is: what types of compression work better for a given model? Or even better: can we improve by combining compressions in a suitable way? We formulate this generally as a problem of optimizing the loss but where the weights are constrained to equal an additive combination of separately compressed parts; and we give an algorithm to learn the corresponding parts' parameters. Experimentally with deep neural nets, we observe that 1) we can find significantly better models in the error-compression space, indicating that different compression types have complementary benefits, and 2) the best type of combination depends exquisitely on the type of neural net. For example, we can compress ResNets and AlexNet using only 1 bit per weight without error degradation at the cost of adding a few floating point weights. However, VGG nets can be better compressed by combining low-rank with a few floating point weights.
Causal discovery from data affected by unobserved variables is an important but difficult problem to solve. The effects that unobserved variables have on the relationships between observed variables are more complex in nonlinear cases than in linear cases. In this study, we focus on causal additive models in the presence of unobserved variables. Causal additive models exhibit structural equations that are additive in the variables and error terms. We take into account the presence of not only unobserved common causes but also unobserved intermediate variables. Our theoretical results show that, when the causal relationships are nonlinear and there are unobserved variables, it is not possible to identify all the causal relationships between observed variables through regression and independence tests. However, our theoretical results also show that it is possible to avoid incorrect inferences. We propose a method to identify all the causal relationships that are theoretically possible to identify without being biased by unobserved variables. The empirical results using artificial data and simulated functional magnetic resonance imaging (fMRI) data show that our method effectively infers causal structures in the presence of unobserved variables.
Deployment of machine learning models in real high-risk settings (e.g. healthcare) often depends not only on the model's accuracy but also on its fairness, robustness, and interpretability. Generalized Additive Models (GAMs) are a class of interpretable models with a long history of use in these high-risk domains, but they lack desirable features of deep learning such as differentiability and scalability. In this work, we propose a neural GAM (NODE-GAM) and neural GA$^2$M (NODE-GA$^2$M) that scale well and perform better than other GAMs on large datasets, while remaining interpretable compared to other ensemble and deep learning models. We demonstrate that our models find interesting patterns in the data. Lastly, we show that we improve model accuracy via self-supervised pre-training, an improvement that is not possible for non-differentiable GAMs.
In this article, we propose a new hypothesis testing method for directed acyclic graph (DAG). While there is a rich class of DAG estimation methods, there is a relative paucity of DAG inference solutions. Moreover, the existing methods often impose some specific model structures such as linear models or additive models, and assume independent data observations. Our proposed test instead allows the associations among the random variables to be nonlinear and the data to be time-dependent. We build the test based on some highly flexible neural networks learners. We establish the asymptotic guarantees of the test, while allowing either the number of subjects or the number of time points for each subject to diverge to infinity. We demonstrate the efficacy of the test through simulations and a brain connectivity network analysis.
Sparse regression and classification estimators that respect group structures have application to an assortment of statistical and machine learning problems, from multitask learning to sparse additive modeling to hierarchical selection. This work introduces structured sparse estimators that combine group subset selection with shrinkage. To accommodate sophisticated structures, our estimators allow for arbitrary overlap between groups. We develop an optimization framework for fitting the nonconvex regularization surface and present finite-sample error bounds for estimation of the regression function. As an application requiring structure, we study sparse semiparametric additive modeling, a procedure that allows the effect of each predictor to be zero, linear, or nonlinear. For this task, the new estimators improve across several metrics on synthetic data compared to alternatives. Finally, we demonstrate their efficacy in modeling supermarket foot traffic and economic recessions using many predictors. These demonstrations suggest sparse semiparametric additive models, fit using the new estimators, are an excellent compromise between fully linear and fully nonparametric alternatives. All of our algorithms are made available in the scalable implementation grpsel.
This paper focuses on building of models of stochastic systems with aleatoric uncertainty. The nature of the considered systems is such that the identical inputs can result in different outputs, i.e. the output is a random variable. The suggested in this paper algorithm targets an identification of multi-modal properties of the output distributions, even when they depend on the inputs and vary significantly throughout the dataset. This ability of the suggested method to recognise complex and not only bell-shaped distributions follows from its construction and is backed up by provided experimental results. In general, the suggested method belongs to the category of boosted ensemble learning techniques, where the single deterministic component can be an arbitrarily-chosen regression model. The algorithm does not require any special properties of the chosen regression model, other than having descriptive capabilities with some expected accuracy for the training data type.
Background: Limited research exists on the association between changes in physical activity levels and injury in children. Objective: To assess how well different variations of the acute:chronic workload ratio (ACWR), a measure of change in activity, predict injury in children. Methods: We conducted a prospective cohort study using data from 1670 Danish schoolchildren measured over 5.5 years (2008 to 2014). Coupled 4-week, uncoupled 4-week, and uncoupled 5-week ACWRs were calculated using activity frequency in the past week as the acute load (numerator), and average weekly activity frequency in the past 4 or 5 weeks as the chronic load (denominator). We modelled the relationship between different ACWR variations and injury using generalized linear and generalized additive models, with and without accounting for repeated measures. Results: The prognostic relationship between the ACWR and injury risk was best represented using a generalized additive mixed model for the uncoupled 5-week ACWR. It predicted an injury risk of ~3% for ACWRs between 0.8 (activity level decreased by 20%) and 1.5 (activity level increased by 50%). When activity decreased by more than 20% (ACWR< 0.8), injury risk was lower (minimum of 1.5% at ACWR=0). When activity increased by more than 50% (ACWR > 1.5), injury risk was higher (maximum of 6% at ACWR = 5). Girls were at significantly higher risk of injury than boys. Conclusion: Increases in physical activity in children are associated with much lower injury risks compared to previous results in adults.
Interpretability in machine learning (ML) is crucial for high stakes decisions and troubleshooting. In this work, we provide fundamental principles for interpretable ML, and dispel common misunderstandings that dilute the importance of this crucial topic. We also identify 10 technical challenge areas in interpretable machine learning and provide history and background on each problem. Some of these problems are classically important, and some are recent problems that have arisen in the last few years. These problems are: (1) Optimizing sparse logical models such as decision trees; (2) Optimization of scoring systems; (3) Placing constraints into generalized additive models to encourage sparsity and better interpretability; (4) Modern case-based reasoning, including neural networks and matching for causal inference; (5) Complete supervised disentanglement of neural networks; (6) Complete or even partial unsupervised disentanglement of neural networks; (7) Dimensionality reduction for data visualization; (8) Machine learning models that can incorporate physics and other generative or causal constraints; (9) Characterization of the "Rashomon set" of good models; and (10) Interpretable reinforcement learning. This survey is suitable as a starting point for statisticians and computer scientists interested in working in interpretable machine learning.
The empirical success of deep convolutional networks on tasks involving high-dimensional data such as images or audio suggests that they can efficiently approximate certain functions that are well-suited for such tasks. In this paper, we study this through the lens of kernel methods, by considering simple hierarchical kernels with two or three convolution and pooling layers, inspired by convolutional kernel networks. These achieve good empirical performance on standard vision datasets, while providing a precise description of their functional space that yields new insights on their inductive bias. We show that the RKHS consists of additive models of interaction terms between patches, and that its norm encourages spatial similarities between these terms through pooling layers. We then provide generalization bounds which illustrate how pooling and patches yield improved sample complexity guarantees when the target function presents such regularities.
Bayesian Optimization (BO) has shown significant success in tackling expensive low-dimensional black-box optimization problems. Many optimization problems of interest are high-dimensional, and scaling BO to such settings remains an important challenge. In this paper, we consider generalized additive models in which low-dimensional functions with overlapping subsets of variables are composed to model a high-dimensional target function. Our goal is to lower the computational resources required and facilitate faster model learning by reducing the model complexity while retaining the sample-efficiency of existing methods. Specifically, we constrain the underlying dependency graphs to tree structures in order to facilitate both the structure learning and optimization of the acquisition function. For the former, we propose a hybrid graph learning algorithm based on Gibbs sampling and mutation. In addition, we propose a novel zooming-based algorithm that permits generalized additive models to be employed more efficiently in the case of continuous domains. We demonstrate and discuss the efficacy of our approach via a range of experiments on synthetic functions and real-world datasets.
Non-parametric, additive models are able to capture complex data dependencies in a flexible, yet interpretable way. However, choosing the format of the additive components often requires non-trivial data exploration. Here, as an alternative, we propose PrAda-net, a one-hidden-layer neural network, trained with proximal gradient descent and adaptive lasso. PrAda-net automatically adjusts the size and architecture of the neural network to reflect the complexity and structure of the data. The compact network obtained by PrAda-net can be translated to additive model components, making it suitable for non-parametric statistical modelling with automatic model selection. We demonstrate PrAda-net on simulated data, where wecompare the test error performance, variable importance and variable subset identification properties of PrAda-net to other lasso-based regularization approaches for neural networks. We also apply PrAda-net to the massive U.K. black smoke data set, to demonstrate how PrAda-net can be used to model complex and heterogeneous data with spatial and temporal components. In contrast to classical, statistical non-parametric approaches, PrAda-net requires no preliminary modeling to select the functional forms of the additive components, yet still results in an interpretable model representation.
In recent years, the field of recommendation systems has attracted increasing attention to developing predictive models that provide explanations of why an item is recommended to a user. The explanations can be either obtained by post-hoc diagnostics after fitting a relatively complex model or embedded into an intrinsically interpretable model. In this paper, we propose the explainable recommendation systems based on a generalized additive model with manifest and latent interactions (GAMMLI). This model architecture is intrinsically interpretable, as it additively consists of the user and item main effects, the manifest user-item interactions based on observed features, and the latent interaction effects from residuals. Unlike conventional collaborative filtering methods, the group effect of users and items are considered in GAMMLI. It is beneficial for enhancing the model interpretability, and can also facilitate the cold-start recommendation problem. A new Python package GAMMLI is developed for efficient model training and visualized interpretation of the results. By numerical experiments based on simulation data and real-world cases, the proposed method is shown to have advantages in both predictive performance and explainable recommendation.
We conducted this study to determine whether fallopian tube anatomy can predict the likelihood of pregnancy and pregnancy outcomes after tubal sterilization reversal. We built a flexible, non-parametric, multivariate model via generalized additive models to assess the effects of the following tubal parameters observed during tubal reparative surgery: tubal lengths; differences in tubal segment location, and diameters at the anastomosis sites; and, fibrosis of the tubal muscularis. In this study population, age and tubal length - in that order - were the primary factors predicting the likelihood of pregnancy. For pregnancy outcomes, tubal length was the most influential predictor of birth and ectopic pregnancy, while age was the primary predictor of miscarriage. Segment location and diameters contributed slightly to the odds of miscarriage and ectopic pregnancy. Tubal muscularis fibrosis had a little apparent effect. This study is the first to show that a statistical learning predictive model based on fallopian tube anatomy can predict pregnancy and pregnancy outcome probabilities after tubal reversal surgery.
Many works on inverse problems in the imaging sciences consider regularization via one or more penalty functions or constraint sets. When the models/images are not easily described using one or a few penalty functions/constraints, additive model descriptions for regularization lead to better imaging results. These include cartoon-texture decomposition, morphological component analysis, and robust principal component analysis; methods that typically rely on penalty functions. We propose a regularization framework, based on the Minkowski set, that merges the strengths of additive models and constrained formulations. We generalize the Minkowski set, such that the model parameters are the sum of two components, each of which is constrained to an intersection of sets. Furthermore, the sum of the components is also an element of another intersection of sets. These generalizations allow us to include multiple pieces of prior knowledge on each of the components, as well as on the sum of components, which is necessary to ensure physical feasibility of partial-differential-equation based parameters estimation problems. We derive the projection operation onto the generalized Minkowski sets and construct an algorithm based on the alternating direction method of multipliers. We illustrate how we benefit from using more prior knowledge in the form of the generalized Minkowski set using seismic waveform inversion and video background-anomaly separation.