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Register a new userBeware of the Simulated DAG! Varsortability in Additive Noise Models
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Additive noise models are a class of causal models in which each variable is defined as a function of its causes plus independent noise. In such models, the ordering of variables by marginal variances may be indicative of the causal order. We introduce varsortability as a measure of agreement between the ordering by marginal variance and the causal order. We show how varsortability dominates the performance of continuous structure learning algorithms on synthetic data. On real-world data, varsortability is an implausible and untestable assumption and we find no indication of high varsortability. We aim to raise awareness that varsortability easily occurs in simulated additive noise models. We provide a baseline method that explicitly exploits varsortability and advocate reporting varsortability in benchmarking data.
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Deep latent variable models (DLVMs) combine the approximation abilities of deep neural networks and the statistical foundations of generative models. Variational methods are commonly used for inference; however, the exact likelihood of these models has been largely overlooked. The purpose of this work is to study the general properties of this quantity and to show how they can be leveraged in practice. We focus on important inferential problems that rely on the likelihood: estimation and missing data imputation. First, we investigate maximum likelihood estimation for DLVMs: in particular, we show that most unconstrained models used for continuous data have an unbounded likelihood function. This problematic behaviour is demonstrated to be a source of mode collapse. We also show how to ensure the existence of maximum likelihood estimates, and draw useful connections with nonparametric mixture models. Finally, we describe an algorithm for missing data imputation using the exact conditional likelihood of a deep latent variable model. On several data sets, our algorithm consistently and significantly outperforms the usual imputation scheme used for DLVMs.
Machine Learning algorithms are increasingly being used in recent years due to their flexibility in model fitting and increased predictive performance. However, the complexity of the models makes them hard for the data analyst to interpret the results and explain them without additional tools. This has led to much research in developing various approaches to understand the model behavior. In this paper, we present the Explainable Neural Network (xNN), a structured neural network designed especially to learn interpretable features. Unlike fully connected neural networks, the features engineered by the xNN can be extracted from the network in a relatively straightforward manner and the results displayed. With appropriate regularization, the xNN provides a parsimonious explanation of the relationship between the features and the output. We illustrate this interpretable feature--engineering property on simulated examples.
Single Index Models (SIMs) are simple yet flexible semi-parametric models for classification and regression. Response variables are modeled as a nonlinear, monotonic function of a linear combination of features. Estimation in this context requires learning both the feature weights, and the nonlinear function. While methods have been described to learn SIMs in the low dimensional regime, a method that can efficiently learn SIMs in high dimensions has not been forthcoming. We propose three variants of a computationally and statistically efficient algorithm for SIM inference in high dimensions. We establish excess risk bounds for the proposed algorithms and experimentally validate the advantages that our SIM learning methods provide relative to Generalized Linear Model (GLM) and low dimensional SIM based learning methods.
A generalized additive model (GAM, Hastie and Tibshirani (1987)) is a nonparametric model by the sum of univariate functions with respect to each explanatory variable, i.e., $f({mathbf x}) = sum f_j(x_j)$, where $x_jinmathbb{R}$ is $j$-th component of a sample ${mathbf x}in mathbb{R}^p$. In this paper, we introduce the total variation (TV) of a function as a measure of the complexity of functions in $L^1_{rm c}(mathbb{R})$-space. Our analysis shows that a GAM based on TV-regularization exhibits a Rademacher complexity of $O(sqrt{frac{log p}{m}})$, which is tight in terms of both $m$ and $p$ in the agnostic case of the classification problem. In result, we obtain generalization error bounds for finite samples according to work by Bartlett and Mandelson (2002).
Additive models form a widely popular class of regression models which represent the relation between covariates and response variables as the sum of low-dimensional transfer functions. Besides flexibility and accuracy, a key benefit of these models is their interpretability: the transfer functions provide visual means for inspecting the models and identifying domain-specific relations between inputs and outputs. However, in large-scale problems involving the prediction of many related tasks, learning independently additive models results in a loss of model interpretability, and can cause overfitting when training data is scarce. We introduce a novel multi-task learning approach which provides a corpus of accurate and interpretable additive models for a large number of related forecasting tasks. Our key idea is to share transfer functions across models in order to reduce the model complexity and ease the exploration of the corpus. We establish a connection with sparse dictionary learning and propose a new efficient fitting algorithm which alternates between sparse coding and transfer function updates. The former step is solved via an extension of Orthogonal Matching Pursuit, whose properties are analyzed using a novel recovery condition which extends existing results in the literature. The latter step is addressed using a traditional dictionary update rule. Experiments on real-world data demonstrate that our approach compares favorably to baseline methods while yielding an interpretable corpus of models, revealing structure among the individual tasks and being more robust when training data is scarce. Our framework therefore extends the well-known benefits of additive models to common regression settings possibly involving thousands of tasks.
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