We propose a new sparsity-smoothness penalty for high-dimensional generalized additive models. The combination of sparsity and smoothness is crucial for mathematical theory as well as performance for finite-sample data. We present a computationally efficient algorithm, with provable numerical convergence properties, for optimizing the penalized likelihood. Furthermore, we provide oracle results which yield asymptotic optimality of our estimator for high dimensional but sparse additive models. Finally, an adaptive version of our sparsity-smoothness penalized approach yields large additional performance gains.
Sparse generalized additive models (GAMs) are an extension of sparse generalized linear models which allow a model's prediction to vary non-linearly with an input variable. This enables the data analyst build more accurate models, especially when the linearity assumption is known to be a poor approximation of reality. Motivated by reluctant interaction modeling (Yu et al. 2019), we propose a multi-stage algorithm, called $\textit{reluctant generalized additive modeling (RGAM)}$, that can fit sparse generalized additive models at scale. It is guided by the principle that, if all else is equal, …
Extrapolation Guarantees for Perturbation Modeling Under the Additive Latent Shift Assumption
Seminar for Statistics, ETH Zurich（统计系，苏黎世联邦理工学院）；ETH Zurich（苏黎世联邦理工学院）；DZNE, Bonn, Germany（波恩德国DZNE）；Department of Statistics, University of Washington, Seattle, USA（华盛顿大学统计系，美国西雅图）
Cluster-Based Generalized Additive Models Informed by Random Fourier Features
Department of Mathematics and Mathematical Statistics, Umeå University（数学与统计学系，乌梅大学）；Department of Statistics, The Pennsylvania State University（统计系，宾夕法尼亚州立大学）
Gaussian Process (GP) models are a class of flexible non-parametric models that have rich representational power. By using a Gaussian process with additive structure, complex responses can be modelled whilst retaining interpretability. Previous work showed that additive Gaussian process models require high-dimensional interaction terms. We propose the orthogonal additive kernel (OAK), which imposes an orthogonality constraint on the additive functions, enabling an identifiable, low-dimensional representation of the functional relationship. We connect the OAK kernel to functional ANOVA decompos…
The University of Sydney（悉尼大学）；Tsinghua University（清华大学）；National Institute of Informatics（国立信息学研究所）
We present the Additive Poisson Process (APP), a novel framework that can model the higher-order interaction effects of the intensity functions in stochastic processes using lower dimensional projections. Our model combines the techniques in information geometry to model higher-order interactions on a statistical manifold and in generalized additive models to use lower-dimensional projections to overcome the effects from the curse of dimensionality. Our approach solves a convex optimization problem by minimizing the KL divergence from a sample distribution in lower dimensional projections to t…
National University of Singapore（新加坡国立大学）；Indian Institute of Technology (BHU) Varanasi（印度理工学院（巴纳拉斯印度教大学）瓦拉纳西分校）
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-effici…
CERAMATHS（CERAMATHS实验室）；Univ. Polytechnique Hauts-de-France（上法兰西综合理工大学）；IMT, UMR5219 CNRS（图卢兹数学研究所（法国国家科学研究中心混合研究单位5219））；Université Paul Sabatier（保罗·萨巴蒂埃大学）；INSA（国立应用科学学院）
We introduce an additive Gaussian process framework accounting for monotonicity constraints and scalable to high dimensions. Our contributions are threefold. First, we show that our framework enables to satisfy the constraints everywhere in the input space. We also show that more general componentwise linear inequality constraints can be handled similarly, such as componentwise convexity. Second, we propose the additive MaxMod algorithm for sequential dimension reduction. By sequentially maximizing a squared-norm criterion, MaxMod identifies the active input dimensions and refines the most imp…
Wire-feed laser additive manufacturing (WLAM) is gaining wide interest due to its high level of automation, high deposition rates, and good quality of printed parts. In-process monitoring and feedback controls that would reduce the uncertainty in the quality of the material are in the early stages of development. Machine learning promises the ability to accelerate the adoption of new processes and property design in additive manufacturing by making process-structure-property connections between process setting inputs and material quality outcomes. The molten pool dimensional information and te…
EPFL（洛桑联邦理工学院）；National University of Singapore（新加坡国立大学）
Bayesian optimization (BO) is a popular technique for sequential black-box function optimization, with applications including parameter tuning, robotics, environmental monitoring, and more. One of the most important challenges in BO is the development of algorithms that scale to high dimensions, which remains a key open problem despite recent progress. In this paper, we consider the approach of Kandasamy et al. (2015), in which the high-dimensional function decomposes as a sum of lower-dimensional functions on subsets of the underlying variables. In particular, we significantly generalize this…
Cornell University（康奈尔大学）；New York University（纽约大学）
Gaussian processes (GPs) provide flexible distributions over functions, with inductive biases controlled by a kernel. However, in many applications Gaussian processes can struggle with even moderate input dimensionality. Learning a low dimensional projection can help alleviate this curse of dimensionality, but introduces many trainable hyperparameters, which can be cumbersome, especially in the small data regime. We use additive sums of kernels for GP regression, where each kernel operates on a different random projection of its inputs. Surprisingly, we find that as the number of random projec…
Motivated by the sampling problems and heterogeneity issues common in high- dimensional big datasets, we consider a class of discordant additive index models. We propose method of moments based procedures for estimating the indices of such discordant additive index models in both low and high-dimensional settings. Our estimators are based on factorizing certain moment tensors and are also applicable in the overcomplete setting, where the number of indices is more than the dimensionality of the datasets. Furthermore, we provide rates of convergence of our estimator in both high and low-dimensio…
