We introduce GAMSEL (Generalized Additive Model Selection), a penalized likelihood approach for fitting sparse generalized additive models in high dimension. Our method interpolates between null, linear and additive models by allowing the effect of each variable to be estimated as being either zero, linear, or a low-complexity curve, as determined by the data. We present a blockwise coordinate descent procedure for efficiently optimizing the penalized likelihood objective over a dense grid of the tuning parameter, producing a regularization path of additive models. We demonstrate the performance of our method on both real and simulated data examples, and compare it with existing techniques for additive model selection.
Predicting the execution time of computer programs is an important but challenging problem in the community of computer systems. Existing methods require experts to perform detailed analysis of program code in order to construct predictors or select important features. We recently developed a new system to automatically extract a large number of features from program execution on sample inputs, on which prediction models can be constructed without expert knowledge. In this paper we study the construction of predictive models for this problem. We propose the SPORE (Sparse POlynomial REgression) methodology to build accurate prediction models of program performance using feature data collected from program execution on sample inputs. Our two SPORE algorithms are able to build relationships between responses (e.g., the execution time of a computer program) and features, and select a few from hundreds of the retrieved features to construct an explicitly sparse and non-linear model to predict the response variable. The compact and explicitly polynomial form of the estimated model could reveal important insights into the computer program (e.g., features and their non-linear combinations that dominate the execution time), enabling a better understanding of the program’s behavior. Our evaluation on three widely used computer programs shows that SPORE methods can give accurate prediction with relative error less than 7% by using a moderate number of training data samples. In addition, we compare SPORE algorithms to state-of-the-art sparse regression algorithms, and show that SPORE methods, motivated by real applications, outperform the other methods in terms of both interpretability and prediction accuracy.
Additive nonparametric regression models provide an attractive tool for variable selection in high dimensions when the relationship between the response and predictors is complex. They offer greater flexibility compared to parametric non-linear regression models and better interpretability and scalability than the non-parametric regression models. However, achieving sparsity simultaneously in the number of nonparametric components as well as in the variables within each nonparametric component poses a stiff computational challenge. In this article, we develop a novel Bayesian additive regression model using a combination of hard and soft shrinkages to separately control the number of additive components and the variables within each component. An efficient algorithm is developed to select the importance variables and estimate the interaction network. Excellent performance is obtained in simulated and real data examples.
The generalized partially linear additive model (GPLAM) is a flexible and interpretable approach to building predictive models. It combines features in an additive manner, allowing each to have either a linear or nonlinear effect on the response. However, the choice of which features to treat as linear or nonlinear is typically assumed known. Thus, to make a GPLAM a viable approach in situations in which little is known $a~priori$ about the features, one must overcome two primary model selection challenges: deciding which features to include in the model and determining which of these features to treat nonlinearly. We introduce the sparse partially linear additive model (SPLAM), which combines model fitting and $both$ of these model selection challenges into a single convex optimization problem. SPLAM provides a bridge between the lasso and sparse additive models. Through a statistical oracle inequality and thorough simulation, we demonstrate that SPLAM can outperform other methods across a broad spectrum of statistical regimes, including the high-dimensional ($p\gg N$) setting. We develop efficient algorithms that are applied to real data sets with half a million samples and over 45,000 features with excellent predictive performance.
We extend the adaptive regression spline model by incorporating saturation, the natural requirement that a function extend as a constant outside a certain range. We fit saturating splines to data using a convex optimization problem over a space of measures, which we solve using an efficient algorithm based on the conditional gradient method. Unlike many existing approaches, our algorithm solves the original infinite-dimensional (for splines of degree at least two) optimization problem without pre-specified knot locations. We then adapt our algorithm to fit generalized additive models with saturating splines as coordinate functions and show that the saturation requirement allows our model to simultaneously perform feature selection and nonlinear function fitting. Finally, we briefly sketch how the method can be extended to higher order splines and to different requirements on the extension outside the data range.