Scaling regression to large datasets is a common problem in many application areas. We propose a two step approach to scaling regression to large datasets. Using a regression tree (CART) to segment the large dataset constitutes the first step of this approach. The second step of this approach is to develop a suitable regression model for each segment. Since segment sizes are not very large, we have the ability to apply sophisticated regression techniques if required. A nice feature of this two step approach is that it can yield models that have good explanatory power as well as good predictive performance. Ensemble methods like Gradient Boosted Trees can offer excellent predictive performance but may not provide interpretable models. In the experiments reported in this study, we found that the predictive performance of the proposed approach matched the predictive performance of Gradient Boosted Trees.

Relational probabilistic models have the challenge of aggregation, where one variable depends on a population of other variables. Consider the problem of predicting gender from movie ratings; this is challenging because the number of movies per user and users per movie can vary greatly. Surprisingly, aggregation is not well understood. In this paper, we show that existing relational models (implicitly or explicitly) either use simple numerical aggregators that lose great amounts of information, or correspond to naive Bayes, logistic regression, or noisy-OR that suffer from overconfidence. We propose new simple aggregators and simple modifications of existing models that empirically outperform the existing ones. The intuition we provide on different (existing or new) models and their shortcomings plus our empirical findings promise to form the foundation for future representations.

IBM Data Science Experience (DSX) is an interactive, collaborative, cloud-based environment where data scientists can use multiple tools to achieve insights. Data scientists can use the best of open source, tap into IBM's unique features, grow their skills, and collaborate with teams. One of the many features of DSX provides the capability to create and train a machine learning model in DSX with little to no coding. This model can subsequently be saved and deployed to Watson Machine Learning on IBM Bluemix and called for scoring in real-time. This tutorial is a continuation of the following logistic regression analysis, which creates, trains, saves and deploys a logistic regression model that predicts the possibility for a tent purchase based on age, sex, marital status, and job profession for an individual.

We consider a sparse linear regression model Y=X\beta^{*}+W where X has a Gaussian entries, W is the noise vector with mean zero Gaussian entries, and \beta^{*} is a binary vector with support size (sparsity) k. Using a novel conditional second moment method we obtain a tight up to a multiplicative constant approximation of the optimal squared error \min_{\beta}\|Y-X\beta\|_{2}, where the minimization is over all k-sparse binary vectors \beta. The approximation reveals interesting structural properties of the underlying regression problem. In particular, a) We establish that n^*=2k\log p/\log (2k/\sigma^{2}+1) is a phase transition point with the following "all-or-nothing" property. When n exceeds n^{*}, (2k)^{-1}\|\beta_{2}-\beta^*\|_0\approx 0, and when n is below n^{*}, (2k)^{-1}\|\beta_{2}-\beta^*\|_0\approx 1, where \beta_2 is the optimal solution achieving the smallest squared error. With this we prove that n^{*} is the asymptotic threshold for recovering \beta^* information theoretically. b) We compute the squared error for an intermediate problem \min_{\beta}\|Y-X\beta\|_{2} where minimization is restricted to vectors \beta with \|\beta-\beta^{*}\|_0=2k \zeta, for \zeta\in [0,1]. We show that a lower bound part \Gamma(\zeta) of the estimate, which corresponds to the estimate based on the first moment method, undergoes a phase transition at three different thresholds, namely n_{\text{inf,1}}=\sigma^2\log p, which is information theoretic bound for recovering \beta^* when k=1 and \sigma is large, then at n^{*} and finally at n_{\text{LASSO/CS}}. c) We establish a certain Overlap Gap Property (OGP) on the space of all binary vectors \beta when n\le ck\log p for sufficiently small constant c. We conjecture that OGP is the source of algorithmic hardness of solving the minimization problem \min_{\beta}\|Y-X\beta\|_{2} in the regime n

Quantitatively predicting phenotype variables by the expression changes in a set of candidate genes is of great interest in molecular biology but it is also a challenging task for several reasons. First, the collected biological observations might be heterogeneous and correspond to different biological mechanisms. Secondly, the gene expression variables used to predict the phenotype are potentially highly correlated since genes interact though unknown regulatory networks. In this paper, we present a novel approach designed to predict quantitative trait from transcriptomic data, taking into account the heterogeneity in biological samples and the hidden gene regulatory networks underlying different biological mechanisms. The proposed model performs well on prediction but it is also fully parametric, which facilitates the downstream biological interpretation. The model provides clusters of individuals based on the relation between gene expression data and the phenotype, and also leads to infer a gene regulatory network specific for each cluster of individuals. We perform numerical simulations to demonstrate that our model is competitive with other prediction models, and we demonstrate the predictive performance and the interpretability of our model to predict alcohol sensitivity from transcriptomic data on real data from Drosophila Melanogaster Genetic Reference Panel (DGRP).

The positivity assumption, or the experimental treatment assignment (ETA) assumption, is important for identifiability in causal inference. Even if the positivity assumption holds, practical violations of this assumption may jeopardize the finite sample performance of the causal estimator. One of the consequences of practical violations of the positivity assumption is extreme values in the estimated propensity score (PS). A common practice to address this issue is truncating the PS estimate when constructing PS-based estimators. In this study, we propose a novel adaptive truncation method, Positivity-C-TMLE, based on the collaborative targeted maximum likelihood estimation (C-TMLE) methodology. We demonstrate the outstanding performance of our novel approach in a variety of simulations by comparing it with other commonly studied estimators. Results show that by adaptively truncating the estimated PS with a more targeted objective function, the Positivity-C-TMLE estimator achieves the best performance for both point estimation and confidence interval coverage among all estimators considered.

Bayesian optimization has recently attracted the attention of the automatic machine learning community for its excellent results in hyperparameter tuning. BO is characterized by the sample efficiency with which it can optimize expensive black-box functions. The efficiency is achieved in a similar fashion to the learning to learn methods: surrogate models (typically in the form of Gaussian processes) learn the target function and perform intelligent sampling. This surrogate model can be applied even in the presence of noise; however, as with most regression methods, it is very sensitive to outlier data. This can result in erroneous predictions and, in the case of BO, biased and inefficient exploration. In this work, we present a GP model that is robust to outliers which uses a Student-t likelihood to segregate outliers and robustly conduct Bayesian optimization. We present numerical results evaluating the proposed method in both artificial functions and real problems.

We present a parallel AND/OR Branch-and-Bound scheme that uses the power of a computational grid to push the boundaries of feasibility for combinatorial optimization. Two variants of the scheme are described, one of which aims to use machine learning techniques for parallel load balancing. In-depth analysis identifies two inherent sources of parallel search space redundancies that, together with general parallel execution overhead, can impede parallelization and render the problem far from embarrassingly parallel. We conduct extensive empirical evaluation on hundreds of CPUs, the first of its kind, with overall positive results. In a significant number of cases parallel speedup is close to the theoretical maximum and we are able to solve many very complex problem instances orders of magnitude faster than before; yet analysis of certain results also serves to demonstrate the inherent limitations of the approach due to the aforementioned redundancies.

Linear Mixed Models (LMMs) are important tools in statistical genetics. When used for feature selection, they allow to find a sparse set of genetic traits that best predict a continuous phenotype of interest, while simultaneously correcting for various confounding factors such as age, ethnicity and population structure. Formulated as models for linear regression, LMMs have been restricted to continuous phenotypes. We introduce the Sparse Probit Linear Mixed Model (Probit-LMM), where we generalize the LMM modeling paradigm to binary phenotypes. As a technical challenge, the model no longer possesses a closed-form likelihood function. In this paper, we present a scalable approximate inference algorithm that lets us fit the model to high-dimensional data sets. We show on three real-world examples from different domains that in the setup of binary labels, our algorithm leads to better prediction accuracies and also selects features which show less correlation with the confounding factors.

In this article, we are going to learn how the logistic regression model works in machine learning. The logistic regression model is one member of the supervised classification algorithm family. The building block concepts of logistic regression can be helpful in deep learning while building the neural networks. Logistic regression classifier is more like a linear classifier which uses the calculated logits (score) to predict the target class. If you are not familiar with the concepts of the logits, don't frighten. We are going to learn each and every block of logistic regression by the end of this post.