We study the behavior of a fundamental tool in sparse statistical modeling --the best-subset selection procedure (aka "best-subsets"). Assuming that the underlying linear model is sparse, it is well known, both in theory and in practice, that the best-subsets procedure works extremely well in terms of several statistical metrics (prediction, estimation and variable selection) when the signal to noise ratio (SNR) is high. However, its performance degrades substantially when the SNR is low -- it is outperformed in predictive accuracy by continuous shrinkage methods, such as ridge regression and the Lasso. We explain why this behavior should not come as a surprise, and contend that the original version of the classical best-subsets procedure was, perhaps, not designed to be used in the low SNR regimes. We propose a close cousin of best-subsets, namely, its $\ell_{q}$-regularized version, for $q \in\{1, 2\}$, which (a) mitigates, to a large extent, the poor predictive performance of best-subsets in the low SNR regimes; (b) performs favorably and generally delivers a substantially sparser model when compared to the best predictive models available via ridge regression and the Lasso. Our estimator can be expressed as a solution to a mixed integer second order conic optimization problem and, hence, is amenable to modern computational tools from mathematical optimization. We explore the theoretical properties of the predictive capabilities of the proposed estimator and complement our findings via several numerical experiments.

Churn prediction is big business. It minimizes customer defection by predicting which customers are likely to cancel a subscription to a service. Though originally used within the telecommunications industry, it has become common practice across banks, ISPs, insurance firms, and other verticals. The prediction process is heavily data-driven and often utilizes advanced machine learning techniques. In this post, we'll take a look at what types of customer data are typically used, do some preliminary analysis of the data, and generate churn prediction models -- all with Spark and its machine learning frameworks.

Noisy PN learning is the problem of binary classification when training examples may be mislabeled (flipped) uniformly with noise rate rho1 for positive examples and rho0 for negative examples. We propose Rank Pruning (RP) to solve noisy PN learning and the open problem of estimating the noise rates, i.e. the fraction of wrong positive and negative labels. Unlike prior solutions, RP is time-efficient and general, requiring O(T) for any unrestricted choice of probabilistic classifier with T fitting time. We prove RP has consistent noise estimation and equivalent expected risk as learning with uncorrupted labels in ideal conditions, and derive closed-form solutions when conditions are non-ideal. RP achieves state-of-the-art noise estimation and F1, error, and AUC-PR for both MNIST and CIFAR datasets, regardless of the amount of noise and performs similarly impressively when a large portion of training examples are noise drawn from a third distribution. To highlight, RP with a CNN classifier can predict if an MNIST digit is a "one"or "not" with only 0.25% error, and 0.46 error across all digits, even when 50% of positive examples are mislabeled and 50% of observed positive labels are mislabeled negative examples.

We introduce canonical correlation forests (CCFs), a new decision tree ensemble method for classification and regression. Individual canonical correlation trees are binary decision trees with hyperplane splits based on local canonical correlation coefficients calculated during training. Unlike axis-aligned alternatives, the decision surfaces of CCFs are not restricted to the coordinate system of the inputs features and therefore more naturally represent data with correlated inputs. CCFs naturally accommodate multiple outputs, provide a similar computational complexity to random forests, and inherit their impressive robustness to the choice of input parameters. As part of the CCF training algorithm, we also introduce projection bootstrapping, a novel alternative to bagging for oblique decision tree ensembles which maintains use of the full dataset in selecting split points, often leading to improvements in predictive accuracy. Our experiments show that, even without parameter tuning, CCFs out-perform axis-aligned random forests and other state-of-the-art tree ensemble methods on both classification and regression problems, delivering both improved predictive accuracy and faster training times. We further show that they outperform all of the 179 classifiers considered in a recent extensive survey.

Welcome to the fourth article in a five-part series about machine learning. In this article, we will take a deeper dive into model evaluation and performance metrics, and potential prediction-related errors that one may encounter. Before digging deeper into model performance and error types, we must first discuss the concept of residuals and errors for regression, positive and negative classifications for classification problems, and in-sample versus out-of-sample measurements. Any reference to models, metrics, or errors computed with respect to the data used to train, validate, or tune a predictive model (i.e., data you have) is called in-sample. Conversely, reference to test data metrics and errors, or new data in general is called out-of-sample (i.e., data you don't have). Recall that regression involves predicting a continuous valued output (response) based on some set of input variables (features/predictors).

In recent years, randomized methods for numerical linear algebra have received growing interest as a general approach to large-scale problems. Typically, the essential ingredient of these methods is some form of randomized dimension reduction, which accelerates computations, but also creates random approximation error. In this way, the dimension reduction step encodes a tradeoff between cost and accuracy. However, the exact numerical relationship between cost and accuracy is typically unknown, and consequently, it may be difficult for the user to precisely know (1) how accurate a given solution is, or (2) how much computation is needed to achieve a given level of accuracy. In the current paper, we study randomized matrix multiplication (sketching) as a prototype setting for addressing these general problems. As a solution, we develop a bootstrap method for {directly estimating} the accuracy as a function of the reduced dimension (as opposed to deriving worst-case bounds on the accuracy in terms of the reduced dimension). From a computational standpoint, the proposed method does not substantially increase the cost of standard sketching methods, and this is made possible by an "extrapolation" technique. In addition, we provide both theoretical and empirical results to demonstrate the effectiveness of the proposed method.

We consider the problem of efficient "on the fly" tuning of existing, or {\it legacy}, Artificial Intelligence (AI) systems. The legacy AI systems are allowed to be of arbitrary class, albeit the data they are using for computing interim or final decision responses should posses an underlying structure of a high-dimensional topological real vector space. The tuning method that we propose enables dealing with errors without the need to re-train the system. Instead of re-training a simple cascade of perceptron nodes is added to the legacy system. The added cascade modulates the AI legacy system's decisions. If applied repeatedly, the process results in a network of modulating rules "dressing up" and improving performance of existing AI systems. Mathematical rationale behind the method is based on the fundamental property of measure concentration in high dimensional spaces. The method is illustrated with an example of fine-tuning a deep convolutional network that has been pre-trained to detect pedestrians in images.

Early identification of mood episodes enabling timely mood stabilisation is an important clinical goal. Recent technological advances allow the prospective reporting of mood in real time enabling more accurate, efficient data capture. The complex nature of these data streams in combination with challenge of deriving meaning from missing data mean pose a significant analytic challenge. The signature method is derived from stochastic analysis and has the ability to capture important properties of complex ordered time series data. Objective: To explore whether the onset of episodes of mania and depression can be identified using self-reported mood data.

This paper presents a Neural Aggregation Network (NAN) for video face recognition. The network takes a face video or face image set of a person with a variable number of face images as its input, and produces a compact, fixed-dimension feature representation for recognition. The whole network is composed of two modules. The feature embedding module is a deep Convolutional Neural Network (CNN) which maps each face image to a feature vector. The aggregation module consists of two attention blocks which adaptively aggregate the feature vectors to form a single feature inside the convex hull spanned by them. Due to the attention mechanism, the aggregation is invariant to the image order. Our NAN is trained with a standard classification or verification loss without any extra supervision signal, and we found that it automatically learns to advocate high-quality face images while repelling low-quality ones such as blurred, occluded and improperly exposed faces. The experiments on IJB-A, YouTube Face, Celebrity-1000 video face recognition benchmarks show that it consistently outperforms naive aggregation methods and achieves the state-of-the-art accuracy.

Quick and accurate medical diagnosis is crucial for the successful treatment of a disease. Using machine learning algorithms, we have built two models to predict a hematologic disease, based on laboratory blood test results. In one predictive model, we used all available blood test parameters and in the other a reduced set, which is usually measured upon patient admittance. Both models produced good results, with a prediction accuracy of 0.88 and 0.86, when considering the list of five most probable diseases, and 0.59 and 0.57, when considering only the most probable disease. Models did not differ significantly from each other, which indicates that a reduced set of parameters contains a relevant fingerprint of a disease, expanding the utility of the model for general practitioner's use and indicating that there is more information in the blood test results than physicians recognize. In the clinical test we showed that the accuracy of our predictive models was on a par with the ability of hematology specialists. Our study is the first to show that a machine learning predictive model based on blood tests alone, can be successfully applied to predict hematologic diseases and could open up unprecedented possibilities in medical diagnosis.