The Morse-Smale complex of f is a partition of K based on the gradient flow induced by f. Roughly speaking, the complex consists of sets, called crystals or cells, comprised of regions where f is increasing or decreasing. Figure 1 shows the Morse-Smale complex for a two-dimensional function. The cells are the intersections of the basins of attractions (under the gradient flow) of the function's maxima and minima. The function f is piecewise monotonic over cells with respect to some directions. In a sense, the Morse-Smale complex provides a generalization of isotonic regression. Because the Morse-Smale complex represents a multivariate function in terms of regions on which the function has simple behavior, the Morse-Smale complex has useful applications in statistics, including in clustering, regression, testing, and visualization. For instance, when f is a density function, the basins of attraction of f's modes are the (population) clusters for density-mode clustering

In this paper, we consider stochastic dual coordinate (SDCA) {\em without} strongly convex assumption or convex assumption. We show that SDCA converges linearly under mild conditions termed restricted strong convexity. This covers a wide array of popular statistical models including Lasso, group Lasso, and logistic regression with $\ell_1$ regularization, corrected Lasso and linear regression with SCAD regularizer. This significantly improves previous convergence results on SDCA for problems that are not strongly convex. As a by product, we derive a dual free form of SDCA that can handle general regularization term, which is of interest by itself.

The intersection of causal inference and machine learning is a rapidly advancing field. We propose a new approach, the method of direct estimation, that draws on both traditions in order to obtain nonparametric estimates of treatment effects. The approach focuses on estimating the effect of fluctuations in a treatment variable on an outcome. A tensor-spline implementation enables rich interactions between functional bases allowing for the approach to capture treatment/covariate interactions. We show how new innovations in Bayesian sparse modeling readily handle the proposed framework, and then document its performance in simulation and applied examples. Furthermore we show how the method of direct estimation can easily extend to structural estimators commonly used in a variety of disciplines, like instrumental variables, mediation analysis, and sequential g-estimation.

We quantitatively investigate how machine learning models leak information about the individual data records on which they were trained. We focus on the basic membership inference attack: given a data record and black-box access to a model, determine if the record was in the model's training dataset. To perform membership inference against a target model, we make adversarial use of machine learning and train our own inference model to recognize differences in the target model's predictions on the inputs that it trained on versus the inputs that it did not train on. We empirically evaluate our inference techniques on classification models trained by commercial "machine learning as a service" providers such as Google and Amazon. Using realistic datasets and classification tasks, including a hospital discharge dataset whose membership is sensitive from the privacy perspective, we show that these models can be vulnerable to membership inference attacks. We then investigate the factors that influence this leakage and evaluate mitigation strategies.

The optimization of algorithm (hyper-)parameters is crucial for achieving peak performance across a wide range of domains, ranging from deep neural networks to solvers for hard combinatorial problems. The resulting algorithm configuration (AC) problem has attracted much attention from the machine learning community. However, the proper evaluation of new AC procedures is hindered by two key hurdles. First, AC benchmarks are hard to set up. Second and even more significantly, they are computationally expensive: a single run of an AC procedure involves many costly runs of the target algorithm whose performance is to be optimized in a given AC benchmark scenario. One common workaround is to optimize cheap-to-evaluate artificial benchmark functions (e.g., Branin) instead of actual algorithms; however, these have different properties than realistic AC problems. Here, we propose an alternative benchmarking approach that is similarly cheap to evaluate but much closer to the original AC problem: replacing expensive benchmarks by surrogate benchmarks constructed from AC benchmarks. These surrogate benchmarks approximate the response surface corresponding to true target algorithm performance using a regression model, and the original and surrogate benchmark share the same (hyper-)parameter space. In our experiments, we construct and evaluate surrogate benchmarks for hyperparameter optimization as well as for AC problems that involve performance optimization of solvers for hard combinatorial problems, drawing training data from the runs of existing AC procedures. We show that our surrogate benchmarks capture overall important characteristics of the AC scenarios, such as high- and low-performing regions, from which they were derived, while being much easier to use and orders of magnitude cheaper to evaluate.

TensorFlow is an open source software library for Machine Intelligence. The independent recipes in this book will teach you how to use TensorFlow for complex data computations and will let you dig deeper and gain more insights into your data than ever before. This guide starts with the fundamentals of the TensorFlow library which includes variables, matrices, and various data sources. Moving ahead, you will get hands-on experience with Linear Regression techniques with TensorFlow. The next chapters cover important high-level concepts such as neural networks, CNN, RNN, and NLP.

We consider additive models built with trend filtering, i.e., additive models whose components are each regularized by the (discrete) total variation of their $(k+1)$st (discrete) derivative, for a chosen integer $k \geq 0$. This results in $k$th degree piecewise polynomial components, (e.g., $k=0$ gives piecewise constant components, $k=1$ gives piecewise linear, $k=2$ gives piecewise quadratic, etc.). In univariate nonparametric regression, the localized nature of the total variation regularizer used by trend filtering has been shown to produce estimates with superior local adaptivity to those from smoothing splines (and linear smoothers, more generally) (Tibshirani [2014]). Further, the structured nature of this regularizer has been shown to lead to highly efficient computational routines for trend filtering (Kim et al. [2009], Ramdas and Tibshirani [2016]). In this paper, we argue that both of these properties carry over to the additive models setting. We derive fast error rates for additive trend filtering estimates, and prove that these rates are minimax optimal when the underlying function is itself additive and has component functions whose derivatives are of bounded variation. We show that such rates are unattainable by additive smoothing splines (and by additive models built from linear smoothers, in general). We argue that backfitting provides an efficient algorithm for additive trend filtering, as it is built around the fast univariate trend filtering solvers; moreover, we describe a modified backfitting procedure whose iterations can be run in parallel. Finally, we conduct experiments to examine the empirical properties of additive trend filtering, and outline some possible extensions.

There is an especially strong need in modern large-scale data analysis to prioritize samples for manual inspection. For example, the inspection could target important mislabeled samples or key vulnerabilities exploitable by an adversarial attack. In order to solve the "needle in the haystack" problem of which samples to inspect, we develop a new scalable version of Cook's distance, a classical statistical technique for identifying samples which unusually strongly impact the fit of a regression model (and its downstream predictions). In order to scale this technique up to very large and high-dimensional datasets, we introduce a new algorithm which we call "influence sketching." Influence sketching embeds random projections within the influence computation; in particular, the influence score is calculated using the randomly projected pseudo-dataset from the post-convergence Generalized Linear Model (GLM). We validate that influence sketching can reliably and successfully discover influential samples by applying the technique to a malware detection dataset of over 2 million executable files, each represented with almost 100,000 features. For example, we find that randomly deleting approximately 10% of training samples reduces predictive accuracy only slightly from 99.47% to 99.45%, whereas deleting the same number of samples with high influence sketch scores reduces predictive accuracy all the way down to 90.24%. Moreover, we find that influential samples are especially likely to be mislabeled. In the case study, we manually inspect the most influential samples, and find that influence sketching pointed us to new, previously unidentified pieces of malware.

We hear the term "machine learning" a lot these days, usually in the context of predictive analysis and artificial intelligence. Machine learning is, more or less, a way for computers to learn things without being specifically programmed. But how does that actually happen? The answer is, in one word, algorithms. Algorithms are sets of rules that a computer is able to follow.

Representing documents is a crucial component in many NLP tasks, for instance predicting aspect ratings in reviews. Previous methods for this task treat documents globally, and do not acknowledge that target categories are often assigned by their authors with generally no indication of the specific sentences that motivate them. To address this issue, we adopt a weakly supervised learning model, which jointly learns to focus on relevant parts of a document according to the context along with a classifier for the target categories. Derived from the weighted multiple-instance regression (MIR) framework, the model learns decomposable document vectors for each individual category and thus overcomes the representational bottleneck in previous methods due to a fixed-length document vector. During prediction, the estimated relevance or saliency weights explicitly capture the contribution of each sentence to the predicted rating, thus offering an explanation of the rating. Our model achieves state-of-the-art performance on multi-aspect sentiment analysis, improving over several baselines. Moreover, the predicted saliency weights are close to human estimates obtained by crowdsourcing, and increase the performance of lexical and topical features for review segmentation and summarization.