In supervised learning, given a sample of n pairs of inputs and outputs, the goal is to estimate a function to be used to predict future outputs based on observing only the corresponding inputs. The quality of an estimate is often measured in terms of the mean-squared prediction error, in which case the regression function is optimal. Since the properties of the function to be estimated are not known a priori, nonparametric techniques, that can adapt their complexity to the problem at hand, are often key to good results. Kernel methods [15, 36] are probably the most common nonparametric approaches to learning. They are based on choosing a reproducing kernel Hilbert space (RKHS) as the hypothesis space in the design of learning algorithms. A classical learning algorithm using kernel methods to perform learning tasks is kernel ridge regression (KRR), which is based on minimizing the sum of a data-fitting term and an explicit penalty term. The penalty term is used for regularization, and controls the complexity of the solution, preventing overfitting. The statistical properties of KRR have been studied extensively, see e.g.

We analyze the learning properties of the stochastic gradient method when multiple passes over the data and mini-batches are allowed. We study how regularization properties are controlled by the step-size, the number of passes and the mini-batch size. In particular, we consider the square loss and show that for a universal step-size choice, the number of passes acts as a regularization parameter, and optimal finite sample bounds can be achieved by early-stopping. Moreover, we show that larger step-sizes are allowed when considering mini-batches. Our analysis is based on a unifying approach, encompassing both batch and stochastic gradient methods as special cases. As a byproduct, we derive optimal convergence results for batch gradient methods (even in the non-attainable cases).

We revisit the classification problem and focus on nonlinear methods for classification on manifolds. For multivariate datasets lying on an embedded nonlinear Riemannian manifold within the higher-dimensional space, our aim is to acquire a classification boundary between the classes with labels. Motivated by the principal flow [Panaretos, Pham and Yao, 2014], a curve that moves along a path of the maximum variation of the data, we introduce the principal boundary. From the classification perspective, the principal boundary is defined as an optimal curve that moves in between the principal flows traced out from two classes of the data, and at any point on the boundary, it maximizes the margin between the two classes. We estimate the boundary in quality with its direction supervised by the two principal flows. We show that the principal boundary yields the usual decision boundary found by the support vector machine, in the sense that locally, the two boundaries coincide. By means of examples, we illustrate how to find, use and interpret the principal boundary.

Upwork) critically rely on the ability to propose adequate matches based on imperfect knowledge of the two parties to be matched. This prompts the following question: Which matching recommendation algorithms can, in the presence of such uncertainty, lead to efficient platform operation? To answer this question, we develop a model of a task / server matching system. For this model, we give a necessary and sufficient condition for an incoming stream of tasks to be manageable by the system. We further identify a so-called back-pressure policy under which the throughput that the system can handle is optimized. We show that this policy achieves strictly larger throughput than a natural greedy policy. Finally, we validate our model and confirm our theoretical findings with experiments based on logs of Math.StackExchange, a StackOverflow forum dedicated to mathematics.

Generalized Linear Model (GLM) helps represent the dependent variable as a linear combination of independent variables. Simple linear regression is the traditional form of GLM. Simple linear regression works well when the dependent variable is normally distributed. The assumption of normally distributed dependent variable is often violated in real situations. For example, consider a case where dependent variable can take only positive values and has fat tail.

This book examines machine learning models including logistic regression, decision trees, and support vector machines, and applies them to common problems such as categorizing documents and classifying images. It begins with the fundamentals of machine learning, introducing you to the supervised-unsupervised spectrum, the uses of training and test data, and evaluating models. You will learn how to use generalized linear models in regression problems, as well as solve problems with text and categorical features. You will be acquainted with the use of logistic regression, regularization, and the various loss functions that are used by generalized linear models. The book will also walk you through an example project that prompts you to label the most uncertain training examples.

Even though predictive analytics has been around for quite some time, interest around this topic has increased over the last couple of years. It is no longer enough for a company to accurately record what has happened. Today, an organization's success depends on its ability to reliably predict what will happen – be it predictions about what a customer is likely to buy next, an asset that could require maintenance, or the best action to take next in a business process. Predictive analytics uses (big) data, statistical algorithms, and machine learning techniques to identify the likelihood of future outcomes based on historical data, enabling both optimization and innovation. Existing processes can be improved – for example by forecasting sales and spikes in demand and enabling the required adjustments to the production planning.

Artificial intelligence has conquered games and image recognition, but will it master investing? The short answer is yes, but how soon and how complete? Machine learning methods have had impressive recent successes. These include defeating humans at chess, Jeopardy, poker and Go, as well as providing superior image and speech recognition. Developers strive to create tools that automate decision making and that can mimic or exceed human performance for specific tasks.

A popular approach to semi-supervised learning proceeds by endowing the input data with a graph structure in order to extract geometric information and incorporate it into a Bayesian framework. We introduce new theory that gives appropriate scalings of graph parameters that provably lead to a well-defined limiting posterior as the size of the unlabeled data set grows. Furthermore, we show that these consistency results have profound algorithmic implications. When consistency holds, carefully designed graph-based Markov chain Monte Carlo algorithms are proved to have a uniform spectral gap, independent of the number of unlabeled inputs. Several numerical experiments corroborate both the statistical consistency and the algorithmic scalability established by the theory.

The Wasserstein distance received a lot of attention recently in the community of machine learning, especially for its principled way of comparing distributions. It has found numerous applications in several hard problems, such as domain adaptation, dimensionality reduction or generative models. However, its use is still limited by a heavy computational cost. Our goal is to alleviate this problem by providing an approximation mechanism that allows to break its inherent complexity. It relies on the search of an embedding where the Euclidean distance mimics the Wasserstein distance. We show that such an embedding can be found with a siamese architecture associated with a decoder network that allows to move from the embedding space back to the original input space. Once this embedding has been found, computing optimization problems in the Wasserstein space (e.g. barycenters, principal directions or even archetypes) can be conducted extremely fast. Numerical experiments supporting this idea are conducted on image datasets, and show the wide potential benefits of our method.