In this paper, the online variants of the classical Frank-Wolfe algorithm are considered. We consider minimizing the regret with a stochastic cost. The online algorithms only require simple iterative updates and a non-adaptive step size rule, in contrast to the hybrid schemes commonly considered in the literature. Several new results are derived for convex and non-convex losses. With a strongly convex stochastic cost and when the optimal solution lies in the interior of the constraint set or the constraint set is a polytope, the regret bound and anytime optimality are shown to be ${\cal O}( \log^3 T / T )$ and ${\cal O}( \log^2 T / T)$, respectively, where $T$ is the number of rounds played. These results are based on an improved analysis on the stochastic Frank-Wolfe algorithms. Moreover, the online algorithms are shown to converge even when the loss is non-convex, i.e., the algorithms find a stationary point to the time-varying/stochastic loss at a rate of ${\cal O}(\sqrt{1/T})$. Numerical experiments on realistic data sets are presented to support our theoretical claims.

This paper considers \emph{volume minimization} (VolMin)-based structured matrix factorization (SMF). VolMin is a factorization criterion that decomposes a given data matrix into a basis matrix times a structured coefficient matrix via finding the minimum-volume simplex that encloses all the columns of the data matrix. Recent work showed that VolMin guarantees the identifiability of the factor matrices under mild conditions that are realistic in a wide variety of applications. This paper focuses on both theoretical and practical aspects of VolMin. On the theory side, exact equivalence of two independently developed sufficient conditions for VolMin identifiability is proven here, thereby providing a more comprehensive understanding of this aspect of VolMin. On the algorithm side, computational complexity and sensitivity to outliers are two key challenges associated with real-world applications of VolMin. These are addressed here via a new VolMin algorithm that handles volume regularization in a computationally simple way, and automatically detects and {iteratively downweights} outliers, simultaneously. Simulations and real-data experiments using a remotely sensed hyperspectral image and the Reuters document corpus are employed to showcase the effectiveness of the proposed algorithm.

In this post I refer to lyrics of certain bands as being "Metal". I know some people have strong feelings about how genres are defined, and would probably disagree with me about some of the bands I call metal in this post. I call these band "Metal" here for the sake of brevity only, and I apologise in advance. It is all around us, and the rate at which it is produced in written, stored form is only increasing. It is also quite unlike any sort of data I have worked with before. Natural language is made up of sequences of discrete characters arranged into hierarchical groupings: words, sentences and documents, each with both syntactic structure and semantic meaning. Not only is the space of possible strings huge, but the interpretation of a small sections of a document can take on vastly different meanings depending on what context surround it.

Unsupervised nearest neighbors is the foundation of many other learning methods, notably manifold learning and spectral clustering. Supervised neighbors-based learning comes in two flavors: classification for data with discrete labels, and regression for data with continuous labels. The principle behind nearest neighbor methods is to find a predefined number of training samples closest in distance to the new point, and predict the label from these. The number of samples can be a user-defined constant (k-nearest neighbor learning), or vary based on the local density of points (radius-based neighbor learning). The distance can, in general, be any metric measure: standard Euclidean distance is the most common choice. Neighbors-based methods are known as non-generalizing machine learning methods, since they simply "remember" all of its training data (possibly transformed into a fast indexing structure such as a Ball Tree or KD Tree.). Despite its simplicity, nearest neighbors has been successful in a large number of classification and regression problems, including handwritten digits or satellite image scenes.

Machine learning is a fascinating study, but how do you actually use it on your own problems? You may be confused as to how best prepare your data for machine learning, which algorithms to use or how to choose one model over another. In this post you will discover a 14-part crash course into applied machine learning using the Weka platform without a single mathematical equation or line of programming code. Applied Machine Learning With Weka Mini-Course Photo by Leon Yaakov, some rights reserved. Before we get started, let's make sure you are in the right place.

In the previous article (Part I), we introduced the general ideas behind model evaluation in supervised machine learning. We discussed the holdout method, which helps us to deal with real world limitations such as limited access to new, labeled data for model evaluation. Using the holdout method, we split our dataset into two parts: A training and a test set. First, we provide the training data to a supervised learning algorithm. The learning algorithm builds a model from the training set of labeled observations.

Least squares regression can cause impossible estimates such as probabilities that are less than zero and greater than 1.So, when the predicted value is measured as a probability, use Logistic Regression We use the log of the odds rather than the odds directly because an odds ratio cannot be a negative number--but its log can be negative. Notice that we have randomly initialized our coefficients for income and other predictors. These will be adjusted by Solver based on a likelihood function.We will cover them later Column H tells us the predicted probability of the borrower's actual behavior, whether that behavior is repayment or default--not simply, as in Column G, the predicted probability of defaulting on the loan. One property of logarithms is that their sum equals the logarithm of the product of the numbers on which they're based The logarithms of probabilities are always negative numbers, but the closer a probability is to 1.0, the closer its logarithm is to 0.0. I haven't covered cross-validation, which is commonly used to validate a logistic regression equation.If you don't always have a large number of cases to work with, a different approach is to use statistical inference.

This post is part of a series covering the exercises from Andrew Ng's machine learning class on Coursera. The original code, exercise text, and data files for this post are available here. One of the pivotal moments in my professional development this year came when I discovered Coursera. I'd heard of the "MOOC" phenomenon but had not had the time to dive in and take a class. Earlier this year I finally pulled the trigger and signed up for Andrew Ng's Machine Learning class.

This repo contains various IPython notebooks I've created to experiment with libraries and work through exercises, and explore subjects that I find interesting. I've included notebook viewer links below. Click the link to see a live rendering of the notebook. These notebooks contain introductory content such as an overview of the language and a review of IPython's functionality. Implementations of the exercises presented in Andrew Ng's "Machine Learning" class on Coursera.

Following the seminal idea of Tukey, data depth is a function that measures how close an arbitrary point of the space is located to an implicitly defined center of a data cloud. Having undergone theoretical and computational developments, it is now employed in numerous applications with classification being the most popular one. The R-package ddalpha is a software directed to fuse experience of the applicant with recent achievements in the area of data depth and depth-based classification. ddalpha provides an implementation for exact and approximate computation of most reasonable and widely applied notions of data depth. These can be further used in the depth-based multivariate and functional classifiers implemented in the package, where the $DD\alpha$-procedure is in the main focus. The package is expandable with user-defined custom depth methods and separators. The implemented functions for depth visualization and the built-in benchmark procedures may also serve to provide insights into the geometry of the data and the quality of pattern recognition.