We develop a variant of multiclass logistic regression that achieves three properties: i) We minimize a non-convex surrogate loss which makes the method robust to outliers, ii) our method allows transitioning between non-convex and convex losses by the choice of the parameters, iii) the surrogate loss is Bayes consistent, even in the non-convex case. The algorithm has one weight vector per class and the surrogate loss is a function of the linear activations (one per class). The surrogate loss of an example with linear activation vector $\mathbf{a}$ and class $c$ has the form $-\log_{t_1} \exp_{t_2} (a_c - G_{t_2}(\mathbf{a}))$ where the two temperatures $t_1$ and $t_2$ "temper" the $\log$ and $\exp$, respectively, and $G_{t_2}$ is a generalization of the log-partition function. We motivate this loss using the Tsallis divergence. As the temperature of the logarithm becomes smaller than the temperature of the exponential, the surrogate loss becomes "more quasi-convex". Various tunings of the temperatures recover previous methods and tuning the degree of non-convexity is crucial in the experiments. The choice $t_1<1$ and $t_2>1$ performs best experimentally. We explain this by showing that $t_1 < 1$ caps the surrogate loss and $t_2 >1$ makes the predictive distribution have a heavy tail.

Recently, (Blanchet, Kang, and Murhy 2016) showed that several machine learning algorithms, such as square-root Lasso, Support Vector Machines, and regularized logistic regression, among many others, can be represented exactly as distributionally robust optimization (DRO) problems. The distributional uncertainty is defined as a neighborhood centered at the empirical distribution. We propose a methodology which learns such neighborhood in a natural data-driven way. We show rigorously that our framework encompasses adaptive regularization as a particular case. Moreover, we demonstrate empirically that our proposed methodology is able to improve upon a wide range of popular machine learning estimators.

Automatic music transcription (AMT) aims to infer a latent symbolic representation of a piece of music (piano-roll), given a corresponding observed audio recording. Transcribing polyphonic music (when multiple notes are played simultaneously) is a challenging problem, due to highly structured overlapping between harmonics. We study whether the introduction of physically inspired Gaussian process (GP) priors into audio content analysis models improves the extraction of patterns required for AMT. Audio signals are described as a linear combination of sources. Each source is decomposed into the product of an amplitude-envelope, and a quasi-periodic component process. We introduce the Mat\'ern spectral mixture (MSM) kernel for describing frequency content of singles notes. We consider two different regression approaches. In the sigmoid model every pitch-activation is independently non-linear transformed. In the softmax model several activation GPs are jointly non-linearly transformed. This introduce cross-correlation between activations. We use variational Bayes for approximate inference. We empirically evaluate how these models work in practice transcribing polyphonic music. We demonstrate that rather than encourage dependency between activations, what is relevant for improving pitch detection is to learnt priors that fit the frequency content of the sound events to detect.

Regulators require financial institutions to estimate counterparty default risks from liquid CDS quotes for the valuation and risk management of OTC derivatives. However, the vast majority of counterparties do not have liquid CDS quotes and need proxy CDS rates. Existing methods cannot account for counterparty-specific default risks; we propose to construct proxy CDS rates by associating to illiquid counterparty liquid CDS Proxy based on Machine Learning Techniques. After testing 156 classifiers from 8 most popular classifier families, we found that some classifiers achieve highly satisfactory accuracy rates. Furthermore, we have rank-ordered the performances and investigated performance variations amongst and within the 8 classifier families. This paper is, to the best of our knowledge, the first systematic study of CDS Proxy construction by Machine Learning techniques, and the first systematic classifier comparison study based entirely on financial market data. Its findings both confirm and contrast existing classifier performance literature. Given the typically highly correlated nature of financial data, we investigated the impact of correlation on classifier performance. The techniques used in this paper should be of interest for financial institutions seeking a CDS Proxy method, and can serve for proxy construction for other financial variables. Some directions for future research are indicated.

Forward regression is a statistical model selection and estimation procedure which inductively selects covariates that add predictive power into a working statistical regression model. Once a model is selected, unknown regression parameters are estimated by least squares. This paper analyzes forward regression in high-dimensional sparse linear models. Probabilistic bounds for prediction error norm and number of selected covariates are proved. The analysis in this paper gives sharp rates and does not require beta-min or irrepresentability conditions.

The idea of cross-validation is to split the data into N subsets, to put one subset aside, to estimate parameters of the model from the remaining N-1 subsets, and to use the retained subset to estimate the error of the model. Such a process is repeated N times - with each of the N subsets being used as the validation set . Then the values of the errors obtained in such N steps are combined to provide the final estimate of the model error. The cross-validation is used in various classification and prediction procedures, such as regression analysis, discriminant analysis, neural networks and classification and regression trees (CART) . The goal is to improve the quality of the decision that is made from the outcome of the study on the basis of statistical methods, and to ensure that maximum information is obtained from scarce experimental data.

In this guide, we'll take a practical, concise tour through modern machine learning algorithms. While other such lists exist, they don't really explain the practical tradeoffs of each algorithm, which we hope to do here. We'll discuss the advantages and disadvantages of each algorithm based on our experience. Categorizing machine learning algorithms is tricky, and there are several reasonable approaches; they can be grouped into generative/discriminative, parametric/non-parametric, supervised/unsupervised, and so on. However, from our experience, this isn't always the most practical way to group algorithms.

Principal Component Analysis (PCA) is a very successful dimensionality reduction technique, widely used in predictive modeling. A key factor in its widespread use in this domain is the fact that the projection of a dataset onto its first $K$ principal components minimizes the sum of squared errors between the original data and the projected data over all possible rank $K$ projections. Thus, PCA provides optimal low-rank representations of data for least-squares linear regression under standard modeling assumptions. On the other hand, when the loss function for a prediction problem is not the least-squares error, PCA is typically a heuristic choice of dimensionality reduction -- in particular for classification problems under the zero-one loss. In this paper we target classification problems by proposing a straightforward alternative to PCA that aims to minimize the difference in margin distribution between the original and the projected data. Extensive experiments show that our simple approach typically outperforms PCA on any particular dataset, in terms of classification error, though this difference is not always statistically significant, and despite being a filter method is frequently competitive with Partial Least Squares (PLS) and Lasso on a wide range of datasets.

Guest blog by Kevin Gray.. Kevin is president of Cannon Gray, a marketing science and analytics consultancy. Regression is arguably the workhorse of statistics. Despite its popularity, however, it may also be the most misunderstood. The answer might surprise you: There is no such thing as Regression. The Dependent Variable is something you want to predict or explain.

About this course: Case Study - Predicting Housing Prices In our first case study, predicting house prices, you will create models that predict a continuous value (price) from input features (square footage, number of bedrooms and bathrooms,...). This is just one of the many places where regression can be applied. Other applications range from predicting health outcomes in medicine, stock prices in finance, and power usage in high-performance computing, to analyzing which regulators are important for gene expression. In this course, you will explore regularized linear regression models for the task of prediction and feature selection. You will be able to handle very large sets of features and select between models of various complexity.