While the standard approach to statistical learning theory is based on assumptions chosen largely for their convenience (e.g., i.i.d. or stationary ergodic), in this work we are interested in developing a theory of learning based only on the most fundamental and natural assumptions implicit in the requirements of the learning problem itself. We specifically study universally consistent function learning, where the objective is to obtain low long-run average loss for any target function, when the data follow a given stochastic process. We are then interested in the question of whether there exist learning rules guaranteed to be universally consistent given only the assumption that universally consistent learning is possible for the given data process. The reasoning that motivates this criterion emanates from a kind of optimist's decision theory, and so we refer to such learning rules as being optimistically universal. We study this question in three natural learning settings: inductive, self-adaptive, and online. Remarkably, as our strongest positive result, we find that optimistically universal learning rules do indeed exist in the self-adaptive learning setting. Establishing this fact requires us to develop new approaches to the design of learning algorithms. Along the way, we also identify concise characterizations of the family of processes under which universally consistent learning is possible in the inductive and self-adaptive settings. We additionally pose a number of enticing open problems, particularly for the online learning setting.

Large-scale classification of data where classes are structurally organized in a hierarchy is an important area of research. Top-down approaches that exploit the hierarchy during the learning and prediction phase are efficient for large scale hierarchical classification. However, accuracy of top-down approaches is poor due to error propagation i.e., prediction errors made at higher levels in the hierarchy cannot be corrected at lower levels. One of the main reason behind errors at the higher levels is the presence of inconsistent nodes that are introduced due to the arbitrary process of creating these hierarchies by domain experts. In this paper, we propose two different data-driven approaches (local and global) for hierarchical structure modification that identifies and flattens inconsistent nodes present within the hierarchy. Our extensive empirical evaluation of the proposed approaches on several image and text datasets with varying distribution of features, classes and training instances per class shows improved classification performance over competing hierarchical modification approaches. Specifically, we see an improvement upto 7% in Macro-F1 score with our approach over best TD baseline. SOURCE CODE: http://www.cs.gmu.edu/~mlbio/InconsistentNodeFlattening

Logistic regression is used thousands of times a day to fit data, predict future outcomes, and assess the statistical significance of explanatory variables. When used for the purpose of statistical inference, logistic models produce p-values for the regression coefficients by using an approximation to the distribution of the likelihood-ratio test. Indeed, Wilks' theorem asserts that whenever we have a fixed number $p$ of variables, twice the log-likelihood ratio (LLR) $2\Lambda$ is distributed as a $\chi^2_k$ variable in the limit of large sample sizes $n$; here, $k$ is the number of variables being tested. In this paper, we prove that when $p$ is not negligible compared to $n$, Wilks' theorem does not hold and that the chi-square approximation is grossly incorrect; in fact, this approximation produces p-values that are far too small (under the null hypothesis). Assume that $n$ and $p$ grow large in such a way that $p/n\rightarrow\kappa$ for some constant $\kappa < 1/2$. We prove that for a class of logistic models, the LLR converges to a rescaled chi-square, namely, $2\Lambda~\stackrel{\mathrm{d}}{\rightarrow}~\alpha(\kappa)\chi_k^2$, where the scaling factor $\alpha(\kappa)$ is greater than one as soon as the dimensionality ratio $\kappa$ is positive. Hence, the LLR is larger than classically assumed. For instance, when $\kappa=0.3$, $\alpha(\kappa)\approx1.5$. In general, we show how to compute the scaling factor by solving a nonlinear system of two equations with two unknowns. Our mathematical arguments are involved and use techniques from approximate message passing theory, non-asymptotic random matrix theory and convex geometry. We also complement our mathematical study by showing that the new limiting distribution is accurate for finite sample sizes. Finally, all the results from this paper extend to some other regression models such as the probit regression model.

The Expectation-Maximization (EM) algorithm is a widely used method for maximum likelihood estimation in models with latent variables. For estimating mixtures of Gaussians, its iteration can be viewed as a soft version of the k-means clustering algorithm. Despite its wide use and applications, there are essentially no known convergence guarantees for this method. We provide global convergence guarantees for mixtures of two Gaussians with known covariance matrices. We show that the population version of EM, where the algorithm is given access to infinitely many samples from the mixture, converges geometrically to the correct mean vectors, and provide simple, closed-form expressions for the convergence rate. As a simple illustration, we show that, in one dimension, ten steps of the EM algorithm initialized at infinity result in less than 1\% error estimation of the means. In the finite sample regime, we show that, under a random initialization, $\tilde{O}(d/\epsilon^2)$ samples suffice to compute the unknown vectors to within $\epsilon$ in Mahalanobis distance, where $d$ is the dimension. In particular, the error rate of the EM based estimator is $\tilde{O}\left(\sqrt{d \over n}\right)$ where $n$ is the number of samples, which is optimal up to logarithmic factors.

Multi-task learning (MTL) is a supervised learning paradigm in which the prediction models for several related tasks are learned jointly to achieve better generalization performance. When there are only a few training examples per task, MTL considerably outperforms the traditional Single task learning (STL) in terms of prediction accuracy. In this work we develop an MTL based approach for classifying documents that are archived within dual concept hierarchies, namely, DMOZ and Wikipedia. We solve the multi-class classification problem by defining one-versus-rest binary classification tasks for each of the different classes across the two hierarchical datasets. Instead of learning a linear discriminant for each of the different tasks independently, we use a MTL approach with relationships between the different tasks across the datasets established using the non-parametric, lazy, nearest neighbor approach. We also develop and evaluate a transfer learning (TL) approach and compare the MTL (and TL) methods against the standard single task learning and semi-supervised learning approaches. Our empirical results demonstrate the strength of our developed methods that show an improvement especially when there are fewer number of training examples per classification task.

Large-scale Hierarchical Classification (HC) involves datasets consisting of thousands of classes and millions of training instances with high-dimensional features posing several big data challenges. Feature selection that aims to select the subset of discriminant features is an effective strategy to deal with large-scale HC problem. It speeds up the training process, reduces the prediction time and minimizes the memory requirements by compressing the total size of learned model weight vectors. Majority of the studies have also shown feature selection to be competent and successful in improving the classification accuracy by removing irrelevant features. In this work, we investigate various filter-based feature selection methods for dimensionality reduction to solve the large-scale HC problem. Our experimental evaluation on text and image datasets with varying distribution of features, classes and instances shows upto 3x order of speed-up on massive datasets and upto 45% less memory requirements for storing the weight vectors of learned model without any significant loss (improvement for some datasets) in the classification accuracy. Source Code: https://cs.gmu.edu/~mlbio/featureselection.

As with learning any new skills, the more you practice, the better you become. Practice different algorithms and work with different data sets to have a better understanding of machine learning, and to improve your overall problem-solving skills. Machine learning with Python is a great addition to your technical skillset, and there are lots of free and low-cost online resources available to help. How have you picked up machine learning skills? Leave a comment below, or submit an article proposal to share your story.

In today's blog post, we shall look into time series analysis using R package – forecast. Objective of the post will be explaining the different methods available in forecast package which can be applied while dealing with time series analysis/forecasting. A time series is a collection of observations of well-defined data items obtained through repeated measurements over time. For example, measuring the value of retail sales each month of the year would comprise a time series. My data set contains data of Sales of CARS from Jan-2008 to Dec 2013.

Big data bring new opportunities for methods that efficiently summarize and automatically extract knowledge from such compendia. While both supervised learning algorithms and unsupervised clustering algorithms have been successfully applied to biological data, they are either dependent on known biology or limited to discerning the most significant signals in the data. Here we present denoising autoencoders (DAs), which employ a data-defined learning objective independent of known biology, as a method to identify and extract complex patterns from genomic data. We evaluate the performance of DAs by applying them to a large collection of breast cancer gene expression data. Results show that DAs successfully construct features that contain both clinical and molecular information.

We consider the problem of learning high-dimensional Gaussian graphical models. The graphical lasso is one of the most popular methods for estimating Gaussian graphical models. However, it does not achieve the oracle rate of convergence. In this paper, we propose the graphical nonconvex optimization for optimal estimation in Gaussian graphical models, which is then approximated by a sequence of convex programs. Our proposal is computationally tractable and produces an estimator that achieves the oracle rate of convergence. The statistical error introduced by the sequential approximation using the convex programs are clearly demonstrated via a contraction property. The rate of convergence can be further improved using the notion of sparsity pattern. The proposed methodology is then extended to semiparametric graphical models. We show through numerical studies that the proposed estimator outperforms other popular methods for estimating Gaussian graphical models.