Despite being the standard loss function to train multi-class neural networks, the log-softmax has two potential limitations. First, it involves computations that scale linearly with the number of output classes, which can restrict the size of problems we are able to tackle with current hardware. Second, it remains unclear how close it matches the task loss such as the top-k error rate or other non-differentiable evaluation metrics which we aim to optimize ultimately. In this paper, we introduce an alternative classification loss function, the Z-loss, which is designed to address these two issues. Unlike the log-softmax, it has the desirable property of belonging to the spherical loss family (Vincent et al., 2015), a class of loss functions for which training can be performed very efficiently with a complexity independent of the number of output classes. We show experimentally that it significantly outperforms the other spherical loss functions previously investigated. Furthermore, we show on a word language modeling task that it also outperforms the log-softmax with respect to certain ranking scores, such as top-k scores, suggesting that the Z-loss has the flexibility to better match the task loss. These qualities thus makes the Z-loss an appealing candidate to train very efficiently large output networks such as word-language models or other extreme classification problems. On the One Billion Word (Chelba et al., 2014) dataset, we are able to train a model with the Z-loss 40 times faster than the log-softmax and more than 4 times faster than the hierarchical softmax.

This paper considers the problem of canonical-correlation analysis (CCA) (Hotelling, 1936) and, more broadly, the generalized eigenvector problem for a pair of symmetric matrices. These are two fundamental problems in data analysis and scientific computing with numerous applications in machine learning and statistics (Shi and Malik, 2000; Hardoon et al., 2004; Witten et al., 2009). We provide simple iterative algorithms, with improved runtimes, for solving these problems that are globally linearly convergent with moderate dependencies on the condition numbers and eigenvalue gaps of the matrices involved. We obtain our results by reducing CCA to the top-$k$ generalized eigenvector problem. We solve this problem through a general framework that simply requires black box access to an approximate linear system solver. Instantiating this framework with accelerated gradient descent we obtain a running time of $O(\frac{z k \sqrt{\kappa}}{\rho} \log(1/\epsilon) \log \left(k\kappa/\rho\right))$ where $z$ is the total number of nonzero entries, $\kappa$ is the condition number and $\rho$ is the relative eigenvalue gap of the appropriate matrices. Our algorithm is linear in the input size and the number of components $k$ up to a $\log(k)$ factor. This is essential for handling large-scale matrices that appear in practice. To the best of our knowledge this is the first such algorithm with global linear convergence. We hope that our results prompt further research and ultimately improve the practical running time for performing these important data analysis procedures on large data sets.

High dimensional problems where the regressor belongs to a small number of groups play a critical role in many machine learning and signal processing applications, such as computational biology [9] and multitask learning [14]. In most of these cases, the groups overlap, i.e., the same feature can belong to multiple groups. For example, gene pathways overlap in computational biology applications, and parent-child pairs of wavelet transform coefficients overlap in signal processing applications. The existing state-of-the-art methods for solving such group sparsity structured regression problems can be categorized into two broad classes: a) convex relaxation based methods, b) iterative hard thresholding (IHT) or greedy methods. In practice, IHT methods tend to be significantly more scalable than the (group-)lasso style methods that solve a convex program. But, these methods require a certain projection operator which in general is NPhard to compute and often certain simple heuristics are used with relatively weak theoretical guarantees. Moreover, existing guarantees for both classes of methods require relatively restrictive assumptions on the data, like Restricted Isometry Property or variants thereof [2, 8, 18, 14], that are unlikely to hold in most common applications. In fact, even under such settings, the group sparsity based convex programs offer at most polylogarithmic gains over standard sparsity based methods [18].

Many real-world regression problems demand a measure of the uncertainty associated with each prediction. Standard decision forests deliver efficient state-of-the-art predictive performance, but high-quality uncertainty estimates are lacking. Gaussian processes (GPs) deliver uncertainty estimates, but scaling GPs to large-scale data sets comes at the cost of approximating the uncertainty estimates. We extend Mondrian forests, first proposed by Lakshminarayanan et al. (2014) for classification problems, to the large-scale non-parametric regression setting. Using a novel hierarchical Gaussian prior that dovetails with the Mondrian forest framework, we obtain principled uncertainty estimates, while still retaining the computational advantages of decision forests. Through a combination of illustrative examples, real-world large-scale datasets, and Bayesian optimization benchmarks, we demonstrate that Mondrian forests outperform approximate GPs on large-scale regression tasks and deliver better-calibrated uncertainty assessments than decision-forest-based methods.

We study the model-based sparse clustering problem for discrete data using a mixture model of product distributions [9, 7]. This model has application in many fields, including computational neurosciences, crowdsourcing and bioinformatics, and is interesting because it differs technically from the problem for continuous data, where the well-known Gaussian mixture model has been applied successfully. A fundamental difficulty is that, in high-dimensional datasets, some features can be noisy, redundant or generally uninformative for clustering, and these can push clustering algorithms toward inappropriate or uninteresting results. If these uninformative or noise data points could be eliminated then, we argue, the results should be much more satisfying. This is precisely our goal: to find an informative set of data points and to use these to drive the clustering.

Markov chain Monte Carlo algorithms are a popular and well-studied methodology that can be used to draw samples from posterior distributions. Over the past few years these algorithms have been extended to tackle problems where the model likelihood is intractable (Beaumont, 2003). Andrieu and Roberts (2009) showed that within the Metropolis-Hastings algorithm, if the likelihood is replaced with an unbiased estimate, then the sampler still targets the correct stationary distribution. Andrieu et al. (2010) extended this work further to create a class of 1 Markov chain algorithms that use sequential Monte Carlo methods, also known as particle filters. Current implementations of pseudo-marginal and particle Markov chain Monte Carlo use random-walk proposals to update the parameters (e.g., Golightly and Wilkinson, 2011; Knape and de Valpine, 2012) and shall be referred to herein as particle random-walk Metropolis algorithms. Random walk-based algorithms propose a new value from some symmetric density centred on the current value.

Classification is one of fundamental problems in the fields of machine learning and signal processing. The commonly used classifier assigns a sample or a signal to the class with maximum posterior probability, which usually requires probability density function (PDF) estimation in an either model-driven or data-driven manner [1] [2] [3]. For high-dimensional data sets, it is necessary to perform feature reduction to estimate the PDFs robustly in a lowdimensional feature subspace. However, feature reduction may lose pertinent information for discrimination. For example, data samples from different classes that could be well separated in the raw data space may be overlapped in the feature subspace, causing classification errors. The PDF reconstruction approach provides a solution to address this information loss issue in feature reduction by reconstructing the PDF on raw data and making classification in raw data space, which could improve classification performance. Several approaches have been developed along this track.

The Lasso is a computationally efficient regression regularization procedure that can produce sparse estimators when the number of predictors (p) is large. Oracle inequalities provide probability loss bounds for the Lasso estimator at a deterministic choice of the regularization parameter. These bounds tend to zero if p is appropriately controlled, and are thus commonly cited as theoretical justification for the Lasso and its ability to handle high-dimensional settings. Unfortunately, in practice the regularization parameter is not selected to be a deterministic quantity, but is instead chosen using a random, data-dependent procedure. To address this shortcoming of previous theoretical work, we study the loss of the Lasso estimator when tuned optimally for prediction. Assuming orthonormal predictors and a sparse true model, we prove that the probability that the best possible predictive performance of the Lasso deteriorates as p increases is positive and can be arbitrarily close to one given a sufficiently high signal to noise ratio and sufficiently large p. We further demonstrate empirically that the amount of deterioration in performance can be far worse than the oracle inequalities suggest and provide a real data example where deterioration is observed.

Spot-checking is a way of discovering which algorithms perform well on your machine learning problem. You cannot know which algorithms are best suited to your problem before hand. You must trial a number of methods and focus attention on those that prove themselves the most promising. In this post you will discover 6 machine learning algorithms that you can use when spot checking your classification problem in Python with scikit-learn. You cannot know which algorithm will work best on your dataset before hand.

This is an exciting opportunity for a researcher at post-doctoral level with experience of machine learning and data mining. You will work with senior data scientists based within the local NHS trusts, the University of Manchester Health eResearch Centre, and Health Innovation Manchester to automate data extraction of predetermined features for all patients diagnosed with ovarian and colorectal cancer in the conurbation. Machine learning tools including neural networks, support vector machines and naïve Bayes algorithms will be refined and tested using the datasets accrued and optimised for clinical practice. Accuracy of prediction will be assessed using predefined criteria. Knowledge of cancer treatment would be useful but is not essential, as the team has extensive expertise in this area.