Classical machine learning algorithms are currently used for performing complex computational tasks, such as pattern recognition or classification in large amounts of data, and constitute a crucial part of many modern technologies. The aim of quantum learning algorithms is to bring these features into scenarios where information is in a fully quantum form. The scientists, Alex Monràs at the Autonomous University of Barcelona, Spain; Gael Sentís at the University of the Basque Country, Spain, and the University of Siegen, Germany; and Peter Wittek at ICFO-The Institute of Photonic Science, Spain, and the University of Borås, Sweden, have published a paper on their results in a recent issue of Physical Review Letters. "Our work unveils the structure of a general class of quantum learning algorithms at a very fundamental level," Sentís told Phys.org. "It shows that the potentially very complex operations involved in an optimal quantum setup can be dropped in favor of a much simpler operational scheme, which is analogous to the one used in classical algorithms, and no performance is lost in the process. This finding helps in establishing the ultimate capabilities of quantum learning algorithms, and opens the door to applying key results in statistical learning to quantum scenarios."

Solving symmetric positive definite linear problems is a fundamental computational task in machine learning. The exact solution, famously, is cubicly expensive in the size of the matrix. To alleviate this problem, several linear-time approximations, such as spectral and inducing-point methods, have been suggested and are now in wide use. These are low-rank approximations that choose the low-rank space a priori and do not refine it over time. While this allows linear cost in the data-set size, it also causes a finite, uncorrected approximation error. Authors from numerical linear algebra have explored ways to iteratively refine such low-rank approximations, at a cost of a small number of matrix-vector multiplications. This idea is particularly interesting in the many situations in machine learning where one has to solve a sequence of related symmetric positive definite linear problems. From the machine learning perspective, such deflation methods can be interpreted as transfer learning of a low-rank approximation across a time-series of numerical tasks. We study the use of such methods for our field. Our empirical results show that, on regression and classification problems of intermediate size, this approach can interpolate between low computational cost and numerical precision.

The loss surface of deep neural networks has recently attracted interest in the optimization and machine learning communities as a prime example of high-dimensional non-convex problem. Some insights were recently gained using spin glass models and mean-field approximations, but at the expense of strongly simplifying the nonlinear nature of the model. In this work, we do not make any such assumption and study conditions on the data distribution and model architecture that prevent the existence of bad local minima. Our theoretical work quantifies and formalizes two important \emph{folklore} facts: (i) the landscape of deep linear networks has a radically different topology from that of deep half-rectified ones, and (ii) that the energy landscape in the non-linear case is fundamentally controlled by the interplay between the smoothness of the data distribution and model over-parametrization. Our main theoretical contribution is to prove that half-rectified single layer networks are asymptotically connected, and we provide explicit bounds that reveal the aforementioned interplay. The conditioning of gradient descent is the next challenge we address. We study this question through the geometry of the level sets, and we introduce an algorithm to efficiently estimate the regularity of such sets on large-scale networks. Our empirical results show that these level sets remain connected throughout all the learning phase, suggesting a near convex behavior, but they become exponentially more curvy as the energy level decays, in accordance to what is observed in practice with very low curvature attractors.

Quantile normalisation is a popular normalisation method for data subject to unwanted variations such as images, speech, or genomic data. It applies a monotonic transformation to the feature values of each sample to ensure that after normalisation, they follow the same target distribution for each sample. Choosing a "good" target distribution remains however largely empirical and heuristic, and is usually done independently of the subsequent analysis of normalised data. We propose instead to couple the quantile normalisation step with the subsequent analysis, and to optimise the target distribution jointly with the other parameters in the analysis. We illustrate this principle on the problem of estimating a linear model over normalised data, and show that it leads to a particular low-rank matrix regression problem that can be solved efficiently. We illustrate the potential of our method, which we term SUQUAN, on simulated data, images and genomic data, where it outperforms standard quantile normalisation.

Minimizing the empirical risk is a popular training strategy, but for learning tasks where the data may be noisy or heavy-tailed, one may require many observations in order to generalize well. To achieve better performance under less stringent requirements, we introduce a procedure which constructs a robust approximation of the risk gradient for use in an iterative learning routine. We provide high-probability bounds on the excess risk of this algorithm, by showing that it does not deviate far from the ideal gradient-based update. Empirical tests show that in diverse settings, the proposed procedure can learn more efficiently, using less resources (iterations and observations) while generalizing better.

K-Nearest Neighbours (k-NN) is a popular classification and regression algorithm, yet one of its main limitations is the difficulty in choosing the number of neighbours. We present a Bayesian algorithm to compute the posterior probability distribution for k given a target point within a data-set, efficiently and without the use of Markov Chain Monte Carlo (MCMC) methods or simulation - alongside an exact solution for distributions within the exponential family. The central idea is that data points around our target are generated by the same probability distribution, extending outwards over the appropriate, though unknown, number of neighbours. Once the data is projected onto a distance metric of choice, we can transform the choice of k into a change-point detection problem, for which there is an efficient solution: we recursively compute the probability of the last change-point as we move towards our target, and thus de facto compute the posterior probability distribution over k. Applying this approach to both a classification and a regression UCI data-sets, we compare favourably and, most importantly, by removing the need for simulation, we are able to compute the posterior probability of k exactly and rapidly. As an example, the computational time for the Ripley data-set is a few milliseconds compared to a few hours when using a MCMC approach.

Several techniques for domain adaptation have been proposed to account for differences in the distribution of the data used for training and testing. The majority of this work focuses on a binary domain label. Similar problems occur in a scientific context where there may be a continuous family of plausible data generation processes associated to the presence of systematic uncertainties. Robust inference is possible if it is based on a pivot -- a quantity whose distribution does not depend on the unknown values of the nuisance parameters that parametrize this family of data generation processes. In this work, we introduce and derive theoretical results for a training procedure based on adversarial networks for enforcing the pivotal property (or, equivalently, fairness with respect to continuous attributes) on a predictive model. The method includes a hyperparameter to control the trade-off between accuracy and robustness. We demonstrate the effectiveness of this approach with a toy example and examples from particle physics.

This article discusses a far more general version of the technique described in our article The best kept secret about regression. Here we adapt our methodology so that it applies to data sets with a more complex structure, in particular with highly correlated independent variables. Our goal is to produce a regression tool that can be used as a black box, be very robust and parameter-free, and usable and easy-to-interpret by non-statisticians. It is part of a bigger project: automating many fundamental data science tasks, to make it easy, scalable and cheap for data consumers, not just for data experts. Readers are invited to further formalize the technology outlined here, and challenge my proposed methodology.

Fairness is a desirable property of decision rules applied to a population of individuals. For example, college admissions should be decided on variables describing merit, but may also need to take into account the fact that certain communities are inherently disadvantaged. At the same time, individuals should not feel that another individual in a similar situation obtained an unfair advantage. All this must be taken into account while still caring about optimizing for a decision maker's utility function. In particular, for a given distribution over a population, we wish to derive a decision rule that takes into account a merit variable, but also ensures fairness for members of disadvantaged groups. The problem becomes even more challenging when we take into account potential uncertainties in decision making models, which can even make strict notions of fairness impossible to satisfy. As an example, consider the problem of fair prediction with disparate impact as defined by Chouldechova [2016]. Informally, their formulation defines a statistic a such that true category y (also called outcome or true label) is conditionally independent of a sensitive variable z given the statistic and the model parameters θ, i.e. y

Bayesian $l_0$-regularized least squares provides a variable selection technique for high dimensional predictors. The challenge in $l_0$ regularization is optimizing a non-convex objective function via search over model space consisting of all possible predictor combinations, a NP-hard task. Spike-and-slab (a.k.a. Bernoulli-Gaussian, BG) priors are the gold standard for Bayesian variable selection, with a caveat of computational speed and scalability. We show that a Single Best Replacement (SBR) algorithm is a fast scalable alternative. Although SBR calculates a sparse posterior mode, we show that it possesses a number of equivalences and optimality properties of a posterior mean. To illustrate our methodology, we provide simulation evidence and a real data example on the statistical properties and computational efficiency of SBR versus direct posterior sampling using spike-and-slab priors. Finally, we conclude with directions for future research.