arXiv subjects

Richard Nock

Publications and source records attributed to Richard Nock.

At least 55 records · Page 3Linked to original sources

arXiv2018

The Bregman chord divergence

Distances are fundamental primitives whose choice significantly impacts the performances of algorithms in machine learning and signal processing. However selecting the most appropriate distance for a given task is an endeavor. Instead of testing one by one the entries of an ever-expanding dictionary of {\em ad hoc} distances, one rather prefers to consider parametric classes of distances that are exhaustively characterized by axioms derived from first principles. Bregman divergences are such a class. However fine-tuning a Bregman divergence is delicate since it requires to smoothly adjust a functional generator. In this work, we propose an extension of Bregman divergences called the Bregman chord divergences. This new class of distances does not require gradient calculations, uses two scalar parameters that can be easily tailored in applications, and generalizes asymptotically Bregman divergences.

cs.LG↗

arXiv2018

Lipschitz Networks and Distributional Robustness

Robust risk minimisation has several advantages: it has been studied with regards to improving the generalisation properties of models and robustness to adversarial perturbation. We bound the distributionally robust risk for a model class rich enough to include deep neural networks by a regularised empirical risk involving the Lipschitz constant of the model. This allows us to interpretand quantify the robustness properties of a deep neural network. As an application we show the distributionally robust risk upperbounds the adversarial training risk.

stat.ML↗

arXiv2018

D-PAGE: Diverse Paraphrase Generation

In this paper, we investigate the diversity aspect of paraphrase generation. Prior deep learning models employ either decoding methods or add random input noise for varying outputs. We propose a simple method Diverse Paraphrase Generation (D-PAGE), which extends neural machine translation (NMT) models to support the generation of diverse paraphrases with implicit rewriting patterns. Our experimental results on two real-world benchmark datasets demonstrate that our model generates at least one order of magnitude more diverse outputs than the baselines in terms of a new evaluation metric Jeffrey's Divergence. We have also conducted extensive experiments to understand various properties of our model with a focus on diversity.

cs.CL↗

arXiv2018

Private Text Classification

Confidential text corpora exist in many forms, but do not allow arbitrary sharing. We explore how to use such private corpora using privacy preserving text analytics. We construct typical text processing applications using appropriate privacy preservation techniques (including homomorphic encryption, Rademacher operators and secure computation). We set out the preliminary materials from Rademacher operators for binary classifiers, and then construct basic text processing approaches to match those binary classifiers.

cs.CL↗

arXiv2018

Boosted Density Estimation Remastered

There has recently been a steady increase in the number iterative approaches to density estimation. However, an accompanying burst of formal convergence guarantees has not followed; all results pay the price of heavy assumptions which are often unrealistic or hard to check. The Generative Adversarial Network (GAN) literature --- seemingly orthogonal to the aforementioned pursuit --- has had the side effect of a renewed interest in variational divergence minimisation (notably $f$-GAN). We show that by introducing a weak learning assumption (in the sense of the classical boosting framework) we are able to import some recent results from the GAN literature to develop an iterative boosted density estimation algorithm, including formal convergence results with rates, that does not suffer the shortcomings other approaches. We show that the density fit is an exponential family, and as part of our analysis obtain an improved variational characterisation of $f$-GAN.

cs.LG↗

arXiv2018

Entity Resolution and Federated Learning get a Federated Resolution

Consider two data providers, each maintaining records of different feature sets about common entities. They aim to learn a linear model over the whole set of features. This problem of federated learning over vertically partitioned data includes a crucial upstream issue: entity resolution, i.e. finding the correspondence between the rows of the datasets. It is well known that entity resolution, just like learning, is mistake-prone in the real world. Despite the importance of the problem, there has been no formal assessment of how errors in entity resolution impact learning. In this paper, we provide a thorough answer to this question, answering how optimal classifiers, empirical losses, margins and generalisation abilities are affected. While our answer spans a wide set of losses --- going beyond proper, convex, or classification calibrated ---, it brings simple practical arguments to upgrade entity resolution as a preprocessing step to learning. One of these suggests that entity resolution should be aimed at controlling or minimizing the number of matching errors between examples of distinct classes. In our experiments, we modify a simple token-based entity resolution algorithm so that it indeed aims at avoiding matching rows belonging to different classes, and perform experiments in the setting where entity resolution relies on noisy data, which is very relevant to real world domains. Notably, our approach covers the case where one peer \textit{does not} have classes, or a noisy record of classes. Experiments display that using the class information during entity resolution can buy significant uplift for learning at little expense from the complexity standpoint.

cs.DB↗

arXiv2017

Evolving a Vector Space with any Generating Set

In Valiant's model of evolution, a class of representations is evolvable iff a polynomial-time process of random mutations guided by selection converges with high probability to a representation as $ε$-close as desired from the optimal one, for any required $ε>0$. Several previous positive results exist that can be related to evolving a vector space, but each former result imposes disproportionate representations or restrictions on (re)initialisations, distributions, performance functions and/or the mutator. In this paper, we show that all it takes to evolve a normed vector space is merely a set that generates the space. Furthermore, it takes only $\tilde{O}(1/ε^2)$ steps and it is essentially stable, agnostic and handles target drifts that rival some proven in fairly restricted settings. Our algorithm can be viewed as a close relative to a popular fifty-years old gradient-free optimization method for which little is still known from the convergence standpoint: Nelder-Mead simplex method.

cs.LG↗

arXiv2017

Private federated learning on vertically partitioned data via entity resolution and additively homomorphic encryption

Consider two data providers, each maintaining private records of different feature sets about common entities. They aim to learn a linear model jointly in a federated setting, namely, data is local and a shared model is trained from locally computed updates. In contrast with most work on distributed learning, in this scenario (i) data is split vertically, i.e. by features, (ii) only one data provider knows the target variable and (iii) entities are not linked across the data providers. Hence, to the challenge of private learning, we add the potentially negative consequences of mistakes in entity resolution. Our contribution is twofold. First, we describe a three-party end-to-end solution in two phases ---privacy-preserving entity resolution and federated logistic regression over messages encrypted with an additively homomorphic scheme---, secure against a honest-but-curious adversary. The system allows learning without either exposing data in the clear or sharing which entities the data providers have in common. Our implementation is as accurate as a naive non-private solution that brings all data in one place, and scales to problems with millions of entities with hundreds of features. Second, we provide what is to our knowledge the first formal analysis of the impact of entity resolution's mistakes on learning, with results on how optimal classifiers, empirical losses, margins and generalisation abilities are affected. Our results bring a clear and strong support for federated learning: under reasonable assumptions on the number and magnitude of entity resolution's mistakes, it can be extremely beneficial to carry out federated learning in the setting where each peer's data provides a significant uplift to the other.

cs.LG↗

arXiv2017

f-GANs in an Information Geometric Nutshell

Nowozin \textit{et al} showed last year how to extend the GAN \textit{principle} to all $f$-divergences. The approach is elegant but falls short of a full description of the supervised game, and says little about the key player, the generator: for example, what does the generator actually converge to if solving the GAN game means convergence in some space of parameters? How does that provide hints on the generator's design and compare to the flourishing but almost exclusively experimental literature on the subject? In this paper, we unveil a broad class of distributions for which such convergence happens --- namely, deformed exponential families, a wide superset of exponential families --- and show tight connections with the three other key GAN parameters: loss, game and architecture. In particular, we show that current deep architectures are able to factorize a very large number of such densities using an especially compact design, hence displaying the power of deep architectures and their concinnity in the $f$-GAN game. This result holds given a sufficient condition on \textit{activation functions} --- which turns out to be satisfied by popular choices. The key to our results is a variational generalization of an old theorem that relates the KL divergence between regular exponential families and divergences between their natural parameters. We complete this picture with additional results and experimental insights on how these results may be used to ground further improvements of GAN architectures, via (i) a principled design of the activation functions in the generator and (ii) an explicit integration of proper composite losses' link function in the discriminator.

cs.LG↗

arXiv2017

Generalizing Jensen and Bregman divergences with comparative convexity and the statistical Bhattacharyya distances with comparable means

Comparative convexity is a generalization of convexity relying on abstract notions of means. We define the Jensen divergence and the Jensen diversity from the viewpoint of comparative convexity, and show how to obtain the generalized Bregman divergences as limit cases of skewed Jensen divergences. In particular, we report explicit formula of these generalized Bregman divergences when considering quasi-arithmetic means. Finally, we introduce a generalization of the Bhattacharyya statistical distances based on comparative means using relative convexity.

cs.IT↗

arXiv2017

Making Deep Neural Networks Robust to Label Noise: a Loss Correction Approach

We present a theoretically grounded approach to train deep neural networks, including recurrent networks, subject to class-dependent label noise. We propose two procedures for loss correction that are agnostic to both application domain and network architecture. They simply amount to at most a matrix inversion and multiplication, provided that we know the probability of each class being corrupted into another. We further show how one can estimate these probabilities, adapting a recent technique for noise estimation to the multi-class setting, and thus providing an end-to-end framework. Extensive experiments on MNIST, IMDB, CIFAR-10, CIFAR-100 and a large scale dataset of clothing images employing a diversity of architectures --- stacking dense, convolutional, pooling, dropout, batch normalization, word embedding, LSTM and residual layers --- demonstrate the noise robustness of our proposals. Incidentally, we also prove that, when ReLU is the only non-linearity, the loss curvature is immune to class-dependent label noise.

stat.ML↗

arXiv2017

The Crossover Process: Learnability and Data Protection from Inference Attacks

It is usual to consider data protection and learnability as conflicting objectives. This is not always the case: we show how to jointly control inference --- seen as the attack --- and learnability by a noise-free process that mixes training examples, the Crossover Process (cp). One key point is that the cp~is typically able to alter joint distributions without touching on marginals, nor altering the sufficient statistic for the class. In other words, it saves (and sometimes improves) generalization for supervised learning, but can alter the relationship between covariates --- and therefore fool measures of nonlinear independence and causal inference into misleading ad-hoc conclusions. For example, a cp~can increase / decrease odds ratios, bring fairness or break fairness, tamper with disparate impact, strengthen, weaken or reverse causal directions, change observed statistical measures of dependence. For each of these, we quantify changes brought by a cp, as well as its statistical impact on generalization abilities via a new complexity measure that we call the Rademacher cp~complexity. Experiments on a dozen readily available domains validate the theory.

cs.LG↗

arXiv2017

Semi-parametric Network Structure Discovery Models

We propose a network structure discovery model for continuous observations that generalizes linear causal models by incorporating a Gaussian process (GP) prior on a network-independent component, and random sparsity and weight matrices as the network-dependent parameters. This approach provides flexible modeling of network-independent trends in the observations as well as uncertainty quantification around the discovered network structure. We establish a connection between our model and multi-task GPs and develop an efficient stochastic variational inference algorithm for it. Furthermore, we formally show that our approach is numerically stable and in fact numerically easy to carry out almost everywhere on the support of the random variables involved. Finally, we evaluate our model on three applications, showing that it outperforms previous approaches. We provide a qualitative and quantitative analysis of the structures discovered for domains such as the study of the full genome regulation of the yeast Saccharomyces cerevisiae.

cs.LG↗

arXiv2016

A series of maximum entropy upper bounds of the differential entropy

We present a series of closed-form maximum entropy upper bounds for the differential entropy of a continuous univariate random variable and study the properties of that series. We then show how to use those generic bounds for upper bounding the differential entropy of Gaussian mixture models. This requires to calculate the raw moments and raw absolute moments of Gaussian mixtures in closed-form that may also be handy in statistical machine learning and information theory. We report on our experiments and discuss on the tightness of those bounds.

cs.IT↗

arXiv2016

Large Margin Nearest Neighbor Classification using Curved Mahalanobis Distances

We consider the supervised classification problem of machine learning in Cayley-Klein projective geometries: We show how to learn a curved Mahalanobis metric distance corresponding to either the hyperbolic geometry or the elliptic geometry using the Large Margin Nearest Neighbor (LMNN) framework. We report on our experimental results, and further consider the case of learning a mixed curved Mahalanobis distance. Besides, we show that the Cayley-Klein Voronoi diagrams are affine, and can be built from an equivalent (clipped) power diagrams, and that Cayley-Klein balls have Mahalanobis shapes with displaced centers.

cs.LG↗

arXiv2016

Tsallis Regularized Optimal Transport and Ecological Inference

Optimal transport is a powerful framework for computing distances between probability distributions. We unify the two main approaches to optimal transport, namely Monge-Kantorovitch and Sinkhorn-Cuturi, into what we define as Tsallis regularized optimal transport (\trot). \trot~interpolates a rich family of distortions from Wasserstein to Kullback-Leibler, encompassing as well Pearson, Neyman and Hellinger divergences, to name a few. We show that metric properties known for Sinkhorn-Cuturi generalize to \trot, and provide efficient algorithms for finding the optimal transportation plan with formal convergence proofs. We also present the first application of optimal transport to the problem of ecological inference, that is, the reconstruction of joint distributions from their marginals, a problem of large interest in the social sciences. \trot~provides a convenient framework for ecological inference by allowing to compute the joint distribution --- that is, the optimal transportation plan itself --- when side information is available, which is \textit{e.g.} typically what census represents in political science. Experiments on data from the 2012 US presidential elections display the potential of \trot~in delivering a faithful reconstruction of the joint distribution of ethnic groups and voter preferences.

cs.LG↗

arXiv2016

A scaled Bregman theorem with applications

Bregman divergences play a central role in the design and analysis of a range of machine learning algorithms. This paper explores the use of Bregman divergences to establish reductions between such algorithms and their analyses. We present a new scaled isodistortion theorem involving Bregman divergences (scaled Bregman theorem for short) which shows that certain "Bregman distortions'" (employing a potentially non-convex generator) may be exactly re-written as a scaled Bregman divergence computed over transformed data. Admissible distortions include geodesic distances on curved manifolds and projections or gauge-normalisation, while admissible data include scalars, vectors and matrices. Our theorem allows one to leverage to the wealth and convenience of Bregman divergences when analysing algorithms relying on the aforementioned Bregman distortions. We illustrate this with three novel applications of our theorem: a reduction from multi-class density ratio to class-probability estimation, a new adaptive projection free yet norm-enforcing dual norm mirror descent algorithm, and a reduction from clustering on flat manifolds to clustering on curved manifolds. Experiments on each of these domains validate the analyses and suggest that the scaled Bregman theorem might be a worthy addition to the popular handful of Bregman divergence properties that have been pervasive in machine learning.

cs.LG↗

arXiv2016

Fast $(1+ε)$-approximation of the Löwner extremal matrices of high-dimensional symmetric matrices

Matrix data sets are common nowadays like in biomedical imaging where the Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) modality produces data sets of 3D symmetric positive definite matrices anchored at voxel positions capturing the anisotropic diffusion properties of water molecules in biological tissues. The space of symmetric matrices can be partially ordered using the Löwner ordering, and computing extremal matrices dominating a given set of matrices is a basic primitive used in matrix-valued signal processing. In this letter, we design a fast and easy-to-implement iterative algorithm to approximate arbitrarily finely these extremal matrices. Finally, we discuss on extensions to matrix clustering.

cs.CG↗