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The original article is no longer available. Similar (and more comprehensive) material is available below. You can access this material here.

Empirical analysis is often the first step towards the birth of a conjecture. This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing the rational points on an elliptic curve, one of the most celebrated unsolved problems in mathematics. Here we extend the original empirical approach, to the analysis of the Cremona database of quantities relevant to BSD, inspecting more than 2.5 million elliptic curves by means of the latest techniques in data science, machine-learning and topological data analysis. Key quantities such as rank, Weierstrass coefficients, period, conductor, Tamagawa number, regulator and order of the Tate-Shafarevich group give rise to a high-dimensional point-cloud whose statistical properties we investigate. We reveal patterns and distributions in the rank versus Weierstrass coefficients, as well as the Beta distribution of the BSD ratio of the quantities. Via gradient boosted trees, machine learning is applied in finding inter-correlation amongst the various quantities. We anticipate that our approach will spark further research on the statistical properties of large datasets in Number Theory and more in general in pure Mathematics.

Despite their tremendous success in a wide range of applications, deep neural network (DNN) models are inherently vulnerable to two types of malicious manipulations: adversarial inputs, which are crafted samples that deceive target DNNs, and backdoored models, which are forged DNNs that misbehave on trigger-embedded inputs. While prior work has intensively studied the two attack vectors in parallel, there is still a lack of understanding about their fundamental connection, which is critical for assessing the holistic vulnerability of DNNs deployed in realistic settings. In this paper, we bridge this gap by conducting the first systematic study of the two attack vectors within a unified framework. More specifically, (i) we develop a new attack model that integrates both adversarial inputs and backdoored models; (ii) with both analytical and empirical evidence, we reveal that there exists an intricate "mutual reinforcement" effect between the two attack vectors; (iii) we demonstrate that this effect enables a large spectrum for the adversary to optimize the attack strategies, such as maximizing attack evasiveness with respect to various defenses and designing trigger patterns satisfying multiple desiderata; (v) finally, we discuss potential countermeasures against this unified attack and their technical challenges, which lead to several promising research directions.

Modern machine learning models are often so complex that they achieve vanishing classification error on the training set. Max-margin linear classifiers are among the simplest classification methods that have zero training error (with linearly separable data). Despite this simplicity, their high-dimensional behavior is not yet completely understood. We assume to be given i.i.d. data $(y_i,{\boldsymbol x}_i)$, $i\le n$ with ${\boldsymbol x}_i\sim {\sf N}({\boldsymbol 0},{\boldsymbol \Sigma})$ a $p$-dimensional Gaussian feature vector, and $y_i \in\{+1,-1\}$ a label whose distribution depends on a linear combination of the covariates $\langle {\boldsymbol \theta}_*,{\boldsymbol x}_i\rangle$. We consider the proportional asymptotics $n,p\to\infty$ with $p/n\to \psi$, and derive exact expressions for the limiting prediction error. Our asymptotic results match simulations already when $n,p$ are of the order of a few hundreds. We explore several choices for the the pair $({\boldsymbol \theta}_*,{\boldsymbol \Sigma})$, and show that the resulting generalization curve (test error error as a function of the overparametrization ratio $\psi=p/n$) is qualitatively different, depending on this choice. In particular we consider a specific structure of $({\boldsymbol \theta}_*,{\boldsymbol \Sigma})$ that captures the behavior of nonlinear random feature models or, equivalently, two-layers neural networks with random first layer weights. In this case, we observe that the test error is monotone decreasing in the number of parameters. This finding agrees with the recently developed `double descent' phenomenology for overparametrized models.

In variational inference, the complicated target is approximated by a closest member relative to the Kullback-Leibler (KL) divergence in a pre-specified family of tractable densities. In many large-scale machine learning applications including clustering problems [11, 32], image classification [25, 27] and topic models [21, 7], variational inference can be orders of magnitude faster than the traditional sampling based approaches such as Markov Chain Monte Carlo (MCMC). In particular, by turning the integration, or sampling, problem into an optimization problem, variational inference can take advantage of modern optimization tools such as stochastic optimization techniques [20, 17] and distributed optimization architecture [1, 8] for further improving its efficiency. Among various approximating schemes, mean-field approximation is the most common type of variational inference that is conceptually simple, implementation-wise easy and particularly suitable for problems involving large numbers of latent variables. The word "mean-field" is originated from the mean-field theory in physics where despite complex interactions among many particles in a many (infinite) body system, all interactions to any one particle can be approximated by a single averaged effect from a "mean-field". In variational inference, by restricting the approximating family of the mean-field to be all density functions that are fully factorized over (blocks of) unknown variables, the associated optimization problem of finding a closest weih2@illinois.edu

This paper considers the problem of Phase Identification in power distribution systems. In particular, it focuses on improving supervised learning accuracies by focusing on exploiting some of the problem's information theoretic properties. This focus, along with recent advances in Information Theoretic Machine Learning (ITML), helps us to create two new techniques. The first transforms a bound on information losses into a data selection technique. This is important because phase identification data labels are difficult to obtain in practice. The second interprets the properties of distribution systems in the terms of ITML. This allows us to obtain an improvement in the representation learned by any classifier applied to the problem. We tested these two techniques experimentally on real datasets and have found that they yield phenomenal performance in every case. In the most extreme case, they improve phase identification accuracy from $51.7\%$ to $97.3\%$. Furthermore, since many problems share the physical properties of phase identification exploited in this paper, the techniques can be applied to a wide range of similar problems.

Statistical inference of true model parameters based on stochastic gradient descent (SGD) has started receiving attention in recent years. In this paper, we study a simple algorithm to construct asymptotically valid confidence regions for model parameters using the batch means method. The main idea is to cancel out the covariance matrix which is hard/costly to estimate. In the process of developing the algorithm, we establish process-level function central limit theorem for Polyak-Ruppert averaging based SGD estimators. We also extend the batch means method to accommodate more general batch size specifications.

We develop amortized population Gibbs (APG) samplers, a new class of autoencoding variational methods for deep probabilistic models. APG samplers construct high-dimensional proposals by iterating over updates to lower-dimensional blocks of variables. Each conditional update is a neural proposal, which we train by minimizing the inclusive KL divergence relative to the conditional posterior. To appropriately account for the size of the input data, we develop a new parameterization in terms of neural sufficient statistics, resulting in quasi-conjugate variational approximations. Experiments demonstrate that learned proposals converge to the known analytical conditional posterior in conjugate models, and that APG samplers can learn inference networks for highly-structured deep generative models when the conditional posteriors are intractable. Here APG samplers offer a path toward scaling up stochastic variational methods to models in which standard autoencoding architectures fail to produce accurate samples.

We introduce a gradient-based learning method to automatically adapt Markov chain Monte Carlo (MCMC) proposal distributions to intractable targets. We define a maximum entropy regularised objective function, referred to as generalised speed measure, which can be robustly optimised over the parameters of the proposal distribution by applying stochastic gradient optimisation. An advantage of our method compared to traditional adaptive MCMC methods is that the adaptation occurs even when candidate state values are rejected. This is a highly desirable property of any adaptation strategy because the adaptation starts in early iterations even if the initial proposal distribution is far from optimum. We apply the framework for learning multivariate random walk Metropolis and Metropolis-adjusted Langevin proposals with full covariance matrices, and provide empirical evidence that our method can outperform other MCMC algorithms, including Hamiltonian Monte Carlo schemes.