Individual fairness is an intuitive definition of algorithmic fairness that addresses some of the drawbacks of group fairness. Despite its benefits, it depends on a task specific fair metric that encodes our intuition of what is fair and unfair for the ML task at hand, and the lack of a widely accepted fair metric for many ML tasks is the main barrier to broader adoption of individual fairness. In this paper, we present two simple ways to learn fair metrics from a variety of data types. We show empirically that fair training with the learned metrics leads to improved fairness on three machine learning tasks susceptible to gender and racial biases. We also provide theoretical guarantees on the statistical performance of both approaches.

One of the major breakthroughs in deep learning over the past five years has been the Generative Adversarial Network (GAN), a neural network-based generative model which aims to mimic some underlying distribution given a dataset of samples. In contrast to many supervised problems, where one tries to minimize a simple objective function of the parameters, GAN training is formulated as a min-max problem over a pair of network parameters. While empirically GANs have shown impressive success in several domains, researchers have been puzzled by unusual training behavior, including cycling so-called mode collapse. In this paper, we begin by providing a quantitative method to explore some of the challenges in GAN training, and we show empirically how this relates fundamentally to the parametric nature of the discriminator network. We propose a novel approach that resolves many of these issues by relying on a kernel-based non-parametric discriminator that is highly amenable to online training---we call this the Online Kernel-based Generative Adversarial Networks (OKGAN). We show empirically that OKGANs mitigate a number of training issues, including mode collapse and cycling, and are much more amenable to theoretical guarantees. OKGANs empirically perform dramatically better, with respect to reverse KL-divergence, than other GAN formulations on synthetic data; on classical vision datasets such as MNIST, SVHN, and CelebA, show comparable performance.

We consider the problem of designing minimax estimators for estimating the parameters of a probability distribution. Unlike classical approaches such as the MLE and minimum distance estimators, we consider an algorithmic approach for constructing such estimators. We view the problem of designing minimax estimators as finding a mixed strategy Nash equilibrium of a zero-sum game. By leveraging recent results in online learning with non-convex losses, we provide a general algorithm for finding a mixed-strategy Nash equilibrium of general non-convex non-concave zero-sum games. Our algorithm requires access to two subroutines: (a) one which outputs a Bayes estimator corresponding to a given prior probability distribution, and (b) one which computes the worst-case risk of any given estimator. Given access to these two subroutines, we show that our algorithm outputs both a minimax estimator and a least favorable prior. To demonstrate the power of this approach, we use it to construct provably minimax estimators for classical problems such as estimation in the finite Gaussian sequence model, and linear regression.

Instrumental variable methods provide a powerful approach to estimating causal effects in the presence of unobserved confounding. But a key challenge when applying them is the reliance on untestable "exclusion" assumptions that rule out any relationship between the instrument variable and the response that is not mediated by the treatment. In this paper, we show how to perform consistent IV estimation despite violations of the exclusion assumption. In particular, we show that when one has multiple candidate instruments, only a majority of these candidates---or, more generally, the modal candidate-response relationship---needs to be valid to estimate the causal effect. Our approach uses an estimate of the modal prediction from an ensemble of instrumental variable estimators. The technique is simple to apply and is "black-box" in the sense that it may be used with any instrumental variable estimator as long as the treatment effect is identified for each valid instrument independently. As such, it is compatible with recent machine-learning based estimators that allow for the estimation of conditional average treatment effects (CATE) on complex, high dimensional data. Experimentally, we achieve accurate estimates of conditional average treatment effects using an ensemble of deep network-based estimators, including on a challenging simulated Mendelian Randomization problem.

We propose a new embedding method, named Quantile-Quantile Embedding (QQE), for distribution transformation, manifold embedding, and image embedding with the ability to choose the embedding distribution. QQE, which uses the concept of quantile-quantile plot from visual statistical tests, can transform the distribution of data to any theoretical desired distribution or empirical reference sample. Moreover, QQE gives the user a choice of embedding distribution in embedding manifold of data into the low dimensional embedding space. It can also be used for modifying the embedding distribution of different dimensionality reduction methods, either basic or deep ones, for better representation or visualization of data. We propose QQE in both unsupervised and supervised manners. QQE can also transform the distribution to either the exact reference distribution or shape of the reference distribution; and one of its many applications is better discrimination of classes. Our experiments on different synthetic and image datasets show the effectiveness of the proposed embedding method.

Estimating parameters of mixture model has wide applications Determining the number of components in a finite mixture model ranging from classification problems to estimating of complex distributions. is crucial in many application areas such as financial data [16, 31, 35], Most of the current literature on estimating the parameters biomedical studies [17, 36] and low-frequency accident occurrence of the mixture densities are based on iterative Expectation Maximization prediction [27, 32]. Existing literature have witnessed numerous (EM) type algorithms which require the use of either computational methods, and in particular Markov Chain Monte taking expectations over the latent label variables or generating Carlo methods [7, 14, 33] and EM algorithms [20-22] have been samples from the conditional distribution of such latent labels using used with a lot of success. However, either these methods are computationally the Bayes rule. Moreover, when the number of components is demanding and/or these methods are developed under unknown, the problem becomes computationally more demanding the assumption of data being generated from mixtures of densities due to well-known label switching issues [28]. In this paper, we from the exponential family, in part because the family of exponential propose a robust and quick approach based on change-point methods distribution has a sufficient statistic of constant dimension (i.e., to determine the number of mixture components that works the dimension of the sufficient statistic remains fixed for any sample for almost any location-scale families even when the components size) and so the updates of the data augmentation type algorithm are heavy tailed (e.g., Cauchy). We present several numerical illustrations involve their smaller dimensional sufficient statistics [11, 12, 24].

Time-series analyses has become a key instrument for the evaluation of continuously collected data in several domains such as Medicine, Physics and Statistics [Firmino et al., 2014, Box and Jenkins, 2015]. Such analysis generally involves the creation of a model (a regression function or a classifier, for instance) that usually leads to inconsistent results when built over raw data, specially if it contains chaotic behavior [Brock et al., 1992]. In order to reach more reliable results, an alternative is to study time-series trajectories in the phase space, as proposed by the area of Dynamical Systems [Ott, 2002, Alligood et al., 1996]. Besides leading to more robust models, the phase space also allows the inference of other important measures, such as the correlation dimension [Grassberger and Procaccia, 1983, Mandelbrot, 1977, Theiler, 1990, Clark, 1990, Ding et al., 1993] and the Lyapunov exponent [Sano and Sawada, 1985, Kantz and Schreiber, 2004], which support further analyses in modeling. In this context, Takens' embedding theorem [Takens, 1981] is one of the most used methods in the literature to reconstruct phase spaces from time series [Ravindra and Hagedorn, 1998]. Such method relies on two parameters known as embedding dimension m and time delay τ (see Figure 1) that, although Takens proved an arbitrary τ can be used given m is sufficiently large, the minimum-but-sufficient (from now on denoted as optimal) set of embedding parameters is desirable either to optimize phase-space computations as to better understand the analyzed phenomenon. In this context, several methods based on entropy [Han et al., 2012], fractal dimensions [Theiler, 1990] and/or nearest neighbors [Kennel et al., 1992] were proposed to guide the estimation of optimal embeddings.

Anomalies are by definition rare, thus labeled examples are very limited or nonexistent, and likely do not cover unforeseen scenarios. Unsupervised learning methods that don't necessarily encounter anomalies in training would be immensely useful. Generative vision models can be useful in this regard but do not sufficiently represent normal and abnormal data distributions. To this end, we propose constant curvature manifolds for embedding data distributions in unsupervised visual anomaly detection. Through theoretical and empirical explorations of manifold shapes, we develop a novel hyperspherical Variational Auto-Encoder (VAE) via stereographic projections with a gyroplane layer - a complete equivalent to the Poincar\'e VAE. This approach with manifold projections is beneficial in terms of model generalization and can yield more interpretable representations. We present state-of-the-art results on visual anomaly benchmarks in precision manufacturing and inspection, demonstrating real-world utility in industrial AI scenarios. We further demonstrate the approach on the challenging problem of histopathology: our unsupervised approach effectively detects cancerous brain tissue from noisy whole-slide images, learning a smooth, latent organization of tissue types that provides an interpretable decisions tool for medical professionals.

Sum-of-norms clustering is a clustering formulation based on convex optimization that automatically induces hierarchy. Multiple algorithms have been proposed to solve the optimization problem: subgradient descent by Hocking et al.\ \cite{hocking}, ADMM and ADA by Chi and Lange\ \cite{Chi}, stochastic incremental algorithm by Panahi et al.\ \cite{Panahi} and semismooth Newton-CG augmented Lagrangian method by Yuan et al.\ \cite{dsun1}. All algorithms yield approximate solutions, even though an exact solution is demanded to determine the correct cluster assignment. The purpose of this paper is to close the gap between the output from existing algorithms and the exact solution to the optimization problem. We present a clustering test which identifies and certifies the correct cluster assignment from an approximate solution yielded by any primal-dual algorithm. The test may not succeed if the approximation is inaccurate. However, we show the correct cluster assignment is guaranteed to be found by a symmetric primal-dual path following algorithm after sufficiently many iterations, provided that the model parameter $\lambda$ avoids a finite number of bad values. Numerical experiments are implemented to support our results.

Motivated by industrial-scale applications, we consider two specific areas of fairness, one connected to the notion of equality of opportunity, and the other one generally tied to fair model performance. Throughout the paper, we consider only methods that can be scaled to Internet-industry size datasets. With this in mind, we propose a simple post-processing method to achieve equality of opportunity and discuss challenges and some solutions in the specific cases of recommendation systems and rankings. We then discuss a class of model performance fairness measures based on conditional ROC curves. We propose both scalable uncertainty assessment tools (that improve upon recent research) as well as scalable penalized methods to improve fairness with respect to these metrics. We provide fast algorithms with an emphasis on making few passes over the data when possible.