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Geographically weighted regression (GWR) models handle geographical dependence through a spatially varying coefficient model and have been widely used in applied science, but its general Bayesian extension is unclear because it involves a weighted log-likelihood which does not imply a probability distribution on data. We present a Bayesian GWR model and show that its essence is dealing with partial misspecification of the model. Current modularized Bayesian inference models accommodate partial misspecification from a single component of the model. We extend these models to handle partial misspecification in more than one component of the model, as required for our Bayesian GWR model. Information from the various spatial locations is manipulated via a geographically weighted kernel and the optimal manipulation is chosen according to a Kullback-Leibler (KL) divergence. We justify the model via an information risk minimization approach and show the consistency of the proposed estimator in terms of a geographically weighted KL divergence.

This paper studies the problem of statistical inference for genetic relatedness between binary traits based on individual-level genome-wide association data. Specifically, under the high-dimensional logistic regression model, we define parameters characterizing the cross-trait genetic correlation, the genetic covariance and the trait-specific genetic variance. A novel weighted debiasing method is developed for the logistic Lasso estimator and computationally efficient debiased estimators are proposed. The rates of convergence for these estimators are studied and their asymptotic normality is established under mild conditions. Moreover, we construct confidence intervals and statistical tests for these parameters, and provide theoretical justifications for the methods, including the coverage probability and expected length of the confidence intervals, as well as the size and power of the proposed tests. Numerical studies are conducted under both model generated data and simulated genetic data to show the superiority of the proposed methods and their applicability to the analysis of real genetic data. Finally, by analyzing a real data set on autoimmune diseases, we demonstrate the ability to obtain novel insights about the shared genetic architecture between ten pediatric autoimmune diseases.

In machine learning, we traditionally evaluate the performance of a single model, averaged over a collection of test inputs. In this work, we propose a new approach: we measure the performance of a collection of models when evaluated on a $\textit{single input point}$. Specifically, we study a point's $\textit{profile}$: the relationship between models' average performance on the test distribution and their pointwise performance on this individual point. We find that profiles can yield new insights into the structure of both models and data -- in and out-of-distribution. For example, we empirically show that real data distributions consist of points with qualitatively different profiles. On one hand, there are "compatible" points with strong correlation between the pointwise and average performance. On the other hand, there are points with weak and even $\textit{negative}$ correlation: cases where improving overall model accuracy actually $\textit{hurts}$ performance on these inputs. We prove that these experimental observations are inconsistent with the predictions of several simplified models of learning proposed in prior work. As an application, we use profiles to construct a dataset we call CIFAR-10-NEG: a subset of CINIC-10 such that for standard models, accuracy on CIFAR-10-NEG is $\textit{negatively correlated}$ with accuracy on CIFAR-10 test. This illustrates, for the first time, an OOD dataset that completely inverts "accuracy-on-the-line" (Miller, Taori, Raghunathan, Sagawa, Koh, Shankar, Liang, Carmon, and Schmidt 2021)

Employing a forward Markov diffusion chain to gradually map the data to a noise distribution, diffusion probabilistic models learn how to generate the data by inferring a reverse Markov diffusion chain to invert the forward diffusion process. To achieve competitive data generation performance, they demand a long diffusion chain that makes them computationally intensive in not only training but also generation. To significantly improve the computation efficiency, we propose to truncate the forward diffusion chain by abolishing the requirement of diffusing the data to random noise. Consequently, we start the inverse diffusion chain from an implicit generative distribution, rather than random noise, and learn its parameters by matching it to the distribution of the data corrupted by the truncated forward diffusion chain. Experimental results show our truncated diffusion probabilistic models provide consistent improvements over the non-truncated ones in terms of the generation performance and the number of required inverse diffusion steps.

Consider a matrix $\mathbf{F} \in \mathbb{K}^{m \times n}$ of univariate polynomials over a field~$\mathbb{K}$. We study the problem of computing the column rank profile of $\mathbf{F}$. To this end we first give an algorithm which improves the minimal kernel basis algorithm of Zhou, Labahn, and Storjohann (Proceedings ISSAC 2012). We then provide a second algorithm which computes the column rank profile of $\mathbf{F}$ with a rank-sensitive complexity of $O\tilde{~}(r^{\omega-2} n (m+D))$ operations in $\mathbb{K}$. Here, $D$ is the sum of row degrees of $\mathbf{F}$, $\omega$ is the exponent of matrix multiplication, and $O\tilde{~}(\cdot)$ hides logarithmic factors.

Center-based clustering is a pivotal primitive for unsupervised learning and data analysis. A popular variant is undoubtedly the k-means problem, which, given a set $P$ of points from a metric space and a parameter $k<|P|$, requires to determine a subset $S$ of $k$ centers minimizing the sum of all squared distances of points in $P$ from their closest center. A more general formulation, known as k-means with $z$ outliers, introduced to deal with noisy datasets, features a further parameter $z$ and allows up to $z$ points of $P$ (outliers) to be disregarded when computing the aforementioned sum. We present a distributed coreset-based 3-round approximation algorithm for k-means with $z$ outliers for general metric spaces, using MapReduce as a computational model. Our distributed algorithm requires sublinear local memory per reducer, and yields a solution whose approximation ratio is an additive term $O(\gamma)$ away from the one achievable by the best known sequential (possibly bicriteria) algorithm, where $\gamma$ can be made arbitrarily small. An important feature of our algorithm is that it obliviously adapts to the intrinsic complexity of the dataset, captured by the doubling dimension $D$ of the metric space. To the best of our knowledge, no previous distributed approaches were able to attain similar quality-performance tradeoffs for general metrics.

Generative models are typically trained on grid-like data such as images. As a result, the size of these models usually scales directly with the underlying grid resolution. In this paper, we abandon discretized grids and instead parameterize individual data points by continuous functions. We then build generative models by learning distributions over such functions. By treating data points as functions, we can abstract away from the specific type of data we train on and construct models that are agnostic to discretization. To train our model, we use an adversarial approach with a discriminator that acts on continuous signals. Through experiments on a wide variety of data modalities including images, 3D shapes and climate data, we demonstrate that our model can learn rich distributions of functions independently of data type and resolution.

We propose a general and scalable approximate sampling strategy for probabilistic models with discrete variables. Our approach uses gradients of the likelihood function with respect to its discrete inputs to propose updates in a Metropolis-Hastings sampler. We show empirically that this approach outperforms generic samplers in a number of difficult settings including Ising models, Potts models, restricted Boltzmann machines, and factorial hidden Markov models. We also demonstrate the use of our improved sampler for training deep energy-based models on high dimensional discrete data. This approach outperforms variational auto-encoders and existing energy-based models. Finally, we give bounds showing that our approach is near-optimal in the class of samplers which propose local updates.

Alternating Direction Method of Multipliers (ADMM) is a widely used tool for machine learning in distributed settings, where a machine learning model is trained over distributed data sources through an interactive process of local computation and message passing. Such an iterative process could cause privacy concerns of data owners. The goal of this paper is to provide differential privacy for ADMM-based distributed machine learning. Prior approaches on differentially private ADMM exhibit low utility under high privacy guarantee and often assume the objective functions of the learning problems to be smooth and strongly convex. To address these concerns, we propose a novel differentially private ADMM-based distributed learning algorithm called DP-ADMM, which combines an approximate augmented Lagrangian function with time-varying Gaussian noise addition in the iterative process to achieve higher utility for general objective functions under the same differential privacy guarantee. We also apply the moments accountant method to bound the end-to-end privacy loss. The theoretical analysis shows that DP-ADMM can be applied to a wider class of distributed learning problems, is provably convergent, and offers an explicit utility-privacy tradeoff. To our knowledge, this is the first paper to provide explicit convergence and utility properties for differentially private ADMM-based distributed learning algorithms. The evaluation results demonstrate that our approach can achieve good convergence and model accuracy under high end-to-end differential privacy guarantee.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.