Gaussian mixture distributions are commonly employed to represent general probability distributions. Despite the importance of using Gaussian mixtures for uncertainty estimation, the entropy of a Gaussian mixture cannot be analytically calculated. Notably, Gal and Ghahramani [2016] proposed the approximate entropy that is the sum of the entropies of unimodal Gaussian distributions. This approximation is easy to analytically calculate regardless of dimension, but there lack theoretical guarantees. In this paper, we theoretically analyze the approximation error between the true entropy and the approximate one to reveal when this approximation works effectively. This error is controlled by how far apart each Gaussian component of the Gaussian mixture. To measure such separation, we introduce the ratios of the distances between the means to the sum of the variances of each Gaussian component of the Gaussian mixture, and we reveal that the error converges to zero as the ratios tend to infinity. This convergence situation is more likely to occur in higher dimensional spaces. Therefore, our results provide a guarantee that this approximation works well in higher dimension problems, particularly in scenarios such as neural networks that involve a large number of weights.
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Constrained variational inequality problems are recognized for their broad applications across various fields including machine learning and operations research. First-order methods have emerged as the standard approach for solving these problems due to their simplicity and scalability. However, they typically rely on projection or linear minimization oracles to navigate the feasible set, which becomes computationally expensive in practical scenarios featuring multiple functional constraints. Existing efforts to tackle such functional constrained variational inequality problems have centered on primal-dual algorithms grounded in the Lagrangian function. These algorithms along with their theoretical analysis often require the existence and prior knowledge of the optimal Lagrange multipliers. In this work, we propose a simple primal method, termed Constrained Gradient Method (CGM), for addressing functional constrained variational inequality problems, without necessitating any information on the optimal Lagrange multipliers. We establish a non-asymptotic convergence analysis of the algorithm for variational inequality problems with monotone operators under smooth constraints. Remarkably, our algorithms match the complexity of projection-based methods in terms of operator queries for both monotone and strongly monotone settings, while utilizing significantly cheaper oracles based on quadratic programming. Furthermore, we provide several numerical examples to evaluate the efficacy of our algorithms.
Generative models are invaluable in many fields of science because of their ability to capture high-dimensional and complicated distributions, such as photo-realistic images, protein structures, and connectomes. How do we evaluate the samples these models generate? This work aims to provide an accessible entry point to understanding popular notions of statistical distances, requiring only foundational knowledge in mathematics and statistics. We focus on four commonly used notions of statistical distances representing different methodologies: Using low-dimensional projections (Sliced-Wasserstein; SW), obtaining a distance using classifiers (Classifier Two-Sample Tests; C2ST), using embeddings through kernels (Maximum Mean Discrepancy; MMD), or neural networks (Fr\'echet Inception Distance; FID). We highlight the intuition behind each distance and explain their merits, scalability, complexity, and pitfalls. To demonstrate how these distances are used in practice, we evaluate generative models from different scientific domains, namely a model of decision making and a model generating medical images. We showcase that distinct distances can give different results on similar data. Through this guide, we aim to help researchers to use, interpret, and evaluate statistical distances for generative models in science.
We show that, in many settings, the worst-case performance of a distributed optimization algorithm is independent of the number of agents in the system, and can thus be computed in the fundamental case with just two agents. This result relies on a novel approach that systematically exploits symmetries in worst-case performance computation, framed as Semidefinite Programming (SDP) via the Performance Estimation Problem (PEP) framework. Harnessing agent symmetries in the PEP yields compact problems whose size is independent of the number of agents in the system. When all agents are equivalent in the problem, we establish the explicit conditions under which the resulting worst-case performance is independent of the number of agents and is therefore equivalent to the basic case with two agents. Our compact PEP formulation also allows the consideration of multiple equivalence classes of agents, and its size only depends on the number of equivalence classes. This enables practical and automated performance analysis of distributed algorithms in numerous complex and realistic settings, such as the analysis of the worst agent performance. We leverage this new tool to analyze the performance of the EXTRA algorithm in advanced settings and its scalability with the number of agents, providing a tighter analysis and deeper understanding of the algorithm performance.
1. Joint species distribution models (JSDMs) have gained considerable traction among ecologists over the past decade, due to their capacity to answer a wide range of questions at both the species- and the community-level. The family of generalized linear latent variable models in particular has proven popular for building JSDMs, being able to handle many response types including presence-absence data, biomass, overdispersed and/or zero-inflated counts. 2. We extend latent variable models to handle percent cover data, with vegetation, sessile invertebrate, and macroalgal cover data representing the prime examples of such data arising in community ecology. 3. Sparsity is a commonly encountered challenge with percent cover data. Responses are typically recorded as percentages covered per plot, though some species may be completely absent or present, i.e., have 0% or 100% cover respectively, rendering the use of beta distribution inadequate. 4. We propose two JSDMs suitable for percent cover data, namely a hurdle beta model and an ordered beta model. We compare the two proposed approaches to a beta distribution for shifted responses, transformed presence-absence data, and an ordinal model for percent cover classes. Results demonstrate the hurdle beta JSDM was generally the most accurate at retrieving the latent variables and predicting ecological percent cover data.
Previous stance detection studies typically concentrate on evaluating stances within individual instances, thereby exhibiting limitations in effectively modeling multi-party discussions concerning the same specific topic, as naturally transpire in authentic social media interactions. This constraint arises primarily due to the scarcity of datasets that authentically replicate real social media contexts, hindering the research progress of conversational stance detection. In this paper, we introduce a new multi-turn conversation stance detection dataset (called \textbf{MT-CSD}), which encompasses multiple targets for conversational stance detection. To derive stances from this challenging dataset, we propose a global-local attention network (\textbf{GLAN}) to address both long and short-range dependencies inherent in conversational data. Notably, even state-of-the-art stance detection methods, exemplified by GLAN, exhibit an accuracy of only 50.47\%, highlighting the persistent challenges in conversational stance detection. Furthermore, our MT-CSD dataset serves as a valuable resource to catalyze advancements in cross-domain stance detection, where a classifier is adapted from a different yet related target. We believe that MT-CSD will contribute to advancing real-world applications of stance detection research. Our source code, data, and models are available at \url{//github.com/nfq729/MT-CSD}.
Solutions to differential equations, which are used to model physical systems, are computed numerically by solving a set of discretized equations. This set of discretized equations is reduced to a large linear system, whose solution is typically found using an iterative solver. We start with an initial guess, $x_0$, and iterate the algorithm to obtain a sequence of solution vectors, $x_k$, which are approximations to the exact solution of the linear system, $x$. The iterative algorithm is said to converge to $x$, in the field of reals, if and only if $x_k$ converges to $x$ in the limit of $k \to \infty$. In this paper, we formally prove the asymptotic convergence of a particular class of iterative methods called the stationary iterative methods, in the Coq theorem prover. We formalize the necessary and sufficient conditions required for the iterative convergence, and extend this result to two classical iterative methods: the Gauss--Seidel method and the Jacobi method. For the Gauss--Seidel method, we also formalize a set of easily testable conditions for iterative convergence, called the Reich theorem, for a particular matrix structure, and apply this on a model problem of the one-dimensional heat equation. We also apply the main theorem of iterative convergence to prove convergence of the Jacobi method on the model problem.
As artificial intelligence (AI) models continue to scale up, they are becoming more capable and integrated into various forms of decision-making systems. For models involved in moral decision-making, also known as artificial moral agents (AMA), interpretability provides a way to trust and understand the agent's internal reasoning mechanisms for effective use and error correction. In this paper, we provide an overview of this rapidly-evolving sub-field of AI interpretability, introduce the concept of the Minimum Level of Interpretability (MLI) and recommend an MLI for various types of agents, to aid their safe deployment in real-world settings.
Graphs are important data representations for describing objects and their relationships, which appear in a wide diversity of real-world scenarios. As one of a critical problem in this area, graph generation considers learning the distributions of given graphs and generating more novel graphs. Owing to their wide range of applications, generative models for graphs, which have a rich history, however, are traditionally hand-crafted and only capable of modeling a few statistical properties of graphs. Recent advances in deep generative models for graph generation is an important step towards improving the fidelity of generated graphs and paves the way for new kinds of applications. This article provides an extensive overview of the literature in the field of deep generative models for graph generation. Firstly, the formal definition of deep generative models for the graph generation and the preliminary knowledge are provided. Secondly, taxonomies of deep generative models for both unconditional and conditional graph generation are proposed respectively; the existing works of each are compared and analyzed. After that, an overview of the evaluation metrics in this specific domain is provided. Finally, the applications that deep graph generation enables are summarized and five promising future research directions are highlighted.
As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.
Object detection typically assumes that training and test data are drawn from an identical distribution, which, however, does not always hold in practice. Such a distribution mismatch will lead to a significant performance drop. In this work, we aim to improve the cross-domain robustness of object detection. We tackle the domain shift on two levels: 1) the image-level shift, such as image style, illumination, etc, and 2) the instance-level shift, such as object appearance, size, etc. We build our approach based on the recent state-of-the-art Faster R-CNN model, and design two domain adaptation components, on image level and instance level, to reduce the domain discrepancy. The two domain adaptation components are based on H-divergence theory, and are implemented by learning a domain classifier in adversarial training manner. The domain classifiers on different levels are further reinforced with a consistency regularization to learn a domain-invariant region proposal network (RPN) in the Faster R-CNN model. We evaluate our newly proposed approach using multiple datasets including Cityscapes, KITTI, SIM10K, etc. The results demonstrate the effectiveness of our proposed approach for robust object detection in various domain shift scenarios.
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