In an attempt to better understand structural benefits and generalization power of deep neural networks, we firstly present a novel graph theoretical formulation of neural network models, including fully connected, residual network (ResNet) and densely connected networks (DenseNet). Secondly, we extend the error analysis of the population risk for two layer network \cite{ew2019prioriTwo} and ResNet \cite{e2019prioriRes} to DenseNet, and show further that for neural networks satisfying certain mild conditions, similar estimates can be obtained. These estimates are a priori in nature since they depend sorely on the information prior to the training process, in particular, the bounds for the estimation errors are independent of the input dimension.
We present a method to generate directed acyclic graphs (DAGs) using deep reinforcement learning, specifically deep Q-learning. Generating graphs with specified structures is an important and challenging task in various application fields, however most current graph generation methods produce graphs with undirected edges. We demonstrate that this method is capable of generating DAGs with topology and node types satisfying specified criteria in highly sparse reward environments.
Obtaining continuous representations of structural data such as directed acyclic graphs (DAGs) has gained attention in machine learning and artificial intelligence. However, embedding complex DAGs in which both ancestors and descendants of nodes are exponentially increasing is difficult. Tackling in this problem, we develop Disk Embeddings, which is a framework for embedding DAGs into quasi-metric spaces. Existing state-of-the-art methods, Order Embeddings and Hyperbolic Entailment Cones, are instances of Disk Embedding in Euclidean space and spheres respectively. Furthermore, we propose a nov…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Carnegie Mellon University（卡内基梅隆大学）
The main approach to defining equivalence among acyclic directed causal graphical models is based on the conditional independence relationships in the distributions that the causal models can generate, in terms of the Markov equivalence. However, it is known that when cycles are allowed in the causal structure, conditional independence may not be a suitable notion for equivalence of two structures, as it does not reflect all the information in the distribution that is useful for identification of the underlying structure. In this paper, we present a general, unified notion of equivalence for l…
Causal structures for observational survival data provide crucial information regarding the relationships between covariates and time-to-event. We derive motivation from the information theoretic source coding argument, and show that incorporating the knowledge of the directed acyclic graph (DAG) can be beneficial if suitable source encoders are employed. As a possible source encoder in this context, we derive a variational inference based conditional variational autoencoder for causal structured survival prediction, which we refer to as DAGSurv. We illustrate the performance of DAGSurv on low…
We develop a Bregman proximal gradient method for structure learning on linear structural causal models. While the problem is non-convex, has high curvature and is in fact NP-hard, Bregman gradient methods allow us to neutralize at least part of the impact of curvature by measuring smoothness against a highly nonlinear kernel. This allows the method to make longer steps and significantly improves convergence. Each iteration requires solving a Bregman proximal step which is convex and efficiently solvable for our particular choice of kernel. We test our method on various synthetic and real data…
Johns Hopkins University（约翰斯·霍普金斯大学）；Harvard T. H. Chan School of Public Health（哈佛陈曾熙公共卫生学院）
Missing data is a pervasive problem in data analyses, resulting in datasets that contain censored realizations of a target distribution. Many approaches to inference on the target distribution using censored observed data, rely on missing data models represented as a factorization with respect to a directed acyclic graph. In this paper we consider the identifiability of the target distribution within this class of models, and show that the most general identification strategies proposed so far retain a significant gap in that they fail to identify a wide class of identifiable distributions. To…
City University of Hong Kong（香港城市大学）；Xiamen University（厦门大学）；Shanghai University of Finance and Economics（上海财经大学）
Directed acyclic graph (DAG) models are widely used to represent causal relationships among random variables in many application domains. This paper studies a special class of non-Gaussian DAG models, where the conditional variance of each node given its parents is a quadratic function of its conditional mean. Such a class of non-Gaussian DAG models are fairly flexible and admit many popular distributions as special cases, including Poisson, Binomial, Geometric, Exponential, and Gamma. To facilitate learning, we introduce a novel concept of topological layers, and develop an efficient DAG lear…
Åbo Akademi University（奥布学术大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）；University of Helsinki（赫尔辛基大学）
We introduce a novel class of labeled directed acyclic graph (LDAG) models for finite sets of discrete variables. LDAGs generalize earlier proposals for allowing local structures in the conditional probability distribution of a node, such that unrestricted label sets determine which edges can be deleted from the underlying directed acyclic graph (DAG) for a given context. Several properties of these models are derived, including a generalization of the concept of Markov equivalence classes. Efficient Bayesian learning of LDAGs is enabled by introducing an LDAG-based factorization of the Dirich…
City University of Hong Kong（香港城市大学）；Shanghai University of Finance and Economics（上海财经大学）
Acyclic model, often depicted as a directed acyclic graph (DAG), has been widely employed to represent directional causal relations among collected nodes. In this article, we propose an efficient method to learn linear non-Gaussian DAG in high dimensional cases, where the noises can be of any continuous non-Gaussian distribution. This is in sharp contrast to most existing DAG learning methods assuming Gaussian noise with additional variance assumptions to attain exact DAG recovery. The proposed method leverages a novel concept of topological layer to facilitate the DAG learning. Particularly, …
Graph structured data are abundant in the real world. Among different graph types, directed acyclic graphs (DAGs) are of particular interest to machine learning researchers, as many machine learning models are realized as computations on DAGs, including neural networks and Bayesian networks. In this paper, we study deep generative models for DAGs, and propose a novel DAG variational autoencoder (D-VAE). To encode DAGs into the latent space, we leverage graph neural networks. We propose an asynchronous message passing scheme that allows encoding the computations on DAGs, rather than using exist…
Boston University（波士顿大学）；Systems & Technology Research（系统与技术研究公司）
We study the problem of reducing test-time acquisition costs in classification systems. Our goal is to learn decision rules that adaptively select sensors for each example as necessary to make a confident prediction. We model our system as a directed acyclic graph (DAG) where internal nodes correspond to sensor subsets and decision functions at each node choose whether to acquire a new sensor or classify using the available measurements. This problem can be naturally posed as an empirical risk minimization over training data. Rather than jointly optimizing such a highly coupled and non-convex …
We introduce a new class of identifiable DAG models where the conditional distribution of each node given its parents belongs to a family of generalized hypergeometric distributions (GHD). A family of generalized hypergeometric distributions includes a lot of discrete distributions such as the binomial, Beta-binomial, negative binomial, Poisson, hyper-Poisson, and many more. We prove that if the data drawn from the new class of DAG models, one can fully identify the graph structure. We further present a reliable and polynomial-time algorithm that recovers the graph from finitely many data. We …
In this paper, we explore and detail our experiments in a high-dimensionality, multi-class image classification problem often found in the automatic recognition of Sign Languages. Here, our efforts are directed towards comparing the characteristics, advantages and drawbacks of creating and training Support Vector Machines disposed in a Directed Acyclic Graph and Artificial Neural Networks to classify signs from the Brazilian Sign Language (LIBRAS). We explore how the different heuristics, hyperparameters and multi-class decision schemes affect the performance, efficiency and ease of use for ea…
Johns Hopkins University（约翰斯·霍普金斯大学）；Perimeter Institute for Theoretical Physics（圆周理论物理研究所）；University of Gdańsk（格但斯克大学）；International Centre for Theory of Quantum Technologies（国际量子技术理论中心）
Directed acyclic graphs (DAGs) with hidden variables are often used to characterize causal relations between variables in a system. When some variables are unobserved, DAGs imply a notoriously complicated set of constraints on the distribution of observed variables. In this work, we present entropic inequality constraints that are implied by $e$-separation relations in hidden variable DAGs with discrete observed variables. The constraints can intuitively be understood to follow from the fact that the capacity of variables along a causal pathway to convey information is restricted by their entr…
A Framework for Directed Acyclic Hypergraph Learning
Institute for Financial Services Analytics, University of Delaware（特拉华大学金融服务分析研究所）；Department of Electrical and Computer Engineering, University of Delaware（特拉华大学电气与计算机工程系）；Department of Applied Economics and Statistics, University of Delaware（特拉华大学应用经济学与统计学系）；Department of Electrical and Computer Engineering, University of Kentucky（肯塔基大学电气与计算机工程系）
Meta Reinforcement Learning (RL) methods focus on automating the design of RL algorithms that generalize to a wide range of environments. The framework introduced in (Anonymous, 2020) addresses the problem by representing different RL algorithms as Directed Acyclic Graphs (DAGs), and using an evolutionary meta learner to modify these graphs and find good agent update rules. While the search language used to generate graphs in the paper serves to represent numerous already-existing RL algorithms (e.g., DQN, DDQN), it has limitations when it comes to representing Policy Gradient algorithms. In t…
University College London（伦敦大学学院）；Amazon Web Services（亚马逊云科技）；University of Edinburgh（爱丁堡大学）；Inria（法国国家信息与自动化研究所）；ServiceNow
Recently continuous relaxations have been proposed in order to learn Directed Acyclic Graphs (DAGs) from data by backpropagation, instead of using combinatorial optimization. However, a number of techniques for fully discrete backpropagation could instead be applied. In this paper, we explore that direction and propose DAG-DB, a framework for learning DAGs by Discrete Backpropagation. Based on the architecture of Implicit Maximum Likelihood Estimation [I-MLE, arXiv:2106.01798], DAG-DB adopts a probabilistic approach to the problem, sampling binary adjacency matrices from an implicit probabilit…
Different directed acyclic graphs (DAGs) may be Markov equivalent in the sense that they entail the same conditional independence relations among the observed variables. Meek (1995) characterizes Markov equivalence classes for DAGs (with no latent variables) by presenting a set of orientation rules that can correctly identify all arrow orientations shared by all DAGs in a Markov equivalence class, given a member of that class. For DAG models with latent variables, maximal ancestral graphs (MAGs) provide a neat representation that facilitates model search. Earlier work (Ali et al. 2005) has ide…
University of Washington（华盛顿大学）；Northwestern University（西北大学）
Learning directed acyclic graphs (DAGs) from data is a challenging task both in theory and in practice, because the number of possible DAGs scales superexponentially with the number of nodes. In this paper, we study the problem of learning an optimal DAG from continuous observational data. We cast this problem in the form of a mathematical programming model which can naturally incorporate a super-structure in order to reduce the set of possible candidate DAGs. We use the penalized negative log-likelihood score function with both $\ell_0$ and $\ell_1$ regularizations and propose a new mixed-int…
Numerous approaches have been proposed to discover causal dependencies in machine learning and data mining; among them, the state-of-the-art VAR-LiNGAM (short for Vector Auto-Regressive Linear Non-Gaussian Acyclic Model) is a desirable approach to reveal both the instantaneous and time-lagged relationships. However, all the obtained VAR matrices need to be analyzed to infer the final causal graph, leading to a rise in the number of parameters. To address this issue, we propose the CGP-LiNGAM (short for Causal Graph Process-LiNGAM), which has significantly fewer model parameters and deals with …
A general lack of understanding pertaining to deep feedforward neural networks (DNNs) can be attributed partly to a lack of tools with which to analyze the composition of non-linear functions, and partly to a lack of mathematical models applicable to the diversity of DNN architectures. In this paper, we made a number of basic assumptions pertaining to activation functions, non-linear transformations, and DNN architectures in order to use the un-rectifying method to analyze DNNs via directed acyclic graphs (DAGs). DNNs that satisfy these assumptions are referred to as general DNNs. Our construc…
We study a family of regularized score-based estimators for learning the structure of a directed acyclic graph (DAG) for a multivariate normal distribution from high-dimensional data with $p\gg n$. Our main results establish support recovery guarantees and deviation bounds for a family of penalized least-squares estimators under concave regularization without assuming prior knowledge of a variable ordering. These results apply to a variety of practical situations that allow for arbitrary nondegenerate covariance structures as well as many popular regularizers including the MCP, SCAD, $\ell_{0}…
Bayesian network is a frequently-used method for fault detection and diagnosis in industrial processes. The basis of Bayesian network is structure learning which learns a directed acyclic graph (DAG) from data. However, the search space will scale super-exponentially with the increase of process variables, which makes the data-driven structure learning a challenging problem. To this end, the DAGs with NOTEARs methods are being well studied not only for their conversion of the discrete optimization into continuous optimization problem but also their compatibility with deep learning framework. N…
Washington University in St. Louis（圣路易斯华盛顿大学）；Peking University（北京大学）；Beijing Institute for General Artificial Intelligence（北京通用人工智能研究院）
Optimization of directed acyclic graph (DAG) structures has many applications, such as neural architecture search (NAS) and probabilistic graphical model learning. Encoding DAGs into real vectors is a dominant component in most neural-network-based DAG optimization frameworks. Currently, most DAG encoders use an asynchronous message passing scheme which sequentially processes nodes according to the dependency between nodes in a DAG. That is, a node must not be processed until all its predecessors are processed. As a result, they are inherently not parallelizable. In this work, we propose a Par…
MIT-IBM Watson AI Lab（MIT-IBM沃森人工智能实验室）；IBM Research（IBM研究院）
Graph-structured data ubiquitously appears in science and engineering. Graph neural networks (GNNs) are designed to exploit the relational inductive bias exhibited in graphs; they have been shown to outperform other forms of neural networks in scenarios where structure information supplements node features. The most common GNN architecture aggregates information from neighborhoods based on message passing. Its generality has made it broadly applicable. In this paper, we focus on a special, yet widely used, type of graphs -- DAGs -- and inject a stronger inductive bias -- partial ordering -- in…
Causal relationships among a set of variables are commonly represented by a directed acyclic graph. The orientations of some edges in the causal DAG can be discovered from observational/interventional data. Further edges can be oriented by iteratively applying so-called Meek rules. Inferring edges' orientations from some previously oriented edges, which we call Causal Orientation Learning (COL), is a common problem in various causal discovery tasks. In these tasks, it is often required to solve multiple COL problems and therefore applying Meek rules could be time-consuming. Motivated by Meek r…
ETH Zürich（苏黎世联邦理工学院）；University of Washington（华盛顿大学）
We consider the problem of learning causal information between random variables in directed acyclic graphs (DAGs) when allowing arbitrarily many latent and selection variables. The FCI (Fast Causal Inference) algorithm has been explicitly designed to infer conditional independence and causal information in such settings. However, FCI is computationally infeasible for large graphs. We therefore propose the new RFCI algorithm, which is much faster than FCI. In some situations the output of RFCI is slightly less informative, in particular with respect to conditional independence information. Howe…
