Learning directed acyclic graphs (DAGs) is long known a critical challenge at the core of probabilistic and causal modeling. The NoTears approach of (Zheng et al., 2018), through a differentiable function involving the matrix exponential trace $\mathrm{tr}(\exp(\cdot))$, opens up a way to learning DAGs via continuous optimization, though with a $O(d^3)$ complexity in the number $d$ of nodes. This paper presents a low-complexity model, called LoRAM for Low-Rank Additive Model, which combines low-rank matrix factorization with a sparsification mechanism for the continuous optimization of DAGs. The main contribution of the approach lies in an efficient gradient approximation method leveraging the low-rank property of the model, and its straightforward application to the computation of projections from graph matrices onto the DAG matrix space. The proposed method achieves a reduction from a cubic complexity to quadratic complexity while handling the same DAG characteristic function as NoTears, and scales easily up to thousands of nodes for the projection problem. The experiments show that the LoRAM achieves efficiency gains of orders of magnitude compared to the state-of-the-art at the expense of a very moderate accuracy loss in the considered range of sparse matrices, and with a low sensitivity to the rank choice of the model's low-rank component.
We explore if it is possible to learn a directed acyclic graph (DAG) from data without imposing explicitly the acyclicity constraint. In particular, for Gaussian distributions, we frame structural learning as a sparse matrix factorization problem and we empirically show that solving an $\ell_1$-penalized optimization yields to good recovery of the true graph and, in general, to almost-DAG graphs. Moreover, this approach is computationally efficient and is not affected by the explosion of combinatorial complexity as in classical structural learning algorithms.
University of Helsinki（赫尔辛基大学）；HIIT（赫尔辛基信息技术研究所）；University of Oslo（奥斯陆大学）
We give methods for Bayesian inference of directed acyclic graphs, DAGs, and the induced causal effects from passively observed complete data. Our methods build on a recent Markov chain Monte Carlo scheme for learning Bayesian networks, which enables efficient approximate sampling from the graph posterior, provided that each node is assigned a small number $K$ of candidate parents. We present algorithmic techniques to significantly reduce the space and time requirements, which make the use of substantially larger values of $K$ feasible. Furthermore, we investigate the problem of selecting the …
Stanford University（斯坦福大学）；Facebook AI Research（Facebook人工智能研究院）；University of California, Los Angeles（加州大学洛杉矶分校）
A structural equation model (SEM) is an effective framework to reason over causal relationships represented via a directed acyclic graph (DAG). Recent advances have enabled effective maximum-likelihood point estimation of DAGs from observational data. However, a point estimate may not accurately capture the uncertainty in inferring the underlying graph in practical scenarios, wherein the true DAG is non-identifiable and/or the observed dataset is limited. We propose Bayesian Causal Discovery Nets (BCD Nets), a variational inference framework for estimating a distribution over DAGs characterizi…
A family of Markov blankets in a faithful Bayesian network satisfies the symmetry and consistency properties. In this paper, we draw a bijection between families of consistent Markov blankets and moral graphs. We define the new concepts of weak recursive simpliciality and perfect elimination kits. We prove that they are equivalent to graph morality. In addition, we prove that morality can be decided in polynomial time for graphs with maximum degree less than $5$, but the problem is NP-complete for graphs with higher maximum degrees.
The causal relationships among a set of random variables are commonly represented by a Directed Acyclic Graph (DAG), where there is a directed edge from variable $X$ to variable $Y$ if $X$ is a direct cause of $Y$. From the purely observational data, the true causal graph can be identified up to a Markov Equivalence Class (MEC), which is a set of DAGs with the same conditional independencies between the variables. The size of an MEC is a measure of complexity for recovering the true causal graph by performing interventions. We propose a method for efficient iteration over possible MECs given i…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Sharif University of Technology（谢里夫理工大学）；Georgia Institute of Technology（佐治亚理工学院）；Carnegie Mellon University（卡内基梅隆大学）
A directed acyclic graph (DAG) is the most common graphical model for representing causal relationships among a set of variables. When restricted to using only observational data, the structure of the ground truth DAG is identifiable only up to Markov equivalence, based on conditional independence relations among the variables. Therefore, the number of DAGs equivalent to the ground truth DAG is an indicator of the causal complexity of the underlying structure--roughly speaking, it shows how many interventions or how much additional information is further needed to recover the underlying DAG. I…
LIDS, MIT（麻省理工学院林肯实验室（LIDS））；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）；IBM Research（IBM研究）
A growing body of work has begun to study intervention design for efficient structure learning of causal directed acyclic graphs (DAGs). A typical setting is a causally sufficient setting, i.e. a system with no latent confounders, selection bias, or feedback, when the essential graph of the observational equivalence class (EC) is given as an input and interventions are assumed to be noiseless. Most existing works focus on worst-case or average-case lower bounds for the number of interventions required to orient a DAG. These worst-case lower bounds only establish that the largest clique in the …
Georgia Institute of Technology（佐治亚理工学院）；Carnegie Mellon University（卡内基梅隆大学）；MBZUAI（穆罕默德·本·扎耶德人工智能大学）；BioMap（百图生科）
We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able …
Dept. of Signal Theory and Communications, Rey Juan Carlos University, Madrid, Spain（信号理论与通信系，雷亚尔·卡洛斯大学，马德里，西班牙）；Dept. of Electrical and Computer Engineering, University of Rochester（电气与计算机工程系，罗切斯特大学）
Normalizing flows (NFs) have become a prominent method for deep generative models that allow for an analytic probability density estimation and efficient synthesis. However, a flow-based network is considered to be inefficient in parameter complexity because of reduced expressiveness of bijective mapping, which renders the models unfeasibly expensive in terms of parameters. We present an alternative parameterization scheme called NanoFlow, which uses a single neural density estimator to model multiple transformation stages. Hence, we propose an efficient parameter decomposition method and the …
This paper introduces a framework for systematic complexity scaling of deep neural network(DNN) based MIMO detectors. The model uses a fraction of the DNN inputs by scaling their values through weights that follow monotonically non-increasing functions. This allows for weight scaling across and within the different DNN layers in order to achieve accuracy-vs.-complexity scalability during inference. In order to further improve the performance of our proposal, we introduce a sparsity-inducing regularization constraint in conjunction with trainable weight-scaling functions. In this way, the netwo…
Approximate linear programming (ALP) represents one of the major algorithmic families to solve large-scale Markov decision processes (MDP). In this work, we study a primal-dual formulation of the ALP, and develop a scalable, model-free algorithm called bilinear $π$ learning for reinforcement learning when a sampling oracle is provided. This algorithm enjoys a number of advantages. First, it adopts (bi)linear models to represent the high-dimensional value function and state-action distributions, using given state and action features. Its run-time complexity depends on the number of features, no…
Sharif University of Technology（谢里夫理工大学）；Technical University of Munich（慕尼黑工业大学）；Universite Paris-Est（巴黎东部大学）
The Information Bottleneck method is a learning technique that seeks a right balance between accuracy and generalization capability through a suitable tradeoff between compression complexity, measured by minimum description length, and distortion evaluated under logarithmic loss measure. In this paper, we study a variation of the problem, called scalable information bottleneck, where the encoder outputs multiple descriptions of the observation with increasingly richer features. The problem at hand is motivated by some application scenarios that require varying levels of accuracy depending on t…
The CUR matrix decomposition is an important extension of Nyström approximation to a general matrix. It approximates any data matrix in terms of a small number of its columns and rows. In this paper we propose a novel randomized CUR algorithm with an expected relative-error bound. The proposed algorithm has the advantages over the existing relative-error CUR algorithms that it possesses tighter theoretical bound and lower time complexity, and that it can avoid maintaining the whole data matrix in main memory. Finally, experiments on several real-world datasets demonstrate significant improveme…
Scalable Complexity Control Facilitates Reasoning Ability of LLMs
Institute of Natural Sciences, School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院自然科学院）；MOE-LSC, School of Artificial Intelligence, Shanghai Jiao Tong University（上海交通大学人工智能学院）；Center for Machine Learning Research, School of Mathematical Sciences, Peking University（北京大学数学科学学院机器学习研究中心）；Center for LLM, Institute for Advanced Algorithms Research, Shanghai（上海高级算法研究所大语言模型中心）；MemTensor (Shanghai) Technology Co., Ltd.（MemTensor（上海）科技有限公司）
Sharif University of Technology（谢里夫理工大学）；Technical University of Munich（慕尼黑工业大学）；Universite Paris-Est（巴黎东部大学）
In the context of statistical learning, the Information Bottleneck method seeks a right balance between accuracy and generalization capability through a suitable tradeoff between compression complexity, measured by minimum description length, and distortion evaluated under logarithmic loss measure. In this paper, we study a variation of the problem, called scalable information bottleneck, in which the encoder outputs multiple descriptions of the observation with increasingly richer features. The model, which is of successive-refinement type with degraded side information streams at the decoder…
Duke University（杜克大学）；University of Michigan（密歇根大学）
Linear mixed models (LMMs) are used extensively to model dependecies of observations in linear regression and are used extensively in many application areas. Parameter estimation for LMMs can be computationally prohibitive on big data. State-of-the-art learning algorithms require computational complexity which depends at least linearly on the dimension $p$ of the covariates, and often use heuristics that do not offer theoretical guarantees. We present scalable algorithms for learning high-dimensional LMMs with sublinear computational complexity dependence on $p$. Key to our approach are novel …
Scalable Simulation-Based Model Inference with Test-Time Complexity Control
Machine Learning in Science, University of Tübingen；Tübingen AI Center, Tübingen, Germany；Max Planck Institute for Intelligent Systems, Department Empirical Inference, Tübingen, Germany；Sir Peter Mansfield Imaging Centre, School of Medicine, University of Nottingham, UK；National Institute for Health Research (NIHR) Nottingham Biomedical Research Centre, Queens Medical Centre, Nottingham, United Kingdom
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；Technology Innovation Institute（技术创新研究院）
In deep active learning, it is especially important to choose multiple examples to markup at each step to work efficiently, especially on large datasets. At the same time, existing solutions to this problem in the Bayesian setup, such as BatchBALD, have significant limitations in selecting a large number of examples, associated with the exponential complexity of computing mutual information for joint random variables. We, therefore, present the Large BatchBALD algorithm, which gives a well-grounded approximation to the BatchBALD method that aims to achieve comparable quality while being more c…
Generalized Linear Bandits (GLBs), a natural extension of the stochastic linear bandits, has been popular and successful in recent years. However, existing GLBs scale poorly with the number of rounds and the number of arms, limiting their utility in practice. This paper proposes new, scalable solutions to the GLB problem in two respects. First, unlike existing GLBs, whose per-time-step space and time complexity grow at least linearly with time $t$, we propose a new algorithm that performs online computations to enjoy a constant space and time complexity. At its heart is a novel Generalized Lin…
University of Maryland（马里兰大学）；Microsoft（微软）；HEALTH[at]SCALE；Flatiron Health；Georgia Institute of Technology（佐治亚理工学院）；California Institute of Technology（加州理工学院）
We present an integrated approach for structure and parameter estimation in latent tree graphical models. Our overall approach follows a "divide-and-conquer" strategy that learns models over small groups of variables and iteratively merges onto a global solution. The structure learning involves combinatorial operations such as minimum spanning tree construction and local recursive grouping; the parameter learning is based on the method of moments and on tensor decompositions. Our method is guaranteed to correctly recover the unknown tree structure and the model parameters with low sample compl…
Rensselaer Polytechnic Institute（伦斯勒理工学院）；IBM Research AI（IBM研究院）
This paper considers the problem of estimating the unknown intervention targets in a causal directed acyclic graph from observational and interventional data. The focus is on soft interventions in linear structural equation models (SEMs). Current approaches to causal structure learning either work with known intervention targets or use hypothesis testing to discover the unknown intervention targets even for linear SEMs. This severely limits their scalability and sample complexity. This paper proposes a scalable and efficient algorithm that consistently identifies all intervention targets. The …
Norwegian University of Science and Technology（挪威科技大学）
Single-person human pose estimation facilitates markerless movement analysis in sports, as well as in clinical applications. Still, state-of-the-art models for human pose estimation generally do not meet the requirements of real-life applications. The proliferation of deep learning techniques has resulted in the development of many advanced approaches. However, with the progresses in the field, more complex and inefficient models have also been introduced, which have caused tremendous increases in computational demands. To cope with these complexity and inefficiency challenges, we propose a no…
