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RNN-Attention Based Deep Learning for Solving Inverse Boundary Problems in Nonlinear Marshak Waves 基于rnn -注意力的深度学习求解非线性马沙克波反边界问题
IF 6 3区 计算机科学 Q1 Mathematics Pub Date : 2023-04-01 DOI: 10.4208/jml.221209
Di Zhao, Weiming Li, Wengu Chen, Peng Song, and Han Wang null
. Radiative transfer, described by the radiative transfer equation (RTE), is one of the dominant energy exchange processes in the inertial confinement fusion (ICF) experiments. The Marshak wave problem is an important benchmark for time-dependent RTE. In this work, we present a neural network architecture termed RNN-attention deep learning (RADL) as a surrogate model to solve the inverse boundary problem of the nonlinear Marshak wave in a data-driven fashion. We train the surrogate model by numerical simulation data of the forward problem, and then solve the inverse problem by minimizing the distance between the target solution and the surrogate predicted solution concerning the boundary condition. This minimization is made efficient because the surrogate model by-passes the expensive numerical solution, and the model is differentiable so the gradient-based optimization algorithms are adopted. The effectiveness of our approach is demonstrated by solving the inverse boundary problems of the Marshak wave benchmark in two case studies: where the transport process is modeled by RTE and where it is modeled by its nonlinear diffusion approximation (DA). Last but not least, the importance of using both the RNN and the factor-attention blocks in the RADL model is illustrated, and the data efficiency of our model is investigated in this work.
。辐射传递是惯性约束聚变(ICF)实验中主要的能量交换过程之一,用辐射传递方程(RTE)来描述。马沙克波问题是时变RTE的一个重要基准。在这项工作中,我们提出了一种称为rnn -注意力深度学习(RADL)的神经网络架构作为代理模型,以数据驱动的方式解决非线性马沙克波的逆边界问题。我们利用正演问题的数值模拟数据训练代理模型,然后在边界条件下通过最小化目标解与代理预测解之间的距离来求解逆问题。由于替代模型绕过了昂贵的数值解,并且模型是可微的,因此采用了基于梯度的优化算法,从而使这种最小化变得高效。通过在两个案例研究中解决马沙克波基准的逆边界问题,我们的方法的有效性得到了证明:其中输运过程是由RTE建模的,而它是由其非线性扩散近似(DA)建模的。最后,说明了在RADL模型中同时使用RNN和因子注意块的重要性,并对我们的模型的数据效率进行了研究。
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引用次数: 0
Inference for Gaussian Processes with Matérn Covariogram on Compact Riemannian Manifolds. 紧凑黎曼曼形上具有马特恩协方差的高斯过程推理
IF 6 3区 计算机科学 Q1 Mathematics Pub Date : 2023-03-01
Didong Li, Wenpin Tang, Sudipto Banerjee

Gaussian processes are widely employed as versatile modelling and predictive tools in spatial statistics, functional data analysis, computer modelling and diverse applications of machine learning. They have been widely studied over Euclidean spaces, where they are specified using covariance functions or covariograms for modelling complex dependencies. There is a growing literature on Gaussian processes over Riemannian manifolds in order to develop richer and more flexible inferential frameworks for non-Euclidean data. While numerical approximations through graph representations have been well studied for the Matérn covariogram and heat kernel, the behaviour of asymptotic inference on the parameters of the covariogram has received relatively scant attention. We focus on asymptotic behaviour for Gaussian processes constructed over compact Riemannian manifolds. Building upon a recently introduced Matérn covariogram on a compact Riemannian manifold, we employ formal notions and conditions for the equivalence of two Matérn Gaussian random measures on compact manifolds to derive the parameter that is identifiable, also known as the microergodic parameter, and formally establish the consistency of the maximum likelihood estimate and the asymptotic optimality of the best linear unbiased predictor. The circle is studied as a specific example of compact Riemannian manifolds with numerical experiments to illustrate and corroborate the theory.

高斯过程是空间统计学、函数数据分析、计算机建模和机器学习各种应用中广泛使用的通用建模和预测工具。人们对欧几里得空间上的高斯过程进行了广泛的研究,利用协方差函数或协方差图对复杂的依赖关系进行建模。关于黎曼流形上的高斯过程的文献越来越多,以便为非欧几里得数据开发更丰富、更灵活的推理框架。虽然通过图形表示对马特恩协方差和热核的数值近似进行了深入研究,但对协方差参数的渐近推断行为的关注却相对较少。我们重点研究在紧凑黎曼流形上构建的高斯过程的渐近行为。以最近引入的紧凑黎曼流形上的马特恩协变图为基础,我们采用紧凑流形上两个马特恩高斯随机度量等价的形式化概念和条件,推导出可识别的参数(也称为微角参数),并正式建立最大似然估计的一致性和最佳线性无偏预测器的渐近最优性。我们将圆作为紧凑黎曼流形的一个具体实例进行研究,并通过数值实验来说明和证实这一理论。
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引用次数: 0
Bayesian Data Selection. 贝叶斯数据选择。
IF 6 3区 计算机科学 Q1 Mathematics Pub Date : 2023-01-01
Eli N Weinstein, Jeffrey W Miller

Insights into complex, high-dimensional data can be obtained by discovering features of the data that match or do not match a model of interest. To formalize this task, we introduce the "data selection" problem: finding a lower-dimensional statistic-such as a subset of variables-that is well fit by a given parametric model of interest. A fully Bayesian approach to data selection would be to parametrically model the value of the statistic, nonparametrically model the remaining "background" components of the data, and perform standard Bayesian model selection for the choice of statistic. However, fitting a nonparametric model to high-dimensional data tends to be highly inefficient, statistically and computationally. We propose a novel score for performing data selection, the "Stein volume criterion (SVC)", that does not require fitting a nonparametric model. The SVC takes the form of a generalized marginal likelihood with a kernelized Stein discrepancy in place of the Kullback-Leibler divergence. We prove that the SVC is consistent for data selection, and establish consistency and asymptotic normality of the corresponding generalized posterior on parameters. We apply the SVC to the analysis of single-cell RNA sequencing data sets using probabilistic principal components analysis and a spin glass model of gene regulation.

通过发现与感兴趣的模型匹配或不匹配的数据特征,可以获得对复杂高维数据的洞察。为了形式化这个任务,我们引入了“数据选择”问题:找到一个较低维的统计量——比如变量的子集——它与给定的参数模型很好地拟合。数据选择的完全贝叶斯方法是对统计值进行参数化建模,对数据的剩余“背景”成分进行非参数化建模,并对统计值的选择执行标准贝叶斯模型选择。然而,拟合一个非参数模型到高维数据往往是非常低效的,统计和计算。我们提出了一种用于执行数据选择的新评分,即“Stein体积准则(SVC)”,它不需要拟合非参数模型。SVC采用广义边际似然的形式,用核化的Stein差异代替Kullback-Leibler散度。证明了SVC在数据选择上是一致的,并建立了相应的广义后验在参数上的一致性和渐近正态性。我们使用概率主成分分析和基因调控的自旋玻璃模型将SVC应用于单细胞RNA测序数据集的分析。
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引用次数: 0
Inference for a Large Directed Acyclic Graph with Unspecified Interventions. 具有未指定干预的大有向非循环图的推理。
IF 6 3区 计算机科学 Q1 Mathematics Pub Date : 2023-01-01
Chunlin Li, Xiaotong Shen, Wei Pan

Statistical inference of directed relations given some unspecified interventions (i.e., the intervention targets are unknown) is challenging. In this article, we test hypothesized directed relations with unspecified interventions. First, we derive conditions to yield an identifiable model. Unlike classical inference, testing directed relations requires to identify the ancestors and relevant interventions of hypothesis-specific primary variables. To this end, we propose a peeling algorithm based on nodewise regressions to establish a topological order of primary variables. Moreover, we prove that the peeling algorithm yields a consistent estimator in low-order polynomial time. Second, we propose a likelihood ratio test integrated with a data perturbation scheme to account for the uncertainty of identifying ancestors and interventions. Also, we show that the distribution of a data perturbation test statistic converges to the target distribution. Numerical examples demonstrate the utility and effectiveness of the proposed methods, including an application to infer gene regulatory networks. The R implementation is available at https://github.com/chunlinli/intdag.

在给定一些未指明的干预措施(即干预目标未知)的情况下,对定向关系进行统计推断是具有挑战性的。在这篇文章中,我们测试了假设的直接关系与未指明的干预措施。首先,我们导出了产生可识别模型的条件。与经典推理不同,测试定向关系需要识别特定假设的主要变量的祖先和相关干预。为此,我们提出了一种基于节点回归的剥离算法来建立主变量的拓扑顺序。此外,我们证明了剥离算法在低阶多项式时间内产生了一致的估计量。其次,我们提出了一种与数据扰动方案相结合的似然比检验,以解释识别祖先和干预措施的不确定性。此外,我们还证明了数据扰动测试统计量的分布收敛于目标分布。数值例子证明了所提出的方法的实用性和有效性,包括推断基因调控网络的应用。R的实施可在https://github.com/chunlinli/intdag.
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引用次数: 0
Fair Data Representation for Machine Learning at the Pareto Frontier. 帕累托前沿机器学习的公平数据表示
IF 4.3 3区 计算机科学 Q1 AUTOMATION & CONTROL SYSTEMS Pub Date : 2023-01-01
Shizhou Xu, Thomas Strohmer

As machine learning powered decision-making becomes increasingly important in our daily lives, it is imperative to strive for fairness in the underlying data processing. We propose a pre-processing algorithm for fair data representation via which L 2 ( ) -objective supervised learning results in estimations of the Pareto frontier between prediction error and statistical disparity. Particularly, the present work applies the optimal affine transport to approach the post-processing Wasserstein barycenter characterization of the optimal fair L 2 -objective supervised learning via a pre-processing data deformation. Furthermore, we show that the Wasserstein geodesics from learning outcome marginals to their barycenter characterizes the Pareto frontier between L 2 -loss and total Wasserstein distance among the marginals. Numerical simulations underscore the advantages: (1) the pre-processing step is compositive with arbitrary conditional expectation estimation supervised learning methods and unseen data; (2) the fair representation protects the sensitive information by limiting the inference capability of the remaining data with respect to the sensitive data; (3) the optimal affine maps are computationally efficient even for high-dimensional data.

随着机器学习驱动的决策在我们的日常生活中变得越来越重要,在底层数据处理中力求公平势在必行。我们提出了一种用于公平数据表示的预处理算法,通过这种算法,目标监督学习可以估计预测误差和统计差异之间的帕累托前沿。特别是,本研究应用最优仿射传输,通过预处理数据变形,接近最优公平 L 2 目标监督学习的后处理 Wasserstein barycenter 特性。此外,我们还证明了从学习结果边际到其原点的瓦瑟斯坦大地线表征了边际间的 L 2 -损失和总瓦瑟斯坦距离之间的帕累托前沿。数值模拟证明了该方法的优势:(1)预处理步骤与任意条件期望估计监督学习方法和未见数据具有可比性;(2)公平表示通过限制其余数据相对于敏感数据的推理能力来保护敏感信息;(3)即使对于高维数据,最优仿射图的计算效率也很高。
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引用次数: 0
Minimax Estimation for Personalized Federated Learning: An Alternative between FedAvg and Local Training? 个性化联合学习的最小估计:FedAvg 和本地训练之间的替代方案?
IF 4.3 3区 计算机科学 Q1 AUTOMATION & CONTROL SYSTEMS Pub Date : 2023-01-01
Shuxiao Chen, Qinqing Zheng, Qi Long, Weijie J Su

A widely recognized difficulty in federated learning arises from the statistical heterogeneity among clients: local datasets often originate from distinct yet not entirely unrelated probability distributions, and personalization is, therefore, necessary to achieve optimal results from each individual's perspective. In this paper, we show how the excess risks of personalized federated learning using a smooth, strongly convex loss depend on data heterogeneity from a minimax point of view, with a focus on the FedAvg algorithm (McMahan et al., 2017) and pure local training (i.e., clients solve empirical risk minimization problems on their local datasets without any communication). Our main result reveals an approximate alternative between these two baseline algorithms for federated learning: the former algorithm is minimax rate optimal over a collection of instances when data heterogeneity is small, whereas the latter is minimax rate optimal when data heterogeneity is large, and the threshold is sharp up to a constant. As an implication, our results show that from a worst-case point of view, a dichotomous strategy that makes a choice between the two baseline algorithms is rate-optimal. Another implication is that the popular FedAvg following by local fine tuning strategy is also minimax optimal under additional regularity conditions. Our analysis relies on a new notion of algorithmic stability that takes into account the nature of federated learning.

联合学习中一个公认的难题来自于客户之间的统计异质性:本地数据集通常来自不同但并非完全无关的概率分布,因此,要想从每个人的角度获得最佳结果,就必须实现个性化。在本文中,我们从最小化的角度展示了使用平滑、强凸损失的个性化联合学习的超额风险如何取决于数据异质性,重点关注 FedAvg 算法(McMahan 等人,2017 年)和纯本地训练(即客户在不进行任何交流的情况下解决其本地数据集上的经验风险最小化问题)。我们的主要结果揭示了这两种联合学习基线算法之间的近似替代方案:当数据异质性较小时,前一种算法在实例集合上是最小率最优的,而当数据异质性较大且阈值尖锐到一个常数时,后一种算法是最小率最优的。我们的结果表明,从最坏情况的角度来看,在两种基准算法之间做出选择的二分法策略是速率最优的。另一个含义是,在额外的规则性条件下,流行的 FedAvg 跟随局部微调策略也是最小最优的。我们的分析依赖于一个新的算法稳定性概念,它考虑到了联合学习的本质。
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引用次数: 0
Surrogate Assisted Semi-supervised Inference for High Dimensional Risk Prediction. 用于高维风险预测的替代物辅助半监督推理。
IF 6 3区 计算机科学 Q1 Mathematics Pub Date : 2023-01-01
Jue Hou, Zijian Guo, Tianxi Cai

Risk modeling with electronic health records (EHR) data is challenging due to no direct observations of the disease outcome and the high-dimensional predictors. In this paper, we develop a surrogate assisted semi-supervised learning approach, leveraging small labeled data with annotated outcomes and extensive unlabeled data of outcome surrogates and high-dimensional predictors. We propose to impute the unobserved outcomes by constructing a sparse imputation model with outcome surrogates and high-dimensional predictors. We further conduct a one-step bias correction to enable interval estimation for the risk prediction. Our inference procedure is valid even if both the imputation and risk prediction models are misspecified. Our novel way of ultilizing unlabelled data enables the high-dimensional statistical inference for the challenging setting with a dense risk prediction model. We present an extensive simulation study to demonstrate the superiority of our approach compared to existing supervised methods. We apply the method to genetic risk prediction of type-2 diabetes mellitus using an EHR biobank cohort.

由于无法直接观察疾病结果和高维预测因子,利用电子健康记录(EHR)数据进行风险建模具有挑战性。在本文中,我们开发了一种代用数据辅助的半监督学习方法,该方法利用了带有注释结果的小标签数据以及大量未标签的结果代用数据和高维预测因子。我们建议通过利用结果代理和高维预测因子构建稀疏估算模型来估算未观察到的结果。我们还进一步进行了一步纠偏,以实现风险预测的区间估计。即使估算模型和风险预测模型都被错误地指定,我们的推断程序也是有效的。我们采用新颖的方法来充分利用未标注数据,从而能够在具有高密度风险预测模型的挑战性环境中进行高维统计推断。我们进行了广泛的模拟研究,以证明我们的方法与现有的监督方法相比具有优越性。我们利用电子病历生物库队列将该方法应用于 2 型糖尿病遗传风险预测。
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引用次数: 0
Learning Optimal Group-structured Individualized Treatment Rules with Many Treatments. 学习具有多种治疗方法的最佳小组结构个性化治疗规则
IF 6 3区 计算机科学 Q1 Mathematics Pub Date : 2023-01-01
Haixu Ma, Donglin Zeng, Yufeng Liu

Data driven individualized decision making problems have received a lot of attentions in recent years. In particular, decision makers aim to determine the optimal Individualized Treatment Rule (ITR) so that the expected specified outcome averaging over heterogeneous patient-specific characteristics is maximized. Many existing methods deal with binary or a moderate number of treatment arms and may not take potential treatment effect structure into account. However, the effectiveness of these methods may deteriorate when the number of treatment arms becomes large. In this article, we propose GRoup Outcome Weighted Learning (GROWL) to estimate the latent structure in the treatment space and the optimal group-structured ITRs through a single optimization. In particular, for estimating group-structured ITRs, we utilize the Reinforced Angle based Multicategory Support Vector Machines (RAMSVM) to learn group-based decision rules under the weighted angle based multi-class classification framework. Fisher consistency, the excess risk bound, and the convergence rate of the value function are established to provide a theoretical guarantee for GROWL. Extensive empirical results in simulation studies and real data analysis demonstrate that GROWL enjoys better performance than several other existing methods.

近年来,数据驱动的个体化决策问题受到了广泛关注。特别是,决策者的目标是确定最佳个体化治疗规则(ITR),从而最大限度地提高平均于异质性患者特异性特征的预期特定结果。许多现有方法处理二元或中等数量的治疗臂,可能不会考虑潜在的治疗效果结构。然而,当治疗臂数量变多时,这些方法的有效性可能会下降。在本文中,我们提出了 GROWL(Group Outcome Weighted Learning)方法,通过一次优化来估计治疗空间中的潜在结构和最优组结构 ITR。特别是,为了估算组结构 ITR,我们利用基于加强角的多类支持向量机(RAMSVM),在基于加权角的多类分类框架下学习基于组的决策规则。费雪一致性、超额风险约束和价值函数收敛率的建立为 GROWL 提供了理论保证。模拟研究和实际数据分析的大量实证结果表明,GROWL 比其他几种现有方法具有更好的性能。
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引用次数: 0
Conditional Distribution Function Estimation Using Neural Networks for Censored and Uncensored Data. 使用神经网络对有删减和无删减数据进行条件分布函数估计。
IF 6 3区 计算机科学 Q1 Mathematics Pub Date : 2023-01-01
Bingqing Hu, Bin Nan

Most work in neural networks focuses on estimating the conditional mean of a continuous response variable given a set of covariates. In this article, we consider estimating the conditional distribution function using neural networks for both censored and uncensored data. The algorithm is built upon the data structure particularly constructed for the Cox regression with time-dependent covariates. Without imposing any model assumptions, we consider a loss function that is based on the full likelihood where the conditional hazard function is the only unknown nonparametric parameter, for which unconstrained optimization methods can be applied. Through simulation studies, we show that the proposed method possesses desirable performance, whereas the partial likelihood method and the traditional neural networks with L2 loss yields biased estimates when model assumptions are violated. We further illustrate the proposed method with several real-world data sets. The implementation of the proposed methods is made available at https://github.com/bingqing0729/NNCDE.

神经网络方面的大多数研究工作都侧重于在给定一组协变量的情况下估计连续响应变量的条件均值。在本文中,我们将考虑使用神经网络估计有删减和无删减数据的条件分布函数。该算法建立在数据结构的基础上,特别是为具有时间相关协变量的 Cox 回归所构建的数据结构。在不强加任何模型假设的情况下,我们考虑了基于全似然的损失函数,其中条件危险函数是唯一未知的非参数参数,可以应用无约束优化方法。通过模拟研究,我们发现所提出的方法具有理想的性能,而部分似然法和带有 L2 损失的传统神经网络在违反模型假设时会产生有偏差的估计值。我们还用几个真实世界的数据集进一步说明了所提出的方法。建议方法的实现可在 https://github.com/bingqing0729/NNCDE 上获得。
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引用次数: 0
Consistent Second-Order Conic Integer Programming for Learning Bayesian Networks. 学习贝叶斯网络的一致二阶圆锥整数编程
IF 4.3 3区 计算机科学 Q1 AUTOMATION & CONTROL SYSTEMS Pub Date : 2023-01-01
Simge Küçükyavuz, Ali Shojaie, Hasan Manzour, Linchuan Wei, Hao-Hsiang Wu

Bayesian Networks (BNs) represent conditional probability relations among a set of random variables (nodes) in the form of a directed acyclic graph (DAG), and have found diverse applications in knowledge discovery. We study the problem of learning the sparse DAG structure of a BN from continuous observational data. The central problem can be modeled as a mixed-integer program with an objective function composed of a convex quadratic loss function and a regularization penalty subject to linear constraints. The optimal solution to this mathematical program is known to have desirable statistical properties under certain conditions. However, the state-of-the-art optimization solvers are not able to obtain provably optimal solutions to the existing mathematical formulations for medium-size problems within reasonable computational times. To address this difficulty, we tackle the problem from both computational and statistical perspectives. On the one hand, we propose a concrete early stopping criterion to terminate the branch-and-bound process in order to obtain a near-optimal solution to the mixed-integer program, and establish the consistency of this approximate solution. On the other hand, we improve the existing formulations by replacing the linear "big- M " constraints that represent the relationship between the continuous and binary indicator variables with second-order conic constraints. Our numerical results demonstrate the effectiveness of the proposed approaches.

贝叶斯网络(BN)以有向无环图(DAG)的形式表示一组随机变量(节点)之间的条件概率关系,在知识发现领域有着广泛的应用。我们研究的问题是从连续观测数据中学习 BN 的稀疏 DAG 结构。这个核心问题可以建模为一个混合整数程序,其目标函数由一个凸二次损失函数和一个正则化惩罚组成,并受到线性约束。众所周知,该数学程序的最优解在某些条件下具有理想的统计特性。然而,对于中等规模的问题,最先进的优化求解器无法在合理的计算时间内获得现有数学公式的公认最优解。为解决这一难题,我们从计算和统计两个角度着手。一方面,我们提出了一个具体的早期停止准则来终止分支与边界过程,从而获得混合整数程序的近似最优解,并建立了该近似解的一致性。另一方面,我们用二阶圆锥约束取代了表示连续和二进制指标变量之间关系的线性 "big- M "约束,从而改进了现有公式。我们的数值结果证明了所提方法的有效性。
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引用次数: 0
期刊
Journal of Machine Learning Research
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