In experimental design, we are given $n$ vectors in $d$ dimensions, and our goal is to select $k\ll n$ of them to perform expensive measurements, e.g., to obtain labels/responses, for a linear regression task. Many statistical criteria have been proposed for choosing the optimal design, with popular choices including A- and D-optimality. If prior knowledge is given, typically in the form of a $d\times d$ precision matrix $\mathbf A$, then all of the criteria can be extended to incorporate that information via a Bayesian framework. In this paper, we demonstrate a new fundamental connection between Bayesian experimental design and determinantal point processes, the latter being widely used for sampling diverse subsets of data. We use this connection to develop new efficient algorithms for finding $(1+ε)$-approximations of optimal designs under four optimality criteria: A, C, D and V. Our algorithms can achieve this when the desired subset size $k$ is $Ω(\frac{d_{\mathbf A}}ε + \frac{\log 1/ε}{ε^2})$, where $d_{\mathbf A}\leq d$ is the $\mathbf A$-effective dimension, which can often be much smaller than $d$. Our results offer direct improvements over a number of prior works, for both Bayesian and classical experimental design, in terms of algorithm efficiency, approximation quality, and range of applicable criteria.
In Bayesian Optimization (BO) we study black-box function optimization with noisy point evaluations and Bayesian priors. Convergence of BO can be greatly sped up by batching, where multiple evaluations of the black-box function are performed in a single round. The main difficulty in this setting is to propose at the same time diverse and informative batches of evaluation points. In this work, we introduce DPP-Batch Bayesian Optimization (DPP-BBO), a universal framework for inducing batch diversity in sampling based BO by leveraging the repulsive properties of Determinantal Point Processes (DPP…
Semi-supervised clustering is the task of clustering data points into clusters where only a fraction of the points are labelled. The true number of clusters in the data is often unknown and most models require this parameter as an input. Dirichlet process mixture models are appealing as they can infer the number of clusters from the data. However, these models do not deal with high dimensional data well and can encounter difficulties in inference. We present a novel nonparameteric Bayesian kernel based method to cluster data points without the need to prespecify the number of clusters or to mo…
Determinantal point processes (DPPs) are an elegant model for encoding probabilities over subsets, such as shopping baskets, of a ground set, such as an item catalog. They are useful for a number of machine learning tasks, including product recommendation. DPPs are parametrized by a positive semi-definite kernel matrix. Recent work has shown that using a low-rank factorization of this kernel provides remarkable scalability improvements that open the door to training on large-scale datasets and computing online recommendations, both of which are infeasible with standard DPP models that use a fu…
Given a fixed $n\times d$ matrix $\mathbf{X}$, where $n\gg d$, we study the complexity of sampling from a distribution over all subsets of rows where the probability of a subset is proportional to the squared volume of the parallelepiped spanned by the rows (a.k.a. a determinantal point process). In this task, it is important to minimize the preprocessing cost of the procedure (performed once) as well as the sampling cost (performed repeatedly). To that end, we propose a new determinantal point process algorithm which has the following two properties, both of which are novel: (1) a preprocessi…
University of Pennsylvania（宾夕法尼亚大学）；University of Washington（华盛顿大学）；Harvard University（哈佛大学）
Determinantal point processes (DPPs) are well-suited for modeling repulsion and have proven useful in many applications where diversity is desired. While DPPs have many appealing properties, such as efficient sampling, learning the parameters of a DPP is still considered a difficult problem due to the non-convex nature of the likelihood function. In this paper, we propose using Bayesian methods to learn the DPP kernel parameters. These methods are applicable in large-scale and continuous DPP settings even when the exact form of the eigendecomposition is unknown. We demonstrate the utility of o…
Bayesian networks are a class of popular graphical models that encode causal and conditional independence relations among variables by directed acyclic graphs (DAGs). We propose a novel structure learning method, annealing on regularized Cholesky score (ARCS), to search over topological sorts, or permutations of nodes, for a high-scoring Bayesian network. Our scoring function is derived from regularizing Gaussian DAG likelihood, and its optimization gives an alternative formulation of the sparse Cholesky factorization problem from a statistical viewpoint, which is of independent interest. We c…
Selecting diverse and important items, called landmarks, from a large set is a problem of interest in machine learning. As a specific example, in order to deal with large training sets, kernel methods often rely on low rank matrix Nyström approximations based on the selection or sampling of landmarks. In this context, we propose a deterministic and a randomized adaptive algorithm for selecting landmark points within a training data set. These landmarks are related to the minima of a sequence of kernelized Christoffel functions. Beyond the known connection between Christoffel functions and leve…
Gaussian Process bandit optimization has emerged as a powerful tool for optimizing noisy black box functions. One example in machine learning is hyper-parameter optimization where each evaluation of the target function requires training a model which may involve days or even weeks of computation. Most methods for this so-called "Bayesian optimization" only allow sequential exploration of the parameter space. However, it is often desirable to propose batches or sets of parameter values to explore simultaneously, especially when there are large parallel processing facilities at our disposal. Bat…
In the modern age of social media and networks, graph representations of real-world phenomena have become an incredibly useful source to mine insights. Often, we are interested in understanding how entities in a graph are interconnected. The Graph Neural Network (GNN) has proven to be a very useful tool in a variety of graph learning tasks including node classification, link prediction, and edge classification. However, in most of these tasks, the graph data we are working with may be noisy and may contain spurious edges. That is, there is a lot of uncertainty associated with the underlying gr…
Sparse Gaussian Processes are a key component of high-throughput Bayesian Optimisation (BO) loops; however, we show that existing methods for allocating their inducing points severely hamper optimisation performance. By exploiting the quality-diversity decomposition of Determinantal Point Processes, we propose the first inducing point allocation strategy designed specifically for use in BO. Unlike existing methods which seek only to reduce global uncertainty in the objective function, our approach provides the local high-fidelity modelling of promising regions required for precise optimisation…
This paper presents a novel nonlinear regression model for estimating heterogeneous treatment effects from observational data, geared specifically towards situations with small effect sizes, heterogeneous effects, and strong confounding. Standard nonlinear regression models, which may work quite well for prediction, have two notable weaknesses when used to estimate heterogeneous treatment effects. First, they can yield badly biased estimates of treatment effects when fit to data with strong confounding. The Bayesian causal forest model presented in this paper avoids this problem by directly in…
Massachusetts Institute of Technology（麻省理工学院）；University of Michigan（密歇根大学）；Harvard University（哈佛大学）
We propose a novel diverse feature selection method based on determinantal point processes (DPPs). Our model enables one to flexibly define diversity based on the covariance of features (similar to orthogonal matching pursuit) or alternatively based on side information. We introduce our approach in the context of Bayesian sparse regression, employing a DPP as a variational approximation to the true spike and slab posterior distribution. We subsequently show how this variational DPP approximation generalizes and extends mean-field approximation, and can be learned efficiently by exploiting the …
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Washington University in St. Louis（圣路易斯华盛顿大学）
Determinantal point processes (DPPs) are an important concept in random matrix theory and combinatorics. They have also recently attracted interest in the study of numerical methods for machine learning, as they offer an elegant "missing link" between independent Monte Carlo sampling and deterministic evaluation on regular grids, applicable to a general set of spaces. This is helpful whenever an algorithm explores to reduce uncertainty, such as in active learning, Bayesian optimization, reinforcement learning, and marginalization in graphical models. To draw samples from a DPP in practice, exi…
Yale Institute for Network Science（耶鲁网络科学研究所）；Yale University（耶鲁大学）；Expedia Group（亿客行集团）；University of Haifa（海法大学）
In this paper, we propose scalable methods for maximizing a regularized submodular function $f = g - \ell$ expressed as the difference between a monotone submodular function $g$ and a modular function $\ell$. Indeed, submodularity is inherently related to the notions of diversity, coverage, and representativeness. In particular, finding the mode of many popular probabilistic models of diversity, such as determinantal point processes, submodular probabilistic models, and strongly log-concave distributions, involves maximization of (regularized) submodular functions. Since a regularized function…
Sandia National Laboratories（桑迪亚国家实验室）；Xilinx Inc.（赛灵思公司）；Georgia Institute of Technology（佐治亚理工学院）
Bayesian optimization (BO) is an efficient and flexible global optimization framework that is applicable to a very wide range of engineering applications. To leverage the capability of the classical BO, many extensions, including multi-objective, multi-fidelity, parallelization, latent-variable model, have been proposed to improve the limitation of the classical BO framework. In this work, we propose a novel multi-objective (MO) extension, called srMO-BO-3GP, to solve the MO optimization problems in a sequential setting. Three different Gaussian processes (GPs) are stacked together, where each…
Technion – Israel Institute of Technology（以色列理工学院）
In the Bayesian reinforcement learning (RL) setting, a prior distribution over the unknown problem parameters -- the rewards and transitions -- is assumed, and a policy that optimizes the (posterior) expected return is sought. A common approximation, which has been recently popularized as meta-RL, is to train the agent on a sample of $N$ problem instances from the prior, with the hope that for large enough $N$, good generalization behavior to an unseen test instance will be obtained. In this work, we study generalization in Bayesian RL under the probably approximately correct (PAC) framework, …
City University of Hong Kong（香港城市大学）；Arizona State University（亚利桑那州立大学）
This paper develops a novel stochastic tree ensemble method for nonlinear regression, which we refer to as XBART, short for Accelerated Bayesian Additive Regression Trees. By combining regularization and stochastic search strategies from Bayesian modeling with computationally efficient techniques from recursive partitioning approaches, the new method attains state-of-the-art performance: in many settings it is both faster and more accurate than the widely-used XGBoost algorithm. Via careful simulation studies, we demonstrate that our new approach provides accurate point-wise estimates of the m…
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University of California, Santa Cruz（加州大学圣克鲁兹分校）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
Gaussian processes are a flexible Bayesian nonparametric modelling approach that has been widely applied but poses computational challenges. To address the poor scaling of exact inference methods, approximation methods based on sparse Gaussian processes (SGP) are attractive. An issue faced by SGP, especially in latent variable models, is the inefficient learning of the inducing inputs, which leads to poor model prediction. We propose a regularization approach by balancing the reconstruction performance of data and the approximation performance of the model itself. This regularization improves …
Regularized infill criteria for multi-objective Bayesian optimization with application to aircraft design
ISAE-SUPAERO（法国高等航空航天学院）；Université de Toulouse（图卢兹大学）；MINES ALBI（阿尔比矿业学院）；UPS（图卢兹第三大学）；INSA（图卢兹国立应用科学学院）；CNRS（法国国家科学研究中心）；ONERA（法国航空航天研究院）；DLR（德国航空航天中心）
Australian National University（澳大利亚国立大学）；Data61, CSIRO（联邦科学与工业研究组织数据61研究所）
Sequential recommendation is a popular task in academic research and close to real-world application scenarios, where the goal is to predict the next action(s) of the user based on his/her previous sequence of actions. In the training process of recommender systems, the loss function plays an essential role in guiding the optimization of recommendation models to generate accurate suggestions for users. However, most existing sequential recommendation techniques focus on designing algorithms or neural network architectures, and few efforts have been made to tailor loss functions that fit natura…
Distributional Determinantal Point Process for Repulsive Clustering of Distributions
Department of Statistics and Data Sciences, University of Texas at Austin（统计与数据科学系，德克萨斯大学奥斯汀分校）；Department of Mathematics, University of Texas at Austin（数学系，德克萨斯大学奥斯汀分校）；Department of Biostatistics and Informatics, University of Colorado Anschutz（生物统计学与信息学系，科罗拉多大学安舒茨分校）
It is well established that temporal organization is critical to memory, and that the ability to temporally organize information is fundamental to many perceptual, cognitive, and motor processes. While our understanding of how the brain processes the spatial context of memories has advanced considerably, our understanding of their temporal organization lags far behind. In this paper, we propose a new approach for elucidating the neural basis of complex behaviors and temporal organization of memories. More specifically, we focus on neural decoding - the prediction of behavioral or experimental …
Explosive growth in data and availability of cheap computing resources have sparked increasing interest in Big learning, an emerging subfield that studies scalable machine learning algorithms, systems, and applications with Big Data. Bayesian methods represent one important class of statistic methods for machine learning, with substantial recent developments on adaptive, flexible and scalable Bayesian learning. This article provides a survey of the recent advances in Big learning with Bayesian methods, termed Big Bayesian Learning, including nonparametric Bayesian methods for adaptively inferr…
Department of Mathematics, University of California, Los Angeles, CA 90095（加州大学洛杉矶分校数学系）；Department of Mathematics, University of South Carolina, Columbia, SC 29208（南卡罗来纳大学数学系）
