The kernel least mean squares (KLMS) algorithm is a computationally efficient nonlinear adaptive filtering method that "kernelizes" the celebrated (linear) least mean squares algorithm. We demonstrate that the least mean squares algorithm is closely related to the Kalman filtering, and thus, the KLMS can be interpreted as an approximate Bayesian filtering method. This allows us to systematically develop extensions of the KLMS by modifying the underlying state-space and observation models. The resulting extensions introduce many desirable properties such as "forgetting", and the ability to learn from discrete data, while retaining the computational simplicity and time complexity of the original algorithm.
Marginalising over families of Gaussian Process kernels produces flexible model classes with well-calibrated uncertainty estimates. Existing approaches require likelihood evaluations of many kernels, rendering them prohibitively expensive for larger datasets. We propose a Bayesian Quadrature scheme to make this marginalisation more efficient and thereby more practical. Through use of the maximum mean discrepancies between distributions, we define a kernel over kernels that captures invariances between Spectral Mixture (SM) Kernels. Kernel samples are selected by generalising an information-the…
We propose Bayesian extensions of two nonparametric regression methods which are kernel and mutual $k$-nearest neighbor regression methods. Derived based on Gaussian process models for regression, the extensions provide distributions for target value estimates and the framework to select the hyperparameters. It is shown that both the proposed methods asymptotically converge to kernel and mutual $k$-nearest neighbor regression methods, respectively. The simulation results show that the proposed methods can select proper hyperparameters and are better than or comparable to the former methods for…
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所）；Aalto University School of Science（阿尔托大学理工学院）
Multiple kernel learning algorithms are proposed to combine kernels in order to obtain a better similarity measure or to integrate feature representations coming from different data sources. Most of the previous research on such methods is focused on the computational efficiency issue. However, it is still not feasible to combine many kernels using existing Bayesian approaches due to their high time complexity. We propose a fully conjugate Bayesian formulation and derive a deterministic variational approximation, which allows us to combine hundreds or thousands of kernels very efficiently. We …
University of Queensland（昆士兰大学）；University of Cambridge（剑桥大学）
A simple, flexible approach to creating expressive priors in Gaussian process (GP) models makes new kernels from a combination of basic kernels, e.g. summing a periodic and linear kernel can capture seasonal variation with a long term trend. Despite a well-studied link between GPs and Bayesian neural networks (BNNs), the BNN analogue of this has not yet been explored. This paper derives BNN architectures mirroring such kernel combinations. Furthermore, it shows how BNNs can produce periodic kernels, which are often useful in this context. These ideas provide a principled approach to designing …
We revisit Rahimi and Recht (2007)'s kernel random Fourier features (RFF) method through the lens of the PAC-Bayesian theory. While the primary goal of RFF is to approximate a kernel, we look at the Fourier transform as a prior distribution over trigonometric hypotheses. It naturally suggests learning a posterior on these hypotheses. We derive generalization bounds that are optimized by learning a pseudo-posterior obtained from a closed-form expression. Based on this study, we consider two learning strategies: The first one finds a compact landmarks-based representation of the data where each …
We propose a practical Bayesian optimization method over sets, to minimize a black-box function that takes a set as a single input. Because set inputs are permutation-invariant, traditional Gaussian process-based Bayesian optimization strategies which assume vector inputs can fall short. To address this, we develop a Bayesian optimization method with \emph{set kernel} that is used to build surrogate functions. This kernel accumulates similarity over set elements to enforce permutation-invariance, but this comes at a greater computational cost. To reduce this burden, we propose two key componen…
University of Sydney（悉尼大学）；CSIRO（澳大利亚联邦科学与工业研究组织）；NVIDIA（英伟达）
Conditional kernel mean embeddings form an attractive nonparametric framework for representing conditional means of functions, describing the observation processes for many complex models. However, the recovery of the original underlying function of interest whose conditional mean was observed is a challenging inference task. We formalize deconditional kernel mean embeddings as a solution to this inverse problem, and show that it can be naturally viewed as a nonparametric Bayes' rule. Critically, we introduce the notion of task transformed Gaussian processes and establish deconditional kernel …
Trinity College Dublin（都柏林圣三一学院）；Eindhoven University of Technology（埃因霍温理工大学）
A fundamental task in AI is to assess (in)dependence between mixed-type variables (text, image, sound). We propose a Bayesian kernelised correlation test of (in)dependence using a Dirichlet process model. The new measure of (in)dependence allows us to answer some fundamental questions: Based on data, are (mixed-type) variables independent? How likely is dependence/independence to hold? How high is the probability that two mixed-type variables are more than just weakly dependent? We theoretically show the properties of the approach, as well as algorithms for fast computation with it. We empiric…
Virginia Tech（弗吉尼亚理工大学）；Kuwait University（科威特大学）；Department of Industrial and Systems Engineering, Virginia Tech（弗吉尼亚理工大学工业与系统工程系）；Department of Industrial and Management Systems Engineering, Kuwait University（科威特大学工业与管理系统工程系）
Some response surface functions in complex engineering systems are usually highly nonlinear, unformed, and expensive-to-evaluate. To tackle this challenge, Bayesian optimization, which conducts sequential design via a posterior distribution over the objective function, is a critical method used to find the global optimum of black-box functions. Kernel functions play an important role in shaping the posterior distribution of the estimated function. The widely used kernel function, e.g., radial basis function (RBF), is very vulnerable and susceptible to outliers; the existence of outliers is cau…
University of Amsterdam（阿姆斯特丹大学）；Qualcomm AI Research（高通人工智能研究院）
In this work we propose a batch Bayesian optimization method for combinatorial problems on permutations, which is well suited for expensive-to-evaluate objectives. We first introduce LAW, an efficient batch acquisition method based on determinantal point processes using the acquisition weighted kernel. Relying on multiple parallel evaluations, LAW enables accelerated search on combinatorial spaces. We then apply the framework to permutation problems, which have so far received little attention in the Bayesian Optimization literature, despite their practical importance. We call this method LAW2…
University of Edinburgh（爱丁堡大学）；School of Informatics（信息学院）；School of Engineering（工程学院）
Recently, different machine learning methods have been introduced to tackle the challenging few-shot learning scenario that is, learning from a small labeled dataset related to a specific task. Common approaches have taken the form of meta-learning: learning to learn on the new problem given the old. Following the recognition that meta-learning is implementing learning in a multi-level model, we present a Bayesian treatment for the meta-learning inner loop through the use of deep kernels. As a result we can learn a kernel that transfers to new tasks; we call this Deep Kernel Transfer (DKT). Th…
Symmetry-Aware Bayesian Optimization via Max Kernels
School of Computer and Communication Sciences, École polytechnique fédérale de Lausanne (EPFL)（计算机与通信科学学院，瑞士联邦理工学院（EPFL））；Institute of Mathematics, École polytechnique fédérale de Lausanne (EPFL)（数学学院，瑞士联邦理工学院（EPFL））
针对贝叶斯优化中利用对称性但 max kernel 非半正定的问题，提出其 PSD 投影核，在不增加计算复杂度的情况下显著降低合成与真实基准上的遗憾。
Bosch Center for Artificial Intelligence（博世人工智能中心）
Despite recent advances in automated machine learning, model selection is still a complex and computationally intensive process. For Gaussian processes (GPs), selecting the kernel is a crucial task, often done manually by the expert. Additionally, evaluating the model selection criteria for Gaussian processes typically scales cubically in the sample size, rendering kernel search particularly computationally expensive. We propose a novel, efficient search method through a general, structured kernel space. Previous methods solved this task via Bayesian optimization and relied on measuring the di…
Herding and kernel herding are deterministic methods of choosing samples which summarise a probability distribution. A related task is choosing samples for estimating integrals using Bayesian quadrature. We show that the criterion minimised when selecting samples in kernel herding is equivalent to the posterior variance in Bayesian quadrature. We then show that sequential Bayesian quadrature can be viewed as a weighted version of kernel herding which achieves performance superior to any other weighted herding method. We demonstrate empirically a rate of convergence faster than O(1/N). Our resu…
University of Amsterdam（阿姆斯特丹大学）；Canadian Institute for Advanced Research（加拿大高等研究院）
A major challenge in Bayesian Optimization is the boundary issue (Swersky, 2017) where an algorithm spends too many evaluations near the boundary of its search space. In this paper, we propose BOCK, Bayesian Optimization with Cylindrical Kernels, whose basic idea is to transform the ball geometry of the search space using a cylindrical transformation. Because of the transformed geometry, the Gaussian Process-based surrogate model spends less budget searching near the boundary, while concentrating its efforts relatively more near the center of the search region, where we expect the solution to …
Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Gregorio Marañón Health Research Institute（格雷戈里奥·马拉尼翁健康研究所）
Multi-view problems can be faced with latent variable models since they are able to find low-dimensional projections that fairly capture the correlations among the multiple views that characterise each datum. On the other hand, high-dimensionality and non-linear issues are traditionally handled by kernel methods, inducing a (non)-linear function between the latent projection and the data itself. However, they usually come with scalability issues and exposition to overfitting. Here, we propose merging both approaches into single model so that we can exploit the best features of multi-view laten…
University of Amsterdam（阿姆斯特丹大学）；Qualcomm AI Research（高通人工智能研究院）
The sample efficiency of Bayesian optimization(BO) is often boosted by Gaussian Process(GP) surrogate models. However, on mixed variable spaces, surrogate models other than GPs are prevalent, mainly due to the lack of kernels which can model complex dependencies across different types of variables. In this paper, we propose the frequency modulated (FM) kernel flexibly modeling dependencies among different types of variables, so that BO can enjoy the further improved sample efficiency. The FM kernel uses distances on continuous variables to modulate the graph Fourier spectrum derived from discr…
Multiple kernel learning (MKL), structured sparsity, and multi-task learning have recently received considerable attention. In this paper, we show how different MKL algorithms can be understood as applications of either regularization on the kernel weights or block-norm-based regularization, which is more common in structured sparsity and multi-task learning. We show that these two regularization strategies can be systematically mapped to each other through a concave conjugate operation. When the kernel-weight-based regularizer is separable into components, we can naturally consider a generati…
We explore the link between deep ensembles and Gaussian processes (GPs) through the lens of the Neural Tangent Kernel (NTK): a recent development in understanding the training dynamics of wide neural networks (NNs). Previous work has shown that even in the infinite width limit, when NNs become GPs, there is no GP posterior interpretation to a deep ensemble trained with squared error loss. We introduce a simple modification to standard deep ensembles training, through addition of a computationally-tractable, randomised and untrainable function to each ensemble member, that enables a posterior i…
Equation Discovery with Bayesian Spike-and-Slab Priors and Efficient Kernels
The University of Utah（犹他大学）；The University of Sheffield（谢菲尔德大学）；University of California, Berkeley（加利福尼亚大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；International Computer Science Institute（国际计算机科学研究所）
Inference in popular nonparametric Bayesian models typically relies on sampling or other approximations. This paper presents a general methodology for constructing novel tractable nonparametric Bayesian methods by applying the kernel trick to inference in a parametric Bayesian model. For example, Gaussian process regression can be derived this way from Bayesian linear regression. Despite the success of the Gaussian process framework, the kernel trick is rarely explicitly considered in the Bayesian literature. In this paper, we aim to fill this gap and demonstrate the potential of applying the …
National University of Singapore（新加坡国立大学）；Shenzhen University（深圳大学）
This paper presents a variational Bayesian kernel selection (VBKS) algorithm for sparse Gaussian process regression (SGPR) models. In contrast to existing GP kernel selection algorithms that aim to select only one kernel with the highest model evidence, our proposed VBKS algorithm considers the kernel as a random variable and learns its belief from data such that the uncertainty of the kernel can be interpreted and exploited to avoid overconfident GP predictions. To achieve this, we represent the probabilistic kernel as an additional variational variable in a variational inference (VI) framewo…
In practical Bayesian optimization, we must often search over structures with differing numbers of parameters. For instance, we may wish to search over neural network architectures with an unknown number of layers. To relate performance data gathered for different architectures, we define a new kernel for conditional parameter spaces that explicitly includes information about which parameters are relevant in a given structure. We show that this kernel improves model quality and Bayesian optimization results over several simpler baseline kernels.
The University of Sydney（悉尼大学）；CSIRO（澳大利亚联邦科学与工业研究组织）；NVIDIA（英伟达）
In likelihood-free settings where likelihood evaluations are intractable, approximate Bayesian computation (ABC) addresses the formidable inference task to discover plausible parameters of simulation programs that explain the observations. However, they demand large quantities of simulation calls. Critically, hyperparameters that determine measures of simulation discrepancy crucially balance inference accuracy and sample efficiency, yet are difficult to tune. In this paper, we present kernel embedding likelihood-free inference (KELFI), a holistic framework that automatically learns model hyper…
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…
Bayesian Optimization (BO) has become a core method for solving expensive black-box optimization problems. While much research focussed on the choice of the acquisition function, we focus on online length-scale adaption and the choice of kernel function. Instead of choosing hyperparameters in view of maximum likelihood on past data, we propose to use the acquisition function to decide on hyperparameter adaptation more robustly and in view of the future optimization progress. Further, we propose a particular kernel function that includes non-stationarity and local anisotropy and thereby implici…
We propose a vector-valued regression problem whose solution is equivalent to the reproducing kernel Hilbert space (RKHS) embedding of the Bayesian posterior distribution. This equivalence provides a new understanding of kernel Bayesian inference. Moreover, the optimization problem induces a new regularization for the posterior embedding estimator, which is faster and has comparable performance to the squared regularization in kernel Bayes' rule. This regularization coincides with a former thresholding approach used in kernel POMDPs whose consistency remains to be established. Our theoretical …
University College London（伦敦大学学院）；University of Oxford（牛津大学）；University of Amsterdam（阿姆斯特丹大学）
Complicated generative models often result in a situation where computing the likelihood of observed data is intractable, while simulating from the conditional density given a parameter value is relatively easy. Approximate Bayesian Computation (ABC) is a paradigm that enables simulation-based posterior inference in such cases by measuring the similarity between simulated and observed data in terms of a chosen set of summary statistics. However, there is no general rule to construct sufficient summary statistics for complex models. Insufficient summary statistics will "leak" information, which…
Abstract We present PAC-Bayesian bounds for the generalisation error of the K-nearest-neighbour classifier (K-NN). This is achieved by casting the K-NN classifier into a kernel space framework in the limit of vanishing kernel bandwidth. We establish a relation between prior measures over the coefficients in the kernel expansion and the induced measure on the weight vectors in kernel space. Defining a sparse prior over the coefficients allows the application of a PAC-Bayesian folk theorem that leads to a generalisation bound that is a function of the number of redundant training examples: those…
Kernel methods are ubiquitous tools in machine learning. However, there is often little reason for the common practice of selecting a kernel a priori. Even if a universal approximating kernel is selected, the quality of the finite sample estimator may be greatly affected by the choice of kernel. Furthermore, when directly applying kernel methods, one typically needs to compute a $N \times N$ Gram matrix of pairwise kernel evaluations to work with a dataset of $N$ instances. The computation of this Gram matrix precludes the direct application of kernel methods on large datasets, and makes kerne…
We consider multi-objective optimization (MOO) of an unknown vector-valued function in the non-parametric Bayesian optimization (BO) setting, with the aim being to learn points on the Pareto front of the objectives. Most existing BO algorithms do not model the fact that the multiple objectives, or equivalently, tasks can share similarities, and even the few that do lack rigorous, finite-time regret guarantees that capture explicitly inter-task structure. In this work, we address this problem by modelling inter-task dependencies using a multi-task kernel and develop two novel BO algorithms base…
Current neural architecture search (NAS) strategies focus only on finding a single, good, architecture. They offer little insight into why a specific network is performing well, or how we should modify the architecture if we want further improvements. We propose a Bayesian optimisation (BO) approach for NAS that combines the Weisfeiler-Lehman graph kernel with a Gaussian process surrogate. Our method optimises the architecture in a highly data-efficient manner: it is capable of capturing the topological structures of the architectures and is scalable to large graphs, thus making the high-dimen…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Alan Turing Institute（艾伦·图灵研究所）
Clinical prognostic models derived from largescale healthcare data can inform critical diagnostic and therapeutic decisions. To enable off-theshelf usage of machine learning (ML) in prognostic research, we developed AUTOPROGNOSIS: a system for automating the design of predictive modeling pipelines tailored for clinical prognosis. AUTOPROGNOSIS optimizes ensembles of pipeline configurations efficiently using a novel batched Bayesian optimization (BO) algorithm that learns a low-dimensional decomposition of the pipelines high-dimensional hyperparameter space in concurrence with the BO procedure.…
Kernel methods are one of the mainstays of machine learning, but the problem of kernel learning remains challenging, with only a few heuristics and very little theory. This is of particular importance in methods based on estimation of kernel mean embeddings of probability measures. For characteristic kernels, which include most commonly used ones, the kernel mean embedding uniquely determines its probability measure, so it can be used to design a powerful statistical testing framework, which includes nonparametric two-sample and independence tests. In practice, however, the performance of thes…
We consider black box optimization of an unknown function in the nonparametric Gaussian process setting when the noise in the observed function values can be heavy tailed. This is in contrast to existing literature that typically assumes sub-Gaussian noise distributions for queries. Under the assumption that the unknown function belongs to the Reproducing Kernel Hilbert Space (RKHS) induced by a kernel, we first show that an adaptation of the well-known GP-UCB algorithm with reward truncation enjoys sublinear $\tilde{O}(T^{\frac{2 + α}{2(1+α)}})$ regret even with only the $(1+α)$-th moments, $…
In this paper, we propose an outlier-robust regularized kernel-based method for linear system identification. The unknown impulse response is modeled as a zero-mean Gaussian process whose covariance (kernel) is given by the recently proposed stable spline kernel, which encodes information on regularity and exponential stability. To build robustness to outliers, we model the measurement noise as realizations of independent Laplacian random variables. The identification problem is cast in a Bayesian framework, and solved by a new Markov Chain Monte Carlo (MCMC) scheme. In particular, exploiting …
National Institute for Fusion Science（日本核融合科学研究所）；The Institute of Statistical Mathematics（统计数理研究所）；Digital Strategy Center, Daicel Corporation（大赛璐公司数字战略中心）；The Graduate University for Advanced Studies (SOKENDAI)（高级研究大学院大学（SOKENDAI））；RIKEN（理化学研究所）
University of Oxford（牛津大学）；Carnegie Mellon University（卡内基梅隆大学）
The problem of sequentially maximizing the expectation of a function seeks to maximize the expected value of a function of interest without having direct control on its features. Instead, the distribution of such features depends on a given context and an action taken by an agent. In contrast to Bayesian optimization, the arguments of the function are not under agent's control, but are indirectly determined by the agent's action based on a given context. If the information of the features is to be included in the maximization problem, the full conditional distribution of such features, rather …
BOOST: A Data-Driven Framework for the Automated Joint Selection of Kernel and Acquisition Functions in Bayesian Optimization
Department of Chemical and Biomolecular Engineering, Korea Advanced Institute of Science and Technology（化学与生物分子工程系，韩国科学技术院）；Department of AX, Korea Advanced Institute of Science and Technology（AX系，韩国科学技术院）；Saudi Aramco-KAIST CO2 Management Center（沙特阿美-KAIST二氧化碳管理中心）
Kernel Bayesian inference is a principled approach to nonparametric inference in probabilistic graphical models, where probabilistic relationships between variables are learned from data in a nonparametric manner. Various algorithms of kernel Bayesian inference have been developed by combining kernelized basic probabilistic operations such as the kernel sum rule and kernel Bayes' rule. However, the current framework is fully nonparametric, and it does not allow a user to flexibly combine nonparametric and model-based inferences. This is inefficient when there are good probabilistic models (or …
Bayesian Optimization has become the reference method for the global optimization of black box, expensive and possibly noisy functions. Bayesian Op-timization learns a probabilistic model about the objective function, usually a Gaussian Process, and builds, depending on its mean and variance, an acquisition function whose optimizer yields the new evaluation point, leading to update the probabilistic surrogate model. Despite its sample efficiency, Bayesian Optimiza-tion does not scale well with the dimensions of the problem. The optimization of the acquisition function has received less attenti…
This works extends the Random Embedding Bayesian Optimization approach by integrating a warping of the high dimensional subspace within the covariance kernel. The proposed warping, that relies on elementary geometric considerations, allows mitigating the drawbacks of the high extrinsic dimensionality while avoiding the algorithm to evaluate points giving redundant information. It also alleviates constraints on bound selection for the embedded domain, thus improving the robustness, as illustrated with a test case with 25 variables and intrinsic dimension 6.
We extend kernelized matrix factorization with a fully Bayesian treatment and with an ability to work with multiple side information sources expressed as different kernels. Kernel functions have been introduced to matrix factorization to integrate side information about the rows and columns (e.g., objects and users in recommender systems), which is necessary for making out-of-matrix (i.e., cold start) predictions. We discuss specifically bipartite graph inference, where the output matrix is binary, but extensions to more general matrices are straightforward. We extend the state of the art in t…
The paper presents a novel approach to direct covariance function learning for Bayesian optimisation, with particular emphasis on experimental design problems where an existing corpus of condensed knowledge is present. The method presented borrows techniques from reproducing kernel Banach space theory (specifically m-kernels) and leverages them to convert (or re-weight) existing covariance functions into new, problem-specific covariance functions. The key advantage of this approach is that rather than relying on the user to manually select (with some hyperparameter tuning and experimentation) …
NTT Communication Science Laboratories（NTT通信科学实验室）
For Bayesian optimization (BO) on high-dimensional data with complex structure, neural network-based kernels for Gaussian processes (GPs) have been used to learn flexible surrogate functions by the high representation power of deep learning. However, existing methods train neural networks by maximizing the marginal likelihood, which do not directly improve the BO performance. In this paper, we propose a meta-learning method for BO with neural network-based kernels that minimizes the expected gap between the true optimum value and the best value found by BO. We model a policy, which takes the c…
In this paper, we study error bounds for {\em Bayesian quadrature} (BQ), with an emphasis on noisy settings, randomized algorithms, and average-case performance measures. We seek to approximate the integral of functions in a {\em Reproducing Kernel Hilbert Space} (RKHS), particularly focusing on the Matérn-$ν$ and squared exponential (SE) kernels, with samples from the function potentially being corrupted by Gaussian noise. We provide a two-step meta-algorithm that serves as a general tool for relating the average-case quadrature error with the $L^2$-function approximation error. When speciali…
