In Bayesian inference, the posterior distributions are difficult to obtain analytically for complex models such as neural networks. Variational inference usually uses a parametric distribution for approximation, from which we can easily draw samples. Recently discrete approximation by particles has attracted attention because of its high expression ability. An example is Stein variational gradient descent (SVGD), which iteratively optimizes particles. Although SVGD has been shown to be computationally efficient empirically, its theoretical properties have not been clarified yet and no finite sample bound of the convergence rate is known. Another example is the Stein points (SP) method, which minimizes kernelized Stein discrepancy directly. Although a finite sample bound is assured theoretically, SP is computationally inefficient empirically, especially in high-dimensional problems. In this paper, we propose a novel method named maximum mean discrepancy minimization by the Frank-Wolfe algorithm (MMD-FW), which minimizes MMD in a greedy way by the FW algorithm. Our method is computationally efficient empirically and we show that its finite sample convergence bound is in a linear order in finite dimensions.
In this paper, we are assuming that @math is the posterior distribution. On the other hand, if @math is a prior distribution, kernel herding @cite_18 or Bayesian quadrature @cite_27 , are useful. In those methods, @math 's are decided to directly minimize some criterions. For example, the kernel herding method @cite_18 @cite_1 minimizes MMD in a greedy way. The biggest difference from our method is that if @math is the prior distribution, we can sample many particles from @math and thus we can only choose the best particle that decreases the objective function maximally at each iteration. In MMD-FW, on the other hand, we cannot prepare the particles beforehand, and thus, we directly derive particles by gradient descent.
@cite_27 @cite_18 @cite_1
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Three Bayesian related approaches, namely, variational Bayesian (VB), minimum message length (MML) and Bayesian Ying-Yang (BYY) harmony learning, have been applied to automatically determining an appropriate number of components during learning Gaussian mixture model (GMM). This paper aims to provide a comparative investigation on these approaches with not only a Jeffreys prior but also a conjugate Dirichlet-Normal-Wishart (DNW) prior on GMM. In addition to adopting the existing algorithms either directly or with some modifications, the algorithm for VB with Jeffreys prior and the algorithm for BYY with DNW prior are developed in this paper to fill the missing gap. The performances of automatic model selection are evaluated through extensive experiments, with several empirical findings: 1) Considering priors merely on the mixing weights, each of three approaches makes biased mistakes, while considering priors on all the parameters of GMM makes each approach reduce its bias and also improve its performance. 2) As Jeffreys prior is replaced by the DNW prior, all the three approaches improve their performances. Moreover, Jeffreys prior makes MML slightly better than VB, while the DNW prior makes VB better than MML. 3) As the hyperparameters of DNW prior are further optimized by each of its own learning principle, BYY improves its performances while VB and MML deteriorate their performances when there are too many free hyper-parameters. Actually, VB and MML lack a good guide for optimizing the hyper-parameters of DNW prior. 4) BYY considerably outperforms both VB and MML for any type of priors and whether hyper-parameters are optimized. Being different from VB and MML that rely on appropriate priors to perform model selection, BYY does not highly depend on the type of priors. It has model selection ability even without priors and performs already very well with Jeffreys prior, and incrementally improves as Jeffreys prior is replaced by the DNW prior. Finally, all algorithms are applied on the Berkeley segmentation database of real world images. Again, BYY considerably outperforms both VB and MML, especially in detecting the objects of interest from a confusing background. The general perception is that kernel methods are not scalable, and neural nets are the methods of choice for nonlinear learning problems. Or have we simply not tried hard enough for kernel methods? Here we propose an approach that scales up kernel methods using a novel concept called "doubly stochastic functional gradients". Our approach relies on the fact that many kernel methods can be expressed as convex optimization problems, and we solve the problems by making two unbiased stochastic approximations to the functional gradient, one using random training points and another using random functions associated with the kernel, and then descending using this noisy functional gradient. We show that a function produced by this procedure after @math iterations converges to the optimal function in the reproducing kernel Hilbert space in rate @math , and achieves a generalization performance of @math . This doubly stochasticity also allows us to avoid keeping the support vectors and to implement the algorithm in a small memory footprint, which is linear in number of iterations and independent of data dimension. Our approach can readily scale kernel methods up to the regimes which are dominated by neural nets. We show that our method can achieve competitive performance to neural nets in datasets such as 8 million handwritten digits from MNIST, 2.3 million energy materials from MolecularSpace, and 1 million photos from ImageNet. This paper introduces a new directional multilevel algorithm for solving @math -body or @math -point problems with highly oscillatory kernels. These systems often result from the boundary integral formulations of scattering problems and are difficult due to the oscillatory nature of the kernel and the non-uniformity of the particle distribution. We address the problem by first proving that the interaction between a ball of radius @math and a well-separated region has an approximate low rank representation, as long as the well-separated region belongs to a cone with a spanning angle of @math and is at a distance which is at least @math away from from the ball. We then propose an efficient and accurate procedure which utilizes random sampling to generate such a separated, low rank representation. Based on the resulting representations, our new algorithm organizes the high frequency far field computation by a multidirectional and multiscale strategy to achieve maximum efficiency. The algorithm performs well on a large group of highly oscillatory kernels. Our algorithm is proved to have @math computational complexity for any given accuracy when the points are sampled from a two dimensional surface. We also provide numerical results to demonstrate these properties. Given samples from distributions p and q, a two-sample test determines whether to reject the null hypothesis that p = q, based on the value of a test statistic measuring the distance between the samples. One choice of test statistic is the maximum mean discrepancy (MMD), which is a distance between embeddings of the probability distributions in a reproducing kernel Hilbert space. The kernel used in obtaining these embeddings is critical in ensuring the test has high power, and correctly distinguishes unlike distributions with high probability. A means of parameter selection for the two-sample test based on the MMD is proposed. For a given test level (an upper bound on the probability of making a Type I error), the kernel is chosen so as to maximize the test power, and minimize the probability of making a Type II error. The test statistic, test threshold, and optimization over the kernel parameters are obtained with cost linear in the sample size. These properties make the kernel selection and test procedures suited to data streams, where the observations cannot all be stored in memory. In experiments, the new kernel selection approach yields a more powerful test than earlier kernel selection heuristics.