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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We propose a framework for analyzing and comparing distributions, which we use to construct statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS), and is called the maximum mean discrepancy (MMD).We present two distribution free tests based on large deviation bounds for the MMD, and a third test based on the asymptotic distribution of this statistic. The MMD can be computed in quadratic time, although efficient linear time approximations are available. Our statistic is an instance of an integral probability metric, and various classical metrics on distributions are obtained when alternative function classes are used in place of an RKHS. We apply our two-sample tests to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where they perform strongly. Excellent performance is also obtained when comparing distributions over graphs, for which these are the first such tests. We show that the herding procedure of Welling (2009b) takes exactly the form of a standard convex optimization algorithm-- namely a conditional gradient algorithm minimizing a quadratic moment discrepancy. This link enables us to invoke convergence results from convex optimization and to consider faster alternatives for the task of approximating integrals in a reproducing kernel Hilbert space. We study the behavior of the different variants through numerical simulations. Our experiments shed more light on the learning bias of herding: they indicate that while we can improve over herding on the task of approximating integrals, the original herding algorithm approaches more often the maximum entropy distribution.