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.
Lastly, we use FW to establish a greedy algorithm. This enables us to utilize many useful variants of the FW algorithm such as a distributed variant @cite_25 . For details, see Appendix. However, compared with SP, we cannot evaluate the objective function directly, so we resort to other performance measures such as the log likelihood, accuracy, or RMSE in test datasets. For SP, we can directly evaluate KSD at each iteration.
@cite_25
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We extend the Frank-Wolfe (FW) optimization algorithm to solve constrained smooth convex-concave saddle point (SP) problems. Remarkably, the method only requires access to linear minimization oracles. Leveraging recent advances in FW optimization, we provide the first proof of convergence of a FW-type saddle point solver over polytopes, thereby partially answering a 30 year-old conjecture. We also survey other convergence results and highlight gaps in the theoretical underpinnings of FW-style algorithms. Motivating applications without known efficient alternatives are explored through structured prediction with combinatorial penalties as well as games over matching polytopes involving an exponential number of constraints. We present randomized approximation algorithms for multi-criteria Max-TSP. For Max-STSP with k > 1 objective functions, we obtain an approximation ratio of @math for arbitrarily small @math . For Max-ATSP with k objective functions, we obtain an approximation ratio of @math . Recently, there has been a renewed interest in the machine learning community for variants of a sparse greedy approximation procedure for concave optimization known as the Frank-Wolfe (FW) method. In particular, this procedure has been successfully applied to train large-scale instances of non-linear Support Vector Machines (SVMs). Specializing FW to SVM training has allowed to obtain efficient algorithms, but also important theoretical results, including convergence analysis of training algorithms and new characterizations of model sparsity.In this paper, we present and analyze a novel variant of the FW method based on a new way to perform away steps, a classic strategy used to accelerate the convergence of the basic FW procedure. Our formulation and analysis is focused on a general concave maximization problem on the simplex. However, the specialization of our algorithm to quadratic forms is strongly related to some classic methods in computational geometry, namely the Gilbert and MDM algorithms.On the theoretical side, we demonstrate that the method matches the guarantees in terms of convergence rate and number of iterations obtained by using classic away steps. In particular, the method enjoys a linear rate of convergence, a result that has been recently proved for MDM on quadratic forms.On the practical side, we provide experiments on several classification datasets, and evaluate the results using statistical tests. Experiments show that our method is faster than the FW method with classic away steps, and works well even in the cases in which classic away steps slow down the algorithm. Furthermore, these improvements are obtained without sacrificing the predictive accuracy of the obtained SVM model. We consider bottom-k sampling for a set X, picking a sample Sk(X) consisting of the k elements that are smallest according to a given hash function h. With this sample we can estimate the relative size f=|Y| |X| of any subset Y as |Sk(X) intersect Y| k. A standard application is the estimation of the Jaccard similarity f=|A intersect B| |A union B| between sets A and B. Given the bottom-k samples from A and B, we construct the bottom-k sample of their union as Sk(A union B)=Sk(Sk(A) union Sk(B)), and then the similarity is estimated as |Sk(A union B) intersect Sk(A) intersect Sk(B)| k. We show here that even if the hash function is only 2-independent, the expected relative error is O(1√(fk)). For fk=Omega(1) this is within a constant factor of the expected relative error with truly random hashing. For comparison, consider the classic approach of kxmin-wise where we use k hash independent functions h1,...,hk, storing the smallest element with each hash function. For kxmin-wise there is an at least constant bias with constant independence, and it is not reduced with larger k. Recently showed that bottom-k circumvents the bias if the hash function is 8-independent and k is sufficiently large. We get down to 2-independence for any k. Our result is based on a simply union bound, transferring generic concentration bounds for the hashing scheme to the bottom-k sample, e.g., getting stronger probability error bounds with higher independence. For weighted sets, we consider priority sampling which adapts efficiently to the concrete input weights, e.g., benefiting strongly from heavy-tailed input. This time, the analysis is much more involved, but again we show that generic concentration bounds can be applied.