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 study parallel and distributed Frank-Wolfe algorithms; the former on shared memory machines with mini-batching, and the latter in a delayed update framework. In both cases, we perform computations asynchronously whenever possible. We assume block-separable constraints as in Block-Coordinate Frank-Wolfe (BCFW) method (Lacoste-, 2013), but our analysis subsumes BCFW and reveals problemdependent quantities that govern the speedups of our methods over BCFW. A notable feature of our algorithms is that they do not depend on worst-case bounded delays, but only (mildly) on expected delays, making them robust to stragglers and faulty worker threads. We present experiments on structural SVM and Group Fused Lasso, and observe significant speedups over competing state-of-the-art (and synchronous) methods.