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Gradient-based approximate inference methods, such as Stein variational gradient descent (SVGD), provide simple and general-purpose inference engines for differentiable continuous distributions. However, existing forms of SVGD cannot be directly applied to discrete distributions. In this work, we fill this gap by proposing a simple yet general framework that transforms discrete distributions to equivalent piecewise continuous distributions, on which the gradient-free SVGD is applied to perform efficient approximate inference. The empirical results show that our method outperforms traditional algorithms such as Gibbs sampling and discontinuous Hamiltonian Monte Carlo on various challenging benchmarks of discrete graphical models. We demonstrate that our method provides a promising tool for learning ensembles of binarized neural network (BNN), outperforming other widely used ensemble methods on learning binarized AlexNet on CIFAR-10 dataset. In addition, such transform can be straightforwardly employed in gradient-free kernelized Stein discrepancy to perform goodness-of-fit (GOF) test on discrete distributions. Our proposed method outperforms existing GOF test methods for intractable discrete distributions. 

Support Vector Machine (SVM) is originally proposed as a binary classification model, and it has already achieved great success in different applications. In reality, it is more often to solve a problem which has more than two classes. So, it is natural to extend SVM to a multi-class classifier. There have been many works proposed to construct a multi-class classifier based on binary SVM, such as one versus all strategy, one versus one strategy and Weston's multi-class SVM. One versus all strategy and one versus one strategy split the multi-class problem to multiple binary classification subproblems, and we need to train multiple binary classifiers. Weston's multi-class SVM is formed by ensuring risk constraints and imposing a specific regularization, like Frobenius norm. It is not derived by maximizing the margin between hyperplane and training data which is the motivation in SVM. In this paper, we propose a multi-class SVM model from the perspective of maximizing margin between training points and hyperplane, and analyze the relation between our model and other related methods. In the experiment, it shows that our model can get better or compared results when comparing with other related methods.