sophisticated distributions, which are of interest by their own.

-   We discuss the application of the proposed SAGD framework in some important statistical and machine learning problems, including the expectation-maximization algorithm (EM algorithm), and the variational autoencoders (VAE). We show that SAGD is able to automate the EM algorithm for complex models and effectively remove the bias of VAE.

**Notation:** Throughout this article we adopt the following notation. Let $\mathbb{R}^{r}$ be the $r$-dimensional Euclidean space with the inner product $\langle\cdot,\cdot\rangle$ and norm $\Vert\cdot\Vert$. For matrices and higher-order tensors, $\Vert\cdot\Vert$ denotes the operator norm. Let $C\subset\mathbb{R}^{r}$ be a closed convex set, and then the notation $\mathcal{P}_{C}(x)$ means the projection of $x\in\mathbb{R}^{r}$ onto $C$. A mapping $\phi:\mathbb{R}^{r}\to\mathbb{R}^{s}$ is said to have polynomial growth if there exist a constant $C>0$ and an integer $m\ge0$ such that $\Vert\phi(x)\Vert\le C(1+\Vert x\Vert^{m})$ for all $x\in\mathbb{R}^{r}$. The notation $\nabla^{i}\phi$ is used to denote the $i$-th derivative of a multivariate function $\phi:\mathbb{R}^{r}\to\mathbb{R}$, and in particular $\nabla^{0}\phi\equiv\phi$. We use $\mathscr{C}_{poly}^{m}$ to denote the space of mapping $\phi$ such that $\phi$ is $m$-times differentiable, and $\phi$ and its derivatives have polynomial growth. A function $\phi:\mathbb{R}^{r}\to\mathbb{R}$ is said to be $L$-Lipschitz continuous if $|\phi(x)-\phi(y)|\le L\Vert x-y\Vert$ for all $x,y\in\mathbb{R}^{r}$.

Related Work
