Stein Variational Gradient Descent

The main challenge of the modern Bayesian inferences is about how to handle the intractable posterior. In many models, especially in deep neural networks, the posterior distribution consists of billions of parameters in a highly nonlinear and complex relationships. Traditional algorithms such as Markov chain Monte Carlo (MCMC) and variational inference (VI) are usually not suitable; the former lacks scalability and the latter lacks expressive power.

Stein variational gradient descent (SVGD), suggested by Liu & Wang, is a new algorithm for Bayesian models different from the MCMC and VI.

See the visualization of SVGD, © Chi Feng.

Preliminaries: MCMC and VI

Assume that the target distribution \(p\) supported on \(\mathbb{R}^d\) is smooth. Roughly speaking, traditional Bayesian algorithms can be summarized as followed:

  • MCMC samples \(x_i\) from a certain Markov chain of \(p\), and apply Monte Carlo approaches in inferences.
  • VI finds \(q\) from a well-known family of distribution \(\mathcal{Q}\) such that \(q\simeq p\), and use \(q\) in inference instead of \(p\).

Stein's Identity and Kernelized Stein Discrepancy

Consider another smooth distribution \(q\) supported on \(\mathbb{R}^d\). Before approximating \(q\) to \(p\), we must define some discrepancy metric between distributions to measure the quality of approximation. As the name suggest, SVGD use Stein discrepancy as the metric.

Before we move further, note that the following identity holds.

Proposition. Let \(\phi:\mathbb{R}^d\to\mathbb{R}^d\) is smooth. Define the Stein operator \(\mathcal{A}_q \phi(x):=\phi(x)\nabla_x\log q(x)^\top + \nabla_x\phi(x)\). Under some mild conditions, we have \(\mathbb{E}_{x\sim q}\left[\mathcal{A}_q\phi(x)\right]=0\).

This implies that \(\mathbb{E}_{x\sim q}\left[\mathcal{A}_p\phi(x)\right]=0\) relates to the discrepancy between \(p\) and \(q\), which leads to the following definition.

Definition. (Stein discrepancy) For some proper function class \(\mathcal{F}\), the Stein discrepancy is \[ \mathbb{S}(q, p)=\sup_{\phi\in\mathcal{F}}\left[\mathbb{E}_{x\sim q}\text{tr}\mathcal{A}_p\phi(x)\right]^2. \]

The supremum can be evaluated in closed form using a particular choice of \(\mathcal{F}\). For a reproducing kernel \(\kappa: \mathbb{R}^d\times\mathbb{R}^d\to\mathbb{R}\) and the corresponding RKHS \(H_\kappa\), define \(\mathcal{F}_P=\{f\in H_\kappa: \|f\|_{H_\kappa}\leq 1\}\). Then, the optimal solution has been shown to be \[ \mathbb{S}(q, p)=\|\phi^\ast\|_{H_\kappa}^2, \quad\text{where } \phi^\ast(\cdot)\propto\mathbb{E}_{x\sim q}\mathcal{A}_p\kappa(x, \cdot). \]

Stein Variational Gradient Descent

SVGD approximates \(p\) using samples from arbitrary reference distribution \(q_0\), followed by series of certain transforms. In other words, the goal is find \(T_1,\dots,T_n:\mathbb{R}^d\to\mathbb{R}^d\) such that \(T_n\circ\dots\circ T_1\circ q_0 \simeq p\).

To make the objective solvable, consider the following types of transforms only; \[ T(x)=x+\epsilon\phi(x), \] where \(\epsilon>0\) is a perturbation magnitude and \(\phi\) is a perturbation direction. Furthermore, assume that \(T\) is smooth and invertable, thus one-to-one. Then, the following theorem holds:

Theorem. Denote \(q_{[T]}\) as the pushforward distribution of \(q\) through \(T\). Then, \[ \nabla_{\epsilon=0}\text{KL}(q_{[T]}|p)=-\mathbb{E}_{x\sim q}\left[\text{tr}\mathcal{A}_p\phi(x)\right]. \]

The theorem suggests that the perturbation direction \(\phi\) determines the amount of decrease of KL divergence. Since the goal is approximate \(p\) via \(q\) and \(T\), one can apply the steepest descent method here by choosing \(\phi^\ast(\cdot)=\mathbb{E}_{x\sim q}\mathcal{A}_p\kappa(x, \cdot)\) as mentioned above.

In conclusion, the SVGD algorithm is stated as followed:

\[\begin{aligned} &\textbf{Input:}\text{ A target distribution with density function }p(x) \text{ and a set of initial particles }\{x_i^0\}_{i=1}^n\\ &\textbf{for }\text{iteration }\ell\textbf{ do}\\ &\hspace{5mm} x_i^{\ell+1}\gets x_i^\ell + \epsilon_\ell\hat\phi^\ast(x_i^\ell) \\ &\hspace{10mm}\text{where }\hat\phi^\ast(x) = \frac1n\sum_{j=1}^n\left[\kappa(x, x_j^\ell)\nabla_{x_j^\ell}\log p(x_j^\ell)+\nabla_{x_j^\ell}\kappa(x, x_j^\ell)\right]\\ &\textbf{end for}\\ &\textbf{Output:}\text{ A set of particles }\{x_i\}_{i=1}^n \text{ that approximates the target distribution} \end{aligned}\]

Intuitively, the update pushes the particles towards the modes of the target distribution via \(\nabla\log p\), while \(\nabla\kappa(x, x')\) prevents mode collapsing.

Furthermore, the algorithm can be executed more efficiently by using some techniques such as: