SRN – Markovian Score Climbing by Naesseth et al. from NeurIPS 2020

This Sunday Reading Notes post is about a recent article on variational inference (VI). 

In variational inference, we can approximate a posterior distribution {p(z \mid x)} by finding a distribution {q(z ; \lambda^{\star})} that is the `closest’ to {p(z \mid x)} among a collection of functions {Q = \{q(z;\lambda)\}}. Once a divergence between {p} and {q} has been chosen, we can rely on optimization algorithms such as stochastic gradient descent to find {\lambda^{\star}.}

The `exclusive’ Kullback-Leiber (KL) divergence has been popular in VI, due to the ease of working with an expectation with respect to the approximating distribution {q}. This article, however, considers the `inclusive’ KL

\displaystyle \mathbb{E}(p \| q) = \mathbb{E}_{p(z \mid x)} \left[ \log \frac{p(z \mid x)}{q(z ; \lambda)} \right].

Minimizing {\mathrm{KL}(p\| q)} is equivalent to minimizing the cross entropy {L_{\mathrm{KL}} = \mathbb{E}_{p(z \mid )}[ - \log q(z ; \lambda)],} whose gradient is
\displaystyle g_{\mathrm{KL}}(\lambda) := \nabla L_{KL}(\lambda) = \mathbb{E}_{p(z \mid x)}\left[- \nabla_{\lambda} \log q(z; \lambda)\right].

If we can find unbiased estimates of {g_{\mathrm{KL}}(\lambda)}, then with a Robbins-Monroe schedule {\{\varepsilon_k\}_{k=1}^{\infty}}, we can use stochastic gradient descent to approximate {\lambda^{\star}.}

This article propose Markovian Score Climbing (MSC) as another way to approximate {\lambda^{\star}}. Given an Markov kernel {M(z' \mid z;\lambda)} that leases the posterior distribution {p(z \mid x)} invariant, one step of the MSC iterations operates as follows.

(*) Sample {z_k \sim M( \cdot \mid z_{k-1}; \lambda_{k-1})}.
(*) Compute the gradient {\nabla \log q(z_k; \lambda_{k-1}).}
(*) Set {\lambda_{k} = \lambda_{k-1} + \varepsilon_k \nabla \log q(z_k; \lambda_{k-1}).}

The authors prove that {\lambda_k \to \lambda^{\star}} almost surely and illustrate it on the skew normal distribution. One advantage of MSC is that only one sample is required per {\lambda} update. Also, the Markov kernel {M(z' \mid z;\lambda)} provides a systematic way of incorporating information from current sample {z_k} and current parameter {\lambda_k}. As the authors point out, one example of such a proposal is a conditional SMC update [Section 2.4.3 of Andrieu et al., 2010].

While this article definitely provides a general purpose VI method, I am more intrigued by the MCMC samples {z_k}. What can we say about the samples {\{z_k\}}? Can we make use of them?

References:

Naesseth, C. A., Lindsten, F., & Blei, D. (2020). Markovian score climbing: Variational inference with KL (p|| q). arXiv preprint arXiv:2003.10374.

Andrieu, C., Doucet, A., & Holenstein, R. (2010). Particle markov chain monte carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(3), 269-342.

SRN – Distilling Importance Sampling

n this week’s Sunday Reading Notes, I discuss the paper Distilling Importance Sampling by Dennis Prangle. It aims to find the normalising flow $latex q$ that minimizes the distance to the target distribution $latex p$.

In this week’s Sunday Reading Notes, I discuss the paper Distilling Importance Sampling by Dennis Prangle. It aims to find the normalising flow q that minimizes the distance to the target distribution p. DIS uses the inclusive Kulbleck-Leiber divergence (KL divergence from the target distribution p to the approximating distribution q), in order to avoid over-concentration. The process of distilling refers to utilizing tempering distributions p_{\epsilon} that converges to p. In each step, DIS using stochastic gradient descent to minimize KL from p_{\epsilon} to q where the gradients are estimated with self-normalized importance sampling. 

Recently I came across many papers involving both Monte Carlo methods and neural networks. I learned many new concepts, among them, was normalising flow. A normalising flow is a bijective transformation from simple distributions such as normal or uniform. This paper considers non-volume preserving (NVP) normalising flow, particularly the coupling layer. It transforms input vector u into output vector v by fixing d coordinates and shifting and scaling the remaining coordinates by neural network outputs. The resulting family of densities can be sampled rapidly and their gradients are computable.

One question I have while reading this paper is about convergence guarantees. What does the distribution q^{\star} converge to? Numerical experiments suggest that DIS output is very close to the targeting distribution by comparing it to MCMC output. But as Prangle acknowledges, DIS output has less accurate tails than those of MCMC. Intuitively speaking DIS output should converge to the argmin of inclusive KL from target distribution to normalising flows. But does the distilling procedure guarantee convergence, despite the gradient estimates and tempering steps? I also keep thinking about what would be the SMC counterpart of DIS. 

Figure 3 from Prangle, 2019. Comparing MCMC output and DIS output for the M/G/1 queueing example.

Reference:

Prangle, D. (2019). Distilling importance sampling. arXiv preprint arXiv:1910.03632. https://arxiv.org/abs/1910.03632