Tuan Anh Le

Semi-supervised model learning and amortized inference

27 September 2019

This is a note on Kingma et al.’s paper and Siddharth et al.’s paper.

Say we have unsupervised data and some supervised data (where is the marginal of which is not super important here).

Then let’s say we want to learn parameters of a generative model with a latent variable and sometimes-latent variable . We also want to learn an inference network .

To learn , we should maximize the following objective: \begin{align} \mathcal L(\theta, \phi) := \E_{p(x_u)}\left[\mathrm{ELBO}(x_u, \theta, \phi)\right] + \gamma \E_{p(x_s, y_s)}\left[\mathrm{ELBO}(x_s, y_s, \theta, \phi) + \alpha \log q_\phi(y_s \given x_s)\right], \label{eq:obj} \end{align} where \begin{align} \mathrm{ELBO}(x_u, \theta, \phi) &:= \E_{q_\phi(z, y \given x_u)} \left[\log \frac{p_\theta(z, y, x_u)}{q_\phi(z, y \given x_u)}\right], \text{and} \label{eq:elbo1}\\ \mathrm{ELBO}(x_s, y_s, \theta, \phi) &:= \E_{q_\phi(z\given x_s, y_s)} \left[\log \frac{p_\theta(z, y_s, x_s)}{q_\phi(z \given x_s, y_s)}\right]. \label{eq:elbo2} \end{align}

To see why maximizing \eqref{eq:obj} is a good thing to do, rewrite the ELBOs into the logp - KL form and rewrite the logq term as an expected KL: \begin{align} \mathrm{ELBO}(x_u, \theta, \phi) &= \log p_\theta(x_u) - \KL{q_\phi(z, y \given x_u)}{p_\theta(z, y \given x_u)}, \\ \mathrm{ELBO}(x_s, y_s, \theta, \phi) &= \log p_\theta(x_s, y_s) - \KL{q_\phi(z \given x_s, y_s)}{p_\theta(z \given x_s, y_s)}, \\ \E_{p(x_s, y_s)}\left[\log q_\phi(y_s \given x_s)\right] &= -\E_{p(x_s)p(y_s \given x_s)}\left[\log p(y_s \given x_s) - \log q_\phi(y_s \given x_s) - \log p(y_s \given x_s)\right] \nonumber\\ &= -\E_{p(x_s)}\left[\KL{p(y_s \given x_s)}{q_\phi(y_s \given x_s)}\right] - H(p(y_s \given x_s)), \end{align} where is the conditional entropy of .

This allows us to rewrite \eqref{eq:obj} as \begin{align} \mathcal L(\theta, \phi) = \color{blue}{\E_{p(x_u)}\left[\log p_\theta(x_u)\right]} \color{red}{-\E_{p(x_u)}\left[\KL{q_\phi(z, y \given x_u)}{p_\theta(z, y \given x_u)}\right]} + \color{blue}{\gamma\E_{p(x_s, y_s)}\left[\log p_\theta(x_s, y_s)\right]} \color{red}{-\gamma\E_{p(x_s, y_s)}\left[\KL{q_\phi(z \given x_s, y_s)}{p_\theta(z \given x_s, y_s)}\right]} - \color{red}{\gamma\alpha\E_{p(x_s)}\left[\KL{p(y_s \given x_s)}{q_\phi(y_s \given x_s)}\right]} - \gamma H(p(y_s \given x_s)). \end{align} Maximizing the blue terms leads to model learning and minimizing the red terms leads to amortized inference. The H term is not dependent on either .

To estimate gradients of \eqref{eq:obj}, we can sample from and (which are our datasets) and “move the gradients inside the expectations.” How do we estimate the ELBOs and the logq term? If the factorization of q is nice, it is easy (Kingma). Otherwise, we need to use self-normalized importance sampling (Siddharth).

Nice factorization of the inference network

Let’s say the inference network is factorized as \begin{align} q_\phi(z, y \given x) = q_\phi(y \given x) q_\phi(z \given y, x). \end{align} Gradients of both ELBOs in \eqref{eq:elbo1} and \eqref{eq:elbo2} are straightforward to estimate, as long as both and are reparameterizable. The logq term is also easy to evaluate. Kingma et al. use a model where is discrete, however the support is just elements so we can replace the expectation with a sum over ten terms.

Unfavourable factorization of the inference network

Let’s say the inference network is factorized as \begin{align} q_\phi(z, y \given x) = q_\phi(z \given x) q_\phi(y \given z, x). \label{eq:factorization2} \end{align} There are three problems:

  1. the denominator of \eqref{eq:elbo2}, , is difficult to evaluate,
  2. the expectation in \eqref{eq:elbo2} under is difficult to sample from, and
  3. the term is difficult to evaluate.

To solve problem 1, use the identity —where the terms in RHS are only implicitly defined through the factorization \eqref{eq:factorization2}—and rewrite the ELBO in \eqref{eq:elbo2} as \begin{align} \mathrm{ELBO}(x_s, y_s, \theta, \phi) &= \E_{q_\phi(z\given x_s, y_s)} \left[\log \frac{p_\theta(z, y_s, x_s)}{q_\phi(z, y_s \given x_s)}\right] + \log q_\phi(y_s \given x_s). \label{eq:elbo3} \end{align} The extra logq term can be lumped together with the logq term in \eqref{eq:obj} so that we have instead of .

To solve problem 2, we use self-normalized importance sampling where the proposal is and the unnormalized target distribution over is . This now allows us to estimate the expectation in \eqref{eq:elbo3} as \begin{align} \E_{q_\phi(z\given x_s, y_s)} \left[\log \frac{p_\theta(z, y_s, x_s)}{q_\phi(z, y_s \given x_s)}\right] \approx \sum_{k = 1}^K \bar w_k \log \frac{p_\theta(z_k, y_s, x_s)}{q_\phi(z_k, y_s \given x_s)}, \end{align} where , , and .

To solve problem 3, we use evaluate an IWAE-like lower bound on where we also use as the proposal and treat as the unnormalized target distribution corresponding to the normalized . This allows us to use previously sampled and weights in evaluating the stochastic lower bound \begin{align} \widehat{logq} := \log\left(\frac{1}{K}\sum_{k = 1}^K w_k\right) \end{align} whose expectation is a lower bound to .

This allows us to estimate the gradient of \eqref{eq:obj} as \begin{align} \hat g = \nabla_{\theta, \phi} \left(\mathrm{ELBO}(x_u, \theta, \phi) + \gamma \sum_{k = 1}^K \bar w_k \log \frac{p_\theta(z_k, y_s, x_s)}{q_\phi(z_k, y_s \given x_s)} + \gamma(\alpha + 1) \log\left(\frac{1}{K}\sum_{k = 1}^K w_k\right)\right). \end{align} All sampling is reparameterized.

All of this can be generalized to other bad factorizations of the inference network.