References

David Blei has many good write-ups / talks on this topic

• https://youtu.be/Dv86zdWjJKQ
  • Slides
  • "Review" Paper
• https://www.cs.princeton.edu/courses/archive/fall11/cos597C/lectures/variational-inference-i.pdf

Others

• https://ermongroup.github.io/cs228-notes/inference/variational/
• https://turing.ml/dev/docs/for-developers/variational_inference
• TTIC-31230 - Fundamentals of Deep Learning (ELBO)
• CS 285 (Sergey Levine)

Intro

Assumption:

• $x = x_{1:n}$: observations
• $z = z_{1:m}$: hidden variables

We want to model the posterior distribution (notice that this is a form of inference: estimating a hidden variable from observations):

$$p(z|x) = \frac{p(z, x)}{\int_z p(z, x)}$$

The posterior links the data and a model. In most of the interesting problems, calculating denominator is not tractable. $x$ is the evidence about $z$.

The main idea: pick a family of distributions over the latent variables parameterized with its own variation parameters:

$$q(z_{1:m}| \nu)$$

then find a setting of $\nu$ which makes it closest to the posterior of interest.


The closeness can be measured by Kullback-Leibler (KL) divergence:

$$\textrm{KL}(q\|p) = E_{Z\sim q} \big[ \log \frac{q(Z)}{p(Z|x)} \big]$$

We use the arguments ($q$ and $p$) in this order specifically to take expectation over $q$. If you flip the order (i.e., $\textrm{KL}(p\|q)$), that is called expectation propagation. It's a different kind of variational inference and it's more computationally expensive in general.

We cannot minimize this KL divergence directly (why??). But we can minimize a function that is equal to it up to a constant (ELBO).

The Evidence Lower Bound (ELBO)

$$\begin{align*} \textrm{KL}\big(q(z)\|p(z|x) \big) &= E_q \bigg[ \log \frac{q(Z)}{p(Z|x)} \bigg] \\ &= E_q [\log q(Z)] - E_q[\log p(Z|x)] \\ &= E_q [\log q(Z)] - E_q[\log p(Z, x) - \log p(x))] \\ &= E_q [\log q(Z)] - E_q[\log p(Z, x)] + \log p(x) \\ &= - (\underline{E_q[\log p(Z, x)] - E_q [\log q(Z)] }) + \log p(x) \\ \end{align*}$$

Notes:

• The last term $\log p(x)$ is independent of $q$, thus:
• Minimizing KL divergence is equiv to Maximizing ELBO

:::message ELBO derivation Using Jensen's inequality:

$$\begin{align} \log p(x) &= \log \int_z p(x, z) dz \\ &= \log \int_z q(z) \frac{p(x, z)}{q(z)} dz \\ &= \log\bigg( E_q[\frac{p(x, Z)}{q(z)}] \bigg) \\ &\geq \underline{E_q [\log p(x, Z)] - E_q[\log q(Z)]} ~~~(\because \text{Jensen's inequality}). \end{align}$$

Another derivation: forcibly extract KL divergence

$$\begin{align} \log p(x) &= E_q \big[\log p(x)\big] \\ &= E_q \big[\log \{ {\color{blue} p(x)} \cdot \frac{ {\color{blue} p(z|x)} q(z)}{p(z|x) {\color{green} q(z)}}\}\big] ~~~(\text{Stupid technique to make KL term})\\ &= E_q [\log \frac{{\color{blue} p(x, z)}} {\color{green} q(z)} + \log \frac{q(z)}{p(z|x)}] \\ &= E_q [\log p(x, z) - \log q(z)] + E_q [\log \frac{q(z)}{p(z|x)}] \\ &= \underline{E_q [\log p(x, z) - \log q(z)]} + \textrm{KL}\big( q(z)\| p(z|x) \big) \\ &\geq \underline{E_q [\log p(x, Z)] - E_q[\log q(Z)]} ~~~(\because \text{KL divergence is non-negative}). \end{align}$$

The left hand side is called evidence probability. Hence ELBO. :::

The difference between the ELBO and the KL divergence is the log normalizer --- which is what the ELBO bounds (???).

Variational Auto Encoder (Pretty much the same thing)

Latent variable models:

$$\begin{align*} P_{\Phi, \Theta}(z, x) &= P_\Phi(z) P_\Theta(x|z) \\ P_{\Phi, \Theta}(z | x) &= \frac{P_{\Phi, \Theta}(z, x)}{\int_z P_{\Phi, \Theta}(z, x)} \end{align*}$$

We have data population, so we want to estimate $\Phi$ and $\Theta$ based on it:

$$\Phi^*, \Theta^* = \textrm{argmin}_{\Phi, \Theta} E_{x \sim Pop} - \log P_{\Phi, \Theta}(x)$$

The problem is that we can't typically compute $P_{\Phi, \Theta}(x)$.

• $P_{\Phi, \Theta}(x) = \int_z P_\Phi (z) P_\Theta (x|z) dz$ doesn't work as the sum is too large
• The same sum but with importance sampling with $P_{\Phi, \Theta}(z|x)$ is a better idea but doesn't work: (why???)

Variational Bayes sidesteps this with a model $P_\Psi(z|x)$ that approximate $P_{\Phi, \Theta}(z|x)$.

The ELBO:

$$\begin{align*} \log P_{\Phi, \Theta}(x) &\geq E_{z \sim P_{\Psi}} \big[ \log P_{\Phi, \Theta}(z, x) \big] - E_{z \sim P_{\Psi}} \big[ \log P_\Psi(z|x) \big] \\ &= E_{z \sim P_{\Psi}} \big[ \log P_{\Theta}(x|z)P_{\Phi}(z) \big] - E_{z \sim P_{\Psi}} \big[ \log P_\Psi(z|x) \big] \\ &= E_{z \sim P_{\Psi}} \big[ - \bigg( \log \frac{P_\Psi(z|x)}{P_{\Phi}(z)} - \log P_{\Theta}(x|z) \bigg) \big] \end{align*}$$

Thus,

$$\Phi^*, \Theta^*, \Psi^* = \textrm{argmin}~E_{x \sim Pop,~z \sim P_\Psi} \big[ \log \frac{P_\Psi(z|x)}{P_{\Phi}(z)} - P_{\Theta}(x|z) \big]$$

Minor but important: we can't do gradient descent w.r.t. $\Psi$ as there's sampling procedure. We use re-parameterization trick to circumvent this.

• $P_\Phi(z)$: the prior
• $P_\Psi(z|x)$: the encoder
• $P_\Theta(x|z)$: the decoder
• $E[\log P_\Psi(z|x)/P_\Phi(z)]$: rate term, KL-divergence
• $E[- \log P_\Theta (x | z)]$: distortion, Conditional entropy

:::message Something more that are covered in TTIC31230

• EM (Expectation-Maximization) algorithm is indeed a specific instantiation of VAE! EM corresponds to minimiing the VAE objective:
  • First w.r.t. encoder ($\Psi$): Inference step -- E step
  • And then w.r.t. $\Phi$ and $\Theta$, while fixing $\Psi$: Update step -- M step
• VAE is exactly the same as Rate Distortion Autoencoder (RDA) :::

Rate-Distortion Autoencoders (mathematically the same as VAE)

Setting: Image compression where an image $x$ is compressed to $z$.

We assume a stochastic compression algorithm (encoder): $P_\text{enc}(z|x)$

• $H(z)$: The number of bits needed for the compressed file. This is the rate (bits / image) for transmitting compressed images
  • This is modeled with a prior model $P_\text{pri}(z)$
• $H(x|z)$: The number of additional bits needed to exactly recover $x$. This is a measure of the distortion of $x$
  • This is modeled with a decoder model $P_\text{dec}(x|z)$