iclr2020.md

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Generative Ratio Matching Networks

Akash Srivastava$^{\ast,1,2}$, Kai Xu$^{\ast,3}$, Michael U. Gutmann$^{3}$, Charles Sutton$^{3,4,5}$

$\ast$ equal contributions $^1$MIT-IBM Watson AI Lab $^2$IBM Research $^3$University of Edinburgh $^4$Google AI $^5$Alan Turing Institute

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Introduction

Adversarial Generative Models(GANs, MMD-GANs)

• ✅ can generate high-dimensional data such as natural images.
• ❌ are very difficult to train due to the saddle-point optimization problem

GRaM is a stable learning algorithm for implicit deep generative models that does not involve a saddle-point optimization problem and therefore is easy to train 🎉

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Overview

Two steps in the training loop

1. Learn a projection function ($f_\theta$)
   • that projects the data ($p_x$) and the model ($q_x$) densities into a low-dimensional manifold which,
   • preserves the difference between this pair of densities.
   • We use the ratio ($r(x) = \frac{p_x}{q_x}$) of the two densities as the measure of this difference.
2. Train the model ($G_\gamma$) in the low-dimensional manifold
   • using the Maximum Mean Discrepancy criterion as it work very well in low dimensional data.

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GRaM: the algorithm

1️⃣ Learn the manifold projection function $f_\theta(x)$ by minimising the squared difference between the pair of density ratios:

$$ \begin{aligned} D(\theta) &= \int q_x(x) \left( \frac{p_x(x)}{q_x(x)} - \frac{\bar{p}(f_\theta(x))}{\bar{q}(f_\theta(x))} \right)^2 dx \\ &= C - \mathrm{PD}(\bar{q}, \bar{p}) \end{aligned} $$

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GRaM: the algorithm (continued)

2️⃣ Train the generator $G\gamma$ by minimizing the empirical estimator of MMD in the low-dimensional manifold,

$$ \begin{aligned} \min_\gamma \Bigg[&\frac{1}{N^2}\sum_{i=1}^N\sum_{i'=1}^N k(f_\theta(x_i),f_\theta(x_{i'}))

• \frac{2}{NM}\sum_{i=1}^N\sum_{j=1}^M k(f_\theta(x_i), f_\theta(G_\gamma(z_j)))\ &\quad + \frac{1}{M^2}\sum_{j=1}^M\sum_{j'=1}^M k(f_\theta(G_\gamma(z_j)),f_\theta(G_\gamma(z_{j'}))) \Bigg ] \end{aligned} $$

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GRaM: the algorithm (continued)

Pearson divergence maximization and density ratio estimation

Monte Carlo approximation of PD, $$ \mathrm{PD}(\bar{q}, \bar{p}) \approx \frac{1}{N} \sum_{i=1}^N \left( \frac{\bar{p}(f_\theta(x_i))}{\bar{q}(f_\theta(x_i))} \right)^2 - 1 $$ where $x^q_i \sim q_x$.

We use a MMD based density ratio estimator (Sugiyama et al., 2012) under the fixed-design setup: $\hat{r}q = \mathbf{K}^{-1}{q,q} \mathbf{K}_{q,p}\mathbf{1}$.

• $\mathbf{K}{q,q}$ and $\mathbf{K}{q,p}$ are Gram matrices defined by $[\mathbf{K}{q,q}]{i,j} = k(f_\theta(x^q_i),f_\theta(x^q_j))$ and $[\mathbf{K}{q,p}]{i,j} = k(f_\theta(x^q_i),f_\theta(x^p_j)).$

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How do GRAM-nets compare to other methods

GAN	MMD-net	MMD-GAN	GRAM-net

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Illustration of GRAM training

	Blue: data Orange: samples Top: original Bottom: projected

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Evaluations: the stability of models

x-axis = noise dimension and y-axis = generator layer size

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Evaluations: the stability of models (continued)

x-axis = noise dimension and y-axis = generator layer size

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Quantitative results: sample quality

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Qualitative results: random samples

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The end

Extra slides to follow...

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Density ratio estimation via (infinite) moment matching

Maximum mean discrepancy

$$ \textrm{MMD}{\mathcal{F}}(p,q) = \sup{f\in\mathcal{F}} \left(\mathbb{E}_p \lbrack f(x) \rbrack - \mathbb{E}_q \lbrack f(x) \rbrack \right) $$

Gretton et al. (2012) show that it is sufficient to choose $\mathcal{F}$ to be a unit ball in an reproducing kernel Hilbert space $\mathcal{R}$ with a characteristic kernel $k$.

Using this definition of MMD, the density ratio estimator $r(x)$ can be derived as the solution to $$ \min_{r\in\mathcal{R}} \bigg \Vert \int k(x; .)p(x) dx - \int k(x; .)r(x)q(x) dx \bigg \Vert_{\mathcal{R}}^2. $$

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Generator training

The generator $G\gamma$ is trained by minimizing the empirical estimator of MMD

$$ \begin{aligned} \min_\gamma \Bigg[&\frac{1}{N^2}\sum_{i=1}^N\sum_{i'=1}^N k(f_\theta(x_i),f_\theta(x_{i'}))

• \frac{2}{NM}\sum_{i=1}^N\sum_{j=1}^M k(f_\theta(x_i), f_\theta(G_\gamma(z_j)))\ &\quad + \frac{1}{M^2}\sum_{j=1}^M\sum_{j'=1}^M k(f_\theta(G_\gamma(z_j)),f_\theta(G_\gamma(z_{j'}))) \Bigg ] \end{aligned} $$

with respect to it's parameters $\gamma$.