# Overview

## Original GAN

Goodfellow et al. (2014) proposed the following first GAN architecture:

• Prior: $$p_g(\mathbf{z}) = \mathcal{N}(\mathbf{0},\mathbf{I})$$
• Generator: $$G(\mathbf{z})$$
• Discriminator: $$D(\mathbf{x}) \in [0, 1]$$

The overall loss from their theory: $\mathcal{L} = \min_G \max_D \left\{ \mathbb{E}_{p_d(\mathbf{x})}\left[ \log D(\mathbf{x}) \right] + \mathbb{E}_{p_g(\mathbf{z})}\left[ \log(1 - D(G(\mathbf{z}))) \right] \right\}$ The discriminator loss (to maximize): $\mathcal{L}_D = \mathbb{E}_{p_d(\mathbf{x})} \left[ \log D(\mathbf{x}) \right] + \mathbb{E}_{p_g(\mathbf{z})}\left[ \log(1 - D(G(\mathbf{z}))) \right]$

The theoretical generator loss (to minimize): $\mathcal{L}_G = \mathbb{E}_{p_g(\mathbf{z})}\left[ \log(1 - D(G(\mathbf{z}))) \right]$ The actual generator loss (to maximize) in experiments, to avoid saturation in the early stage of training: $\mathcal{L}_G = \mathbb{E}_{p_g(\mathbf{z})}\left[ \log(D(G(\mathbf{z}))) \right]$

### GAN Training Algorithm

The most widely adopted GAN training algorithm, which is proposed in this work, alternates between training the discriminator and the generator. That is, in each training iteration:

• Repeat for n_critics iterations

• Sample $$\mathbf{x}^{(1)}, \dots, \mathbf{x}^{(b)}$$ from $$p_d(\mathbf{x})$$, $$\mathbf{z}^{(1)}, \dots, \mathbf{z}^{(b)}$$ from $$p_g(\mathbf{z})$$.

• Update the discriminator by: $\theta_D = \theta_D + \eta \, \nabla_{\theta_D}\left( \frac{1}{b}\sum_{i=1}^b \left[ \log D(\mathbf{x}^{(i)}) \right] + \frac{1}{b}\sum_{i=1}^b \left[ \log(1 - D(G(\mathbf{z}^{(i)}))) \right] \right)$

• Sample $$\mathbf{z}^{(1)}, \dots, \mathbf{z}^{(b)}$$ from $$p_g(\mathbf{z})$$.

• Update the generator by: $\theta_G = \theta_G - \eta\,\nabla_{\theta_G}\left( \frac{1}{b} \sum_{i=1}^b \log(1 - D(G(\mathbf{z}^{(i)}))) \right)$ or alternatively, $\theta_G = \theta_G + \eta\,\nabla_{\theta_G}\left( \frac{1}{b} \sum_{i=1}^b \log(D(G(\mathbf{z}^{(i)}))) \right)$

# References

Goodfellow, Ian, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. 2014. “Generative Adversarial Nets.” In Advances in Neural Information Processing Systems, 2672–80.

# Evaluation Metrics

## Likelihood

### Negative Log-Likelihood

The negative log-likelihood (NLL) for $$p_{\theta}(\mathbf{x})$$ is defined as:

\begin{align} \text{NLL} &= \mathbb{E}_{p_d(\mathbf{x})} \left[ -\log p_{\theta}(\mathbf{x}) \right] \end{align}

#### Find the Original NLL using Scaled Data

If the original data $$\mathbf{x}$$ is $$k$$-dimensional ,and is scaled by $$\frac{1}{\sigma}$$ at each of its dimensions, such that the data fed into the model is $$\tilde{\mathbf{x}} = \frac{1}{\sigma} \mathbf{x}$$, then: \begin{align} p_d(\mathbf{x}) &= \tilde{p}_d(\tilde{\mathbf{x}}) \left| \det\left( \frac{\mathrm{d}\tilde{\mathbf{x}}}{\mathrm{d}\mathbf{x}} \right) \right| = \tilde{p}_d(\tilde{\mathbf{x}})\left| \det\left( \frac{\mathrm{d}(\mathbf{x} / \sigma)}{\mathrm{d}\mathbf{x}} \right) \right| = \frac{1}{\sigma^k}\,\tilde{p}_d(\tilde{\mathbf{x}}) \\ p_{\theta}(\mathbf{x}) &= \frac{1}{\sigma^k}\, \tilde{p}_{\theta}(\tilde{\mathbf{x}}) \end{align} Thus the computed NLL for $$\mathbf{x}$$ and $$\tilde{\mathbf{x}}$$ has the following relationship: \begin{align} \text{NLL} &= \mathbb{E}_{p_d(\mathbf{x})} \left[ -\log p_{\theta}(\mathbf{x}) \right] \\ &= -\int p_d(\mathbf{x}) \log p_{\theta}(\mathbf{x})\,\mathrm{d}\mathbf{x} \\ &= -\int \frac{1}{\sigma^k}\,\tilde{p}_{d}(\tilde{\mathbf{x}}) \log \left( \frac{1}{\sigma^k}\, \tilde{p}_{\theta}(\tilde{\mathbf{x}}) \right)\left| \det\left( \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}\tilde{\mathbf{x}}} \right) \right|\mathrm{d}\tilde{\mathbf{x}} \\ &= -\int \tilde{p}_{d}(\tilde{\mathbf{x}}) \left[ \log \tilde{p}_{\theta}(\tilde{\mathbf{x}}) - k\log \sigma \right]\mathrm{d}\tilde{\mathbf{x}} \\ &= \mathbb{E}_{\tilde{p}_d(\mathbf{x})} \left[ -\log \tilde{p}_{\theta}(\mathbf{x}) \right] + k\log \sigma \\ &= \widetilde{NLL} + k\log\sigma \end{align}

#### Continuous NLL as an Upper-Bound of Discrete NLL

To train a continuous model upon discrete data (e.g., images), one may add a uniform noise to the data, and obtain an upper-bound of the discrete data NLL with the augmented data.

For pixel integer-valued $$\mathbf{x}$$ ranging from 0 to 255, adding a uniform noise $$\mathbf{u} \sim \mathcal{U}[0, 1)$$, such that $$\tilde{\mathbf{x}} = \mathbf{x} + \mathbf{u}$$, we have (Theis, Oord, and Bethge 2015): \begin{align} -\int \tilde{p}_d(\tilde{\mathbf{x}}) \log \tilde{p}_{\theta}(\tilde{\mathbf{x}}) \,\mathrm{d}\tilde{\mathbf{x}} &= -\sum_{\mathbf{x}} P_d(\mathbf{x}) \int \log \tilde{p}_{\theta}(\mathbf{x} + \mathbf{u}) \,\mathrm{d}\mathbf{u} \\ &\geq -\sum_{\mathbf{x}} P_d(\mathbf{x}) \log \int \tilde{p}_{\theta}(\mathbf{x} + \mathbf{u}) \,\mathrm{d}\mathbf{u} \\ \\ &= -\sum_{\mathbf{x}} P_d(\mathbf{x}) \log P_{\theta}(\mathbf{x}) \\ \end{align} where we define the probability of the true discrete data to be: $P_{\theta}(\mathbf{x}) = \int \tilde{p}_{\theta}(\mathbf{x} + \mathbf{u}) \,\mathrm{d}\mathbf{u}$ That is to say, the NLL of the augmented continuous random variable $$\tilde{\mathbf{x}}$$ can serve as an upper-bound as the true discrete data NLL.

## ROC and AUC

1. There is a one-to-one correspondence between the points on ROC and AUC curves.

2. A curve dominates the ROC (fpr-tpr curve) $$\Leftrightarrow$$ dominates the AUC (recall-precision curve).

3. Interpolation between two points $$A$$ and $$B$$:

1. On ROC: linear interpolation.

2. On AUC: $\left( \frac{TP_A + x}{\text{Total Pos}}, \frac{TP_A + x}{TP_A + x + FP_A + \frac{FP_B - FP_A}{TP_B - TP_A} x} \right)$

4. Compute the area: include the interpolation and use composite trapezoidal method.

1. Incorrect interpoluation for computing AUC-PR will cause over-estimate.
5. Optimize Area under ROC and AUC curves: not exactly the same. (especially when not one algorithm dominates the curve?)

# References

Davis, Jesse, and Mark Goadrich. 2006. “The Relationship Between Precision-Recall and ROC Curves.” In Proceedings of the 23rd International Conference on Machine Learning - ICML ’06, 233–40. Pittsburgh, Pennsylvania: ACM Press. https://doi.org/10.1145/1143844.1143874.

Theis, Lucas, Aäron van den Oord, and Matthias Bethge. 2015. “A Note on the Evaluation of Generative Models.” arXiv Preprint arXiv:1511.01844.

# Energy GAN

## Overview

Given the discriminator $$E_{\theta}(\mathbf{x})$$, a density function can be derived as: \begin{align} p_{\theta}(\mathbf{x}) &= \frac{1}{Z_{\theta}} e^{-E_{\theta}(\mathbf{x})} \\ Z_{\theta} &= \int e^{-E_{\theta}(\mathbf{x})}\,\mathrm{d}\mathbf{x} \end{align}

## Maximum Entropy Generators for Energy-Based Models

Kumar et al. (2019) proposed the following architecture:

• Prior: $$p_z(\mathbf{z})$$
• Generator: $$G_{\omega}(\mathbf{z})$$
• Discriminator: $$E_{\theta}(\mathbf{x}) \in (-\infty, \infty)$$, the energy function
• The density function: $$p_{\theta}(\mathbf{x}) = \frac{1}{Z_{\theta}} e^{-E_{\theta}(\mathbf{x})}$$
• Discriminator for the mutual information estimator: $$T_{\phi}(\mathbf{x},\mathbf{z}) \in (-\infty,\infty)$$

The discriminator loss (to minimize): \begin{align} \mathcal{L}_E &= \mathbb{E}_{p_d(\mathbf{x})}\left[ E_{\theta}(\mathbf{x}) \right] - \mathbb{E}_{p_G(\mathbf{x})}\left[ E_{\theta}(\mathbf{x}) \right] + \Omega \\ \Omega &= \lambda\,\mathbb{E}_{p_d(\mathbf{x})} \left[ \left\| \nabla_{\mathbf{x}} E_{\theta}(\mathbf{x}) \right\|^2 \right] \end{align} The generator loss (to minimize) \begin{align} \mathcal{L}_G &= \mathbb{E}_{p_z(\mathbf{z})}\left[ E_{\theta}(G_{\omega}(\mathbf{z})) \right] - H_{p_G}[X] \\ &= \mathbb{E}_{p_z(\mathbf{z})}\left[ E_{\theta}(G_{\omega}(\mathbf{z})) \right] - I_{p_G}(X;Z) \end{align} where $$H_{p_G}[X] = I_{p_G}(X;Z) + H(G_{\omega}(Z)|Z)$$, and $$H(G_{\omega}(Z)|Z) \equiv 0$$ since $$G_{\omega}(\mathbf{z})$$ is a deterministic mapper.

The mutual information $$I_{p_G}(X;Z)$$ is estimated via the Deep InfoMax estimator by Kumar et al. (2019), formulated as: \begin{align} I_{p_G}(X;Z) &= \mathbb{E}_{p_z(\mathbf{z})}\left[ -\text{sp}(-T_{\phi}(G_{\omega}(\mathbf{z}),\mathbf{z})) \right] - \mathbb{E}_{p_z(\mathbf{z})\times\tilde{p}_z(\tilde{\mathbf{z}})}\left[ \text{sp}(T_{\phi}(G_{\omega}(\mathbf{z}),\tilde{\mathbf{z}})) \right] \\ &= \mathbb{E}_{p_z(\mathbf{z})}\left[ \log \sigma(T_{\phi}(G_{\omega}(\mathbf{z}),\mathbf{z})) \right] - \mathbb{E}_{p_z(\mathbf{z})\times\tilde{p}_z(\tilde{\mathbf{z}})}\left[ \log\left(1 - \sigma(T_{\phi}(G_{\omega}(\mathbf{z}),\tilde{\mathbf{z}}))\right) \right] \end{align}

where $$\text{sp}(a) = \log(1 + e^a)$$ is the SoftPlus function, and $$\sigma(a) = \frac{1}{1 + e^{-a}}$$ is the sigmoid function.

### Training Algorithm

Kumar et al. (2019) proposed the following training algorithm:

• Repeat for n_critics Iterations

• Sample $$\mathbf{x}^{(1)}, \dots, \mathbf{x}^{(b)}$$ from $$p_d(\mathbf{x})$$, $$\mathbf{z}^{(1)}, \dots, \mathbf{z}^{(b)}$$ from $$p_z(\mathbf{z})$$.

• Obtain $$\tilde{\mathbf{x}}^{(i)} = G_{\omega}(\mathbf{z}^{(i)})$$.

• Calculate: $\mathcal{L}_E = \frac{1}{b} \left[ \sum_{i=1}^b E_{\theta}(\mathbf{x}^{(i)}) - \sum_{i=1}^b E_{\theta}(\tilde{\mathbf{x}}^{(i)}) + \lambda\sum_{i=1}^b \left\| \nabla_{\mathbf{x}} E_{\theta}(\mathbf{x}^{(i)}) \right\|^2 \right]$

• Gradient descent: $\theta^{t+1} = \theta^{t} - \eta \, \nabla_{\theta} \mathcal{L}_E$

• Sample $$\mathbf{z}^{(1)}, \dots, \mathbf{z}^{(b)}$$ from $$p_z(\mathbf{z})$$.

• Per-dimensional shuffle of $$\mathbf{z}$$, yielding $$\tilde{\mathbf{z}}^{(1)}, \dots, \tilde{\mathbf{z}}^{(b)}$$.

(Is this really better than re-sampling from the prior $$p_z(\mathbf{z})$$?)

• Obtain $$\tilde{\mathbf{x}}^{(i)} = G_{\omega}(\mathbf{z}^{(i)})$$.

• Calculate: \begin{align} \mathcal{L}_H &= \frac{1}{b} \sum_{i=1}^b \left[ \log \sigma(T_{\phi}(\tilde{\mathbf{x}}^{(i)},\mathbf{z}^{(i)})) - \log\left(1 - \sigma(T_{\phi}(\tilde{\mathbf{x}}^{(i)},\tilde{\mathbf{z}}^{(i)}))\right) \right] \\ \mathcal{L}_G &= \frac{1}{b} \sum_{i=1}^b \left[ E_{\theta}(\tilde{\mathbf{x}}^{(i)}) \right] - \mathcal{L}_H \end{align}

• Gradient descent: \begin{align} \omega^{t+1} &= \omega^{t} - \eta \, \nabla_{\omega} \mathcal{L}_G \\ \phi^{t+1} &= \phi^{t} - \eta \, \nabla_{\phi} \mathcal{L}_H \\ \end{align}

In summary, this algorithm:

• Minimize $$\theta$$ w.r.t. $$\mathcal{L}_E$$ =>
• Minimizes $$E_{\theta}$$ in the discriminator loss $$\mathcal{L}_E$$.
• Minimize $$\phi$$ w.r.t. $$L_H$$ =>
• Minimizes $$T_{\phi}$$ in the mutual information regularizer $$I_{p_G}(X;Z)$$.
• Minimize $$\omega$$ w.r.t. $$L_G$$ =>
• Maximizes $$G_{\omega}$$ in the mutual information regularizer $$I_{p_G}(X;Z)$$;
• Minimizes $$G_{\omega}$$ in the generator loss $$\mathbb{E}_{p_z(\mathbf{z})}\left[ E_{\theta}(G_{\omega}(\mathbf{z})) \right]$$.

### Latent Space MCMC

Kumar et al. (2019) also proposed an MCMC method to refine the $$\mathbf{z}$$ samples obtained from the prior $$p_z(\mathbf{z})$$, according to the energy function on $$\mathbf{z}$$, derived as: $E(\mathbf{z}) = E_{\theta}(G_{\omega}(\mathbf{z}))$ Then Metropolis-adjusted Langevin algorithm is adopted to sample $$\mathbf{z}$$:

• Langevin dynamics: $\mathbf{z}^{(t+1)} = \mathbf{z}^{(t)} - \alpha \frac{\partial E_{\theta}(G_{\omega}(\mathbf{z}^{(t)}))}{\partial \mathbf{z}^{(t)}} + \epsilon \sqrt{2 * \alpha}$ where $$\epsilon \sim \mathcal{N}(\mathbf{0},\mathbf{I})$$.

• Metropolis-Hastings Algorithm: \begin{align} r &= \min\left\{ 1, \frac{p(\mathbf{z}^{(t+1)})}{p(\mathbf{z}^{(t)})} \cdot \frac{q(\mathbf{z}^{(t)}|\mathbf{z}^{(t+1)})}{q(\mathbf{z}^{(t+1)}|\mathbf{z}^{(t)})} \right\} \\ p(\mathbf{z}^{(t)}) &\propto \exp\left\{ -E_{\theta}(G_{\omega}(\mathbf{z}^{(t)})) \right\} \\ q(\mathbf{z}^{(t+1)}|\mathbf{z}^{(t)}) &\propto \exp\left( -\frac{1}{4 \alpha}\left\| \mathbf{z}^{(t+1)} - \mathbf{z}^{(t)} + \alpha \frac{\partial E_{\theta}(G_{\omega}(\mathbf{z}^{(t)}))}{\partial \mathbf{z}^{(t)}} \right\|^2_2 \right) \end{align}

# References

Kumar, Rithesh, Anirudh Goyal, Aaron Courville, and Yoshua Bengio. 2019. “Maximum Entropy Generators for Energy-Based Models.” arXiv:1901.08508 [Cs, Stat], January. http://arxiv.org/abs/1901.08508.