Title: CHAINGAN: A SEQUENTIAL APPROACH TO GANS

Abstract: We propose a new architecture and training methodology for generative adversarial networks. Current approaches attempt to learn the transformation from a noise sample to a generated data sample in one shot. Our proposed generator architecture, called ChainGAN, uses a two-step process. It first attempts to transform a noise vector into a crude sample, similar to a traditional generator. Next, a chain of networks, called editors, attempt to sequentially enhance this sample. We train each of these units independently, instead of with end-to-end backpropagation on the entire chain. Our model is robust, efficient, and flexible as we can apply it to various network architectures. We provide rationale for our choices and experimentally evaluate our model, achieving competitive results on several datasets.

1. INTRODUCTION:

Generative Adversarial Networks (GANs) are a class of generative models that have shown promising results in multiple domains, including image generation, text generation, and style-transfer (Goodfellow et al., 2014; Zhang et al., 2017; Zhu et al., 2017) . GANs use a game-theoretic framework pitting two agents -a discriminator and a generator -against one another to learn a data distribution P d . The generator attempts to transform a noise vector z ∼ N into a sample x that resembles the training data. Meanwhile, the discriminator acts as a classifier trying to distinguish between real samples, x ∼ P d , and generated ones, x ∼ P g . Equation 1is the original minimax objective proposed by Goodfellow et al. (2014) , which can also be viewed as minimizing the Jensen-Shannon divergence between two distributions, P d and P g . These agents are most often implemented as neural networks and Goodfellow et al. (2014) proved that, under suitable conditions, the generator distribution P g will match the data distribution, P d .

min G max D E x∼P d [log(D(x)] + E z∼N [1 -log(D(G(z))]

(1)

Despite their success, GANs are notoriously hard to train and many variants have been proposed to tackle shortcomings. These shortcomings include mode collapse (i.e., the tendency for the generator to produce samples lying on a few modes instead of over the whole data space), training instability, and convergence properties. In most of these modified GANs, however, the generator is realized as a single network that transforms noise into a generated data sample.

In this paper, we propose a sequential generator architecture composed of a number of networks. In the first step, a traditional generator (in this paper called the base generator) is used to transform a noise sample z into a generated sample x0 . The sample is then fed into a network (called editor) which attempts to enhance its quality, as judged by the discriminator. We have a number of these editor networks connected in a chain, such that the output of one is used as the input of the next. Together, the base generator and the editors ahead of it are called the chain generator. Any preexisting generator architecture can be used for the base generator, making it quite flexible. For the editors, architectures designed for sample enhancement work well. The whole chain generator network can be quite small; in fact, it is smaller than most existing generator networks, yet produces equivalent or better results (Section 4.2).

Each network in the chain generator is trained independently based on scores they receive from the discriminator. This is done instead of an end-to-end backpropagation through the whole base generator + editor chain (Section 3.3). This allows each network in the chain to get direct feedback on its output. A similar approach has been quite successful in classification tasks. Huang et al. (2017) showed that in a classification architecture composed of a chain of ResNet blocks, training each of these blocks independently, produces better results than an end-to-end training scheme (He et al., 2015) .

In this paper, we propose a sequential generator architecture and a corresponding training regime for GANs. The benefits of this approach include:

• Better performance on some datasets when compared with relevant models (Table 1 )

• Efficient in memory and in the number of parameters (Section 7.1).

• Base generator itself performs better when trained alongside editors (Figure 4a ).

• It is flexible and can be adapted to existing network architectures 2 BACKGROUND AND RELEVANT WORK

2.1. GAN OBJECTIVES:

The formulation by Goodfellow et al. (2014) (Equation 1) is equivalent to minimizing the Jensen-Shannon divergence between the data distribution P d and the generated distribution P g . However, Arjovsky et al. (2017) showed that, in practice, this leads to a loss that is potentially not continuous, causing training difficulty. To address this, they presented the Wasserstein GAN (WGAN) which minimized the Wasserstein (or 'Earth mover') distance between two distributions. This is a weaker distance than Jensen-Shannon and Arjovsky et al. (2017) showed that, under mild conditions, it resulted in a function that was continuous everywhere and differentiable almost everywhere. The WGAN objective can be expressed as:

EQUATION

where D is the set of all 1-Lipchitz functions. D no longer discerns between real and fake samples but is used to approximate the Wasserstein distance between the two distributions (it is here called a 'critic'). G in turn looks to minimize this approximated distance. Several approaches have been proposed to enforce the Lipchitz constraint on the critic, the most notable being gradient penalty (Gulrajani et al., 2017) . This method penalizes the norm of the critic's gradient with respect to its input and was shown to have stable training behavior on various architectures. Due to its strong theoretical and empirical results, we use a modified version of this objective for our work. The objective for WGAN with gradient penalty can be expressed as:

EQUATION

where P x samples uniformly along straight lines between pair of points sampled from the data distribution P d and the generator distribution P g Gulrajani et al. (2017) . λ is the gradient penalty coefficient.

2.2. GAN ARCHITECTURES:

There have been a diverse range of architectures proposed for the generator and discriminator networks. DCGAN has shown promising results by using a deep convolutional architecture for both the generator and discriminator (Radford et al., 2015) . However, this uses a single large generator network instead of the sequential architecture that we have developed. Our architecture provides similar results as DCGAN while being more efficient, both in network parameters (Section 4.2) and in memory. EnhanceGAN, proposed by Deng et al. (2017) , successfully use GANs for unsupervised image enhancement. In this model, the generator is given an image, and it tries to improve its aesthetic quality. The editor part of our network is similar to this as they are responsible for enhancing images in an unsupervised fashion.

Sequential generator architectures have been proposed for tasks where the underlying data has some inherent sequential structure. TextGAN uses an LSTM architecture for the generator to generate a sentence, one word at a time (Zhang et al., 2017) . Ghosh et al. (2016) used an RNN-style discriminator and generator to solve abstract reasoning problems. This attempts to predict the next logical diagram, given a sequence of past diagrams. C-RNN-GAN is another example that uses GANs on continuous sequential data like music (Mogren, 2016) . Unlike these works, we do not assume the data is inherently sequential nor do we produce a single sample by generating a number of smaller units; rather, the output of any network in the chain generator is a whole sample, and the succeeding ones try to enhance it. Moreover, each editor network has its own set of parameters, unlike LSTMs or RNNs where these are shared. This allows each editor to be independent and makes our model efficient as we do not have to backpropagate across time.

The concept of using multiple generators has been proposed previously. Hoang et al. (2017) used multiple generators which share all but the last layer, trained alongside a single discriminator and a classifier. Their work has shown resistance to mode collapse and achieves state-of-the-art results on some datasets. Their work was influenced by MIX+GAN, which showed strong theoretical and empirical results by using a mixture of GANs instead of a single generator and discriminator (Arora et al., 2017) . Notably, they showed that using a mixture of GANs guarantees existence of approximate equilibrium, leading to stable training. However, a drawback of their work was the large memory cost of using multiple GANs -they suggest a mixture size of 5. Our work is similar to these in that we use multiple generators. The output of each editor is a different transformation on the noise vector z and can be viewed as originating from different generators. However, the generators in our model are not independent; rather, they are connected in a chain and meant to build upon the previous ones.

3. MODEL:



3.1. MOTIVATION:

A clear advantage of a sequential approach is that each output in the sequence is conditioned on the previous one, simplifying the problem of generating a complicated sample by mapping them to a sequence of simpler problems. A sequential approach also means that loss or other latent information in the intermediate steps can be used to guide the training process, which is not possible in one-shot generation.

As such, we propose a sequential generator architecture and an associated training objective for GANs. We observe that the challenge of generating a data sample from noise can be divided into two components: generating a crude representation of what that sample might be, and then iteratively refining it (Figure 1 ). A generator trained on images may first generate a rough picture of a car or a boat, e.g., but it can then be successively edited and enhanced. Intuitively, this is akin to the editorial process wherein a writer produces a rough sketch of an article, which then passes through a number of editors before the finished copy is published. 

3.2. ARCHITECTURE:

We use a WGAN with gradient penalty as the starting point for our architecture due to its theoretical robustness and training stability (Section 2.1). However, instead of using a single generator, we use the chain generator consisting of a base generator, denoted by G b and several editors, denoted by E i . The base generator takes in z ∼ N , and produces a sample x0 attempting to fool the critic, as in the traditional GAN architecture. The editors are connected in a chain where the i th one takes as input a sample x(i-1) and tries to produce an enhanced version as the output, xi (Figure 1 ). Thus, the output of each editor is conditioned on the previous one. Our network can be expressed recursively as:

EQUATION

Each of the intermediate samples in the chain, xi , is an attempt to fool the critic. In the WGAN formulation, the critic is used to approximate the Wasserstein distance between the data distribution P d and the generated distribution P g , where P g is obtained by transforming z ∼ N using the network G. In our formulation, each output xi is obtained by the following transformation on z:

E i (E i-1 (. . . E 1 (G b (z))))

). As such, x i can be viewed as being sampled from distribution P g i and a critic is needed to approximate the Wasserstein distance between P d and P gi (see Section 7 for a detailed explanation). For a chain of n editors plus the base generator, n + 1 critics are needed. For the i th editor and critic, the objective can be expressed as:

L i = E x∼P d [D i (x)] -E xi ∼Pg i [D i (x i )] + λ E x∼P xi [(||∇ xi D i ( x)|| 2 -1) 2 ]

(5)

3.3. TRAINING:

Each network in the chain generator, whether the base generator or any of the editors, is trained independently, based on critic scores for that network's output (Section 7.1). At every iteration, we randomly choose one of them to update, and use their output to also train the corresponding critic, D i . This means that the goal for any network in the chain generator is to ensure its outputs are as realistic as possible. Without loss of generality, we can express the training rule for the chain generator using canonical gradient descent with learning rate α as:

EQUATION

Our training regime is in contrast to performing end-to-end backpropagation (from the last editor all the way to the base generator). The motivation for this is three-fold. First, we want each editor to do its best in generating realistic samples and receive direct feedback on its performance. This way, we can utilize the intermediate samples generated in the chain to guide the training process. So editor i can start with the distribution P gi-1 and use the critic feedback to construct a P gi that is closer to P d . Another benefit of this is that we can possibly use fewer editors during evaluation. For example, we can train with the chain generator composed of G b , E 1 , E 2 , . . . , E n , but in evaluation observe that after editor k (k ≤ n), the quality of the samples saturate or do not improve significantly (Figure 2 ). As such, during evaluation we can cutoff the chain generator at editor k, and use this smaller chain for downstream tasks, making it more compact. Performing end-to-end backpropagation would mean that only the samples generated by the last editor would be viewed by the critic, and the goal of the chain generator would be to generate good samples by the last editor. That network could no longer be made compact at evaluation, and the number of editors to use would be an additional hyper-parameter.

The last rationale behind our training approach is efficiency. Observe that in the canonical GAN architecture, the gradient computation when updating the generator is ∂ ∂θ G D(G(z)), which involves traversing both the generator and critic graphs, which can be expensive for large generator networks. Performing end-to-end backpropagation on our chain generator would cause a similar issue as the entire chain would need to be retained alongside the critic. However, training each of these networks separately means that the gradient computation for an update step becomes ∂ ∂θ E i D i (E i (x i-1 )), so only a single editor or base generator and its critic need to be traversed and loaded on the GPU (Section 7.1). Since these networks are quite small, our model can be made very efficient. Require: initial critic parameters w 0 and initial chain generator parameters {θ 0 0 . . . θ 0 n }, where θ i are the parameters for the i th network in the chain generator. Require: gradient penalty coefficient λ, batch size m, critic iterations per generator iteration n critic , and Adam hyperparametes α, β 1 , β 2 , while θ 0 , ...θ n has not converged do i ∼ Cat(0, n) ; for j = 1, ..., n critic do for k = 1, ..., m do Sample real data, x ∼ P d , and noise, z ∼ N (0, 1); Sample ∼ U(0, 1);

G b E 1 E 2 E 3 E 4 E 5

xi ←E i (E i-1 (...G b (z))); x ← x + (1 -)x i ; L (k) ← D i (x) -D i ( x) + λ(||∇ xD i ( x)|| 2 -1) 2 ; end for w t+1 ← Adam(∇ w t 1 m m k=1 L (k) , w t , α, β 1 , β 2 ); end for Sample a batch of {z} m k=1 ∼ N ; { xi } m k=1 ← {E i (E i-1 (...G b (z k )))} m k=1 ; θ t+1 i ← Adam(∇ θ t i 1 m m k=1 -D i (x i ), θ t i , α, β 1 , β 2 ); end while

4. EXPERIMENTS:

We deploy our model on several datasets that are commonly used to evaluate GANs, including: (i) MNIST -28x28 gray-scale images of hand-written digits (LeCun & Cortes, 2010); (ii) CIFAR10 -32x32 colour images across 10 classes (Krizhevsky et al.) ; (iii) CelebA face dataset -214x110 colour images of celebrity faces (Liu et al., 2015) . For the CIFAR10 dataset, we compute the inception score and compare it against several GAN variants (Salimans et al., 2016) . Our goal is to show that our model can be used with existing generator architectures to provide competitive scores while using a smaller network.

We implement multiple critics by sharing all but the last layer (Section 7.3). To verify our results were not due to this, we also experiment with using a single critic, which did not degrade results. Each editor is composed of a few residual blocks since Deng et al. (2017) showed that such an architecture performs well for unsupervised image enhancement tasks (see Section 7.4 for details). We use this editor architecture for all our experiments and have five such networks connected in a chain. A key goal when designing our networks was efficiency; in fact, all implementations of our chain generator have fewer parameters than comparable models.

4.1. MNIST:

G b E 1 E 2 E 3 E 4 E 5

Epoch 10

Epoch 22

Epoch 54

Epoch 98 Our base generator and the critics use a convolutional architecture similar to DCGAN. The entire chain generator network is quite small, using approximately a third of the number of parameters as the DCGAN generator. Early in the training, the base generator's samples are crude, but the editors work to enhance it significantly (Figure 3 ). As the training progresses, the base generator's samples increase in quality and the editor's effect becomes more subtle. Visually, it appears that the editors collaborate and build upon each other; the class and overall structure of the image remain the same throughout the edit-process. We experiment with different architectures for the base generator. These include variants of DC-GAN and a ResNet architecture proposed by Gulrajani et al. (2017) . We add a chain of five editors to these base generators and train them according to Algorithm 1. We compare them against the original (and larger) versions of these architectures. For original architectures, we use the tuned hyper-parameters recommended in the pertinent paper/implementation. For the ChainGAN variants, we use default parameters instead. We wanted to evaluate the robustness of our model and believe that better results than ours can be obtained by tuning hyper-parameters. All models compared are trained to the same number of epochs using the WGAN formulation with a gradient penalty term (WGAN+GP). Table 1 contains the results of these experiments.

4.2. CIFAR10:

Based on inception scores, our DCGAN variant of the chain generator using multiple critics (model 2 and 4) was able to outperform the base DCGAN model (model 6) while using fewer parameters.

In a related experiment, we train the base generator by itself (i.e., without editors) against the critic (model 1), to note the effect of the editors on the base generator. In this case, the base generator exhibits better inception scores when trained alongside editors than without them, as shown in Fig. 4a . This suggests that the sequential model not only improves the overall sample quality, but may also improve base generator training. To discern the role of our training regime, we run an experiment that uses end-to-end backpropagation on a chain generator with 5 editors (model 5), noting a significant drop in performance. Real Data 11.24±0.12 WGAN (Arjovsky et al., 2017) 3.82±0.06 MIX + WGAN (Arora et al., 2017) 4.04±0.07 DCGAN -WGAN+GP * 5.18±0.05 ALI (Dumoulin et al., 2016) 5.35±0.05 BEGAN (Berthelot et al., 2017) 5.62 Small DCGAN (WGAN+GP) + Editors * 6.05±0.04 Small ResNet (WGAN+GP) + Editors * 6.70±0.02 ResNet -WGAN+GP * 6.86±0.04 EGAN-Ent-VI (Dai et al., 2017) 7.07±0.10 MGAN (Hoang et al., 2017) 8.33±0.10

The variant of ChainGAN using a ResNet base generator also uses a smaller network (model 8), while achieving comparable scores. While performing slightly worse than base on the standard inception score (IV2-TF), we outperform the base model in the IV3-Torch inception model used in Barratt & Sharma (2018) . We also see the same effect as above: results from the base generator are better when in chain and trained alongside editors (model 8) rather than being trained by itself (model 7). Figure 4b shows random samples from Editor 3 of the Small ResNet + Editors model.

Table 2 compares our models against different GAN variants. Entries denoted by * refer to models that we implemented (all using pyTorch). The ResNet -WGAN+GP is the model proposed by Gulrajani et al. (2017) and we closely followed all their recommendations yet were unable to achieve their stated inception score of 7.86.

4.3. CELEBA:

We also trained our model on the celebrity face dataset (Liu et al., 2015) . We do not center-crop the images, which would focus only on the face; rather, we downsample but maintain the whole image. As in the previous experiments, the editors build upon one another rather than working at cross purposes. Section 7.2 shows some samples generated by our model.

5. DISCUSSION AND FUTURE WORK:

In our experiments, we were able to successfully train a sequential GAN architecture to achieve competitive results. Our model uses a smaller network and the training regime is also more efficient than the alternatives (Section 7.1). We trained with 5 editors, but in evaluation needed k ≤ 5 editors to achieve the best scores; this suggests that we can use a smaller network during evaluation or for downstream tasks. Our results yielded diverse samples and we do not notice any visual signs of mode collapse (Figure 4b ). To ensure that our generator architecture and training scheme are responsible for the performance, we run experiments with a single critic and see that results do not degrade (see model 3 in Table 1 ); but with end-to-end backpropagation through the chain generator results degraded significantly (see model 5 in Table 1 ).

We also experimented with inducing a game between the editors themselves, using their loss functions. In one formulation, E i 's loss was the difference of the critic scores between itself and the previous editor scaled by λ, L i = D i ( xi )-λD i-1 (x i-1 ). As such, each editor would compete against the previous one and try to outperform it. In another formulation, each editor's loss was a discounted (by λ) sum of all the critic scores ahead of it,

L i = D i ( xi ) + λD i+1 (x i+1 ) + ... + λ n-i D n (x n ).

This would force earlier editors to take on more responsibility since they influence those further ahead. Although promising, these methods often led to unstable training. Future work could focus on training stability with these approaches as there is merit in exploring strategies between editors.

5.1. OPEN CHALLENGES:

The idea of splitting sample generation into a multi-step process is very flexible and can be extended in a number of ways. Currently, editors are completely unsupervised, but this can be extended to make each editor responsible for a different feature. One approach might build upon InfoGAN, which provides the generator with several latent variables, along with noise, and tries to maximize the mutual information between the generated samples and the latent variables (Chen et al., 2016) . Using a ChainGAN approach, we could use the base generator to simply generate a sample, and each editor would be responsible for a different latent variable, with a goal of maximizing mutual information between that variable and its output.

ChainGAN may also prove effective in text generation -a domain with which GANs struggle. In this context, the base generator could be responsible for generating a bag of words, and the editors would be responsible for reordering them, e.g., for coherence. Editors could similarly be used change the tone or sentiment of text appropriately.

6. CONCLUSION:

In this paper, we present a new sequential approach to GANs. Instead of attempting to generate a sample from noise in one shot, our model first generates a crude sample via the base generator and then successively enhances it through a chain of networks called editors. We use multiple critics, each corresponding to a different network in the chain generator. Our model is efficient and we successfully trained on a number of datasets to achieve competitive results, outperforming some existing models. Furthermore, our scheme is very flexible and there are several avenues for extension. 

where (P d , P g ) is the set of all distributions whose marginals are P g and P d . Arjovsky et al. (2017) show that using the Kantorovich-Rubenstein duality and a slight reframing, this objective can be expressed as:

EQUATION

Here, {f w } w W is the set of all K-Lipchitz functions for some K. We can use a neural network D to approximate the function f w . Now that we have an approximation for the Wasserstein distance between P d and P g , the generator network seeks to minimize this, giving us the WGAN formulation of: min

EQUATION

In our ChainGAN model, the output of each editor can be seen as a different transformation of the noise vector z ∼ N . For example, Editor i's output xi is obtained by the following transformation on z: E i (E i-1 (. . . E 1 (G b (z))))). Thus, to approximate the Wasserstein distance between P d and P gi , we use a network D i . For a base generator and n editors, we have n+1 critics.

:



Figures and Tables:

Figure 1: Chain generator model, with noise z, base generator G b , and editors, E i .

Figure 2: In this example, the first few editors significantly enhance the image of a car, but after Editor 3, they tend to more subtle.

Figure 3: Randomly selected MNIST examples from our DCGAN with editors model at different epochs.

Figure 4: (a) Inception scores across epochs of model 4 vs model 1 (b) Randomly selected CIFAR-10 examples from editor 3 of Model 8.

Figure 5: Whereas backpropagation for traditional GANs (top) require retaining the entire generator graph, ChainGAN requires the graph of one randomly selected editor per iteration (bottom), which has a fraction of the number of parameters. Orange boxes represent the parameters involved in the backpropagation of loss gradients.

Inception scores on unlabelled CIFAR with and without the chain generator approach trained with WGAN+GP. d refers to the depth of the first conv layer in DCGAN architectures. See section 7.4 for details about architecture.

Table content: <table><tr><td colspan="2">Model Generator Arch.</td><td>G b Score</td><td>Best E i Score</td><td>∼Params</td></tr><tr><td>1</td><td>Small DCGAN (d=256)</td><td>5.02±0.05</td><td>N/A</td><td>700,000</td></tr><tr><td>2</td><td>Tiny DCGAN (d=128) + Editors</td><td colspan="3">5.00±0.09 5.67±0.07 (Edit 3) 866,000</td></tr><tr><td>3</td><td>Small DCGAN + Editors (one Critic)</td><td colspan="3">5.24±0.04 6.05±0.08 (Edit 2) 1,250,000</td></tr><tr><td>4</td><td colspan="4">Small DCGAN + Editors (multi Critic) 5.63±0.06 5.86±0.05 (Edit 2) 1,250,000</td></tr><tr><td>5</td><td>Small DCGAN + Editors (end-to-end)</td><td>N/A</td><td>2.56±0.02</td><td>1,250,000</td></tr><tr><td>6</td><td>DCGAN -WGAN+GP (d=512)</td><td>5.18±0.05</td><td>N/A</td><td>1,730,000</td></tr><tr><td>7</td><td>Small ResNet</td><td>5.75±0.05</td><td/><td>720,000</td></tr><tr><td>8</td><td>Small ResNet + Editors</td><td colspan="3">6.35±0.09 6.70±0.03 (Edit 3) 1,000,000</td></tr><tr><td>9</td><td>ResNet -WGAN+GP</td><td>6.86±0.04</td><td>N/A</td><td>1,220,000</td></tr></table>

Comparison of inception scores

Table content: <table><tr><td>Method (unlabelled)</td><td>Inception scores</td></tr></table>

Peer review (iclr2019):
 The paper proposes a GAN variant, called ChainGAN, which expresses the generator as a "base generator" -- which maps the noise vector to a rough model sample -- followed by a sequence of "editors" -- which progressively refine the sample. Each component of the generator is trained independently to fool its own separate discriminator, without backpropagating through the entire chain of editors. The proposed ChainGAN model is trained on MNIST, CIFAR10, and CelebA. The paper presents model samples for all three datasets, as well as Inception scores for CIFAR10. I find the proposed idea simple and elegant but the evaluation lacking, and as such I’m a bit hesitant to outright recommend accepting the paper: - Evaluation is not very extensive or detailed. Inception scores are shown only for CIFAR10 and using two base generator architectures. The Inception score has known limitations, and I would have expected the authors to also provide FID scores. The main takeaway is also not articulated very clearly. As far as I can tell it appears to be that ChainGAN allows to achieve similar performance with less tunable parameters, but Table 1 shows mixed results, where ChainGAN outperforms the baseline DCGAN architecture using fewer parameters but underperforms the baseline ResNet architecture. - The way the experimental section is organized made it difficult for me to find my way around. For example, subsection titles are hard to locate due to the fact that figures and tables were placed immediately underneath them. Overall when the flow of the text is interrupted by a figure, it’s hard to locate where to resume reading. - There is a connection to be made with other sequential generation approaches (not to be confused with sequence generation) such as LAPGAN, DRAW, and Unrolled GANs. Discussing the relationship to those approaches would in my opinion add more depth to the paper.