S PECTRAL N ORMALIZATION FOR G ENERATIVE A DVERSARIAL N ETWORKS
Anonymous authors
Paper under double-blind review
A BSTRACT
One of the challenges in the study of generative adversarial networks is the instability of its training
In this paper, we propose a novel weight normalization tech- nique called spectral normalization to stabilize the training of the discriminator.
Our new normalization technique is computationally light and easy to incorporate into existing implementations.
We tested the efﬁcacy of spectral normalization on
CIFAR10, STL- and ILSVRC2012 dataset
and we experimentally conﬁrmed that spectrally normalized GANs (SN-GANs) is capable of generating images of better or equal quality relative to the previous training stabilization techniques
I NTRODUCTION
Generative adversarial networks (GANs) (Goodfellow et al., 2014) has been enjoying considerable success as a framework of generative models in recent years, and it has been applied to numerous types of tasks and datasets (Radford et al., 2016;
Salimans et al., 2016; Ho & Ermon, 2016;
Li et al., 2017).
In a nutshell, GAN is a framework to produce a model distribution that mimics a given target distribution, and it consists of a generator that produces the model distribution and a discriminator that distinguishes the model distribution from the target.
The concept is to consecutively train the model distribution and the discriminator in turn, with the goal of reducing the difference between the model distribution and the target distribution measured by the best discriminator possible at each step of the training.
GANs have been drawing attention in the machine learning community not only for its ability to learn highly structured probability distribution but also for its theoretically interesting aspects.
For example, (Nowozin et al., 2016; Uehara et al., 2016; Mohamed & Lakshminarayanan, 2017) revealed that the training of the discriminator amounts to the training of a good estimator for the density ratio between the model distribution and the target.
This is a perspective that opens the door to the methods of implicit models (Mohamed & Lakshminarayanan, 2017; Tran et al., 2017) that can be used to carry out variational optimization without the direct knowledge of the density function.
A persisting challenge in the training of GANs is the performance control of the discriminator.
In high dimensional spaces, the density ratio estimation by the discriminator is often inaccurate and unstable during the training, and generator networks fail to learn the multimodal structure of the target distribution.
Even worse, when the support of the model distribution and the support of the target distribution are disjoint, there exists a discriminator that can perfectly distinguish the model distribution from the target (Arjovsky & Bottou, 2017).
Once such discriminator is produced in this situation, the training of the generator comes to complete stop, because the derivative of the so-produced discriminator with respect to the input turns out to be 0.
This motivates us to introduce some form of restriction to the choice of the discriminator.
In this paper, we propose a novel weight normalization method called spectral normalization that can stabilize the training of discriminator networks.
Our normalization enjoys following favorable properties.
• Lipschitz constant is the only hyper-parameter to be tuned, and the algorithm does not require intensive tuning of the only hyper-parameter for satisfactory performance.
• Implementation is simple and the additional computational cost is small.
In fact, our normalization method also functioned well even without tuning Lipschitz constant, which is the only hyper parameter.
In this study, we provide a theoretical explanation of the effec- tiveness of spectral normalization for GANs against other regularization techniques, such as weight normalization (Salimans & Kingma, 2016), weight clipping (Arjovsky et al., 2017), and gradient penalty (Gulrajani et al., 2017).
We also show that, in the absence of complimentary regularization techniques (e.g., batch normalization, weight decay and feature matching on the discriminator), spectral normalization can improve the sheer quality of the generated images better than weight normalization and gradient penalty.
M ETHOD
In this section, we will lay the theoretical groundwork for our proposed method.
Let us consider a simple discriminator made of a neural network of the following form:
[Equation]
where θ := { W 1 , . .
, W L , W L +1 } is the learning parameters set, W l ∈ R d l × d l − 1 , W L +1 ∈ R 1 × d L , and a l is an element-wise non-linear activation function.
We omit the bias term of each layer for simplicity.
The ﬁnal output of the discriminator is given by
[Equation]
where A is an activation function corresponding to the divergence of distance measure of the user’s choice.
The standard formulation of GAN is given by
[Equation]
where min and max of G and D are taken over the set of generator and discriminator functions, respectively.
The conventional form of V ( G, D ) (Goodfellow et al., 2014) is given by
E x ∼ q data [log D ( x )]
+ E x (cid:48) ∼ p G
[log(1 − D ( x (cid:48) ))] , where q data is the data distribution and p G is the (model) generator distribution to be learned through the adversarial min-max optimization.
The activation function A that is used in the D of this expression is some continuous function with range [0 , 1] (e.g, sigmoid function).
It is known that, for a ﬁxed generator G , the optimal discriminator for this form of V ( G, D ) is given by D ∗ G ( x ) :
= q data ( x ) / ( q data ( x ) + p G ( x )) , which is the minimizer of the Jensen-Shannon divergence (Goodfellow et al., 2014).
The machine learning community has been pointing out recently that the function space from which the discriminators are selected crucially affects the performance of GANs.
A number of works (Uehara et al., 2016; Qi, 2017; Gulrajani et al., 2017) advocate the importance of Lipschitz continuity in assuring the boundedness of statistics.
For example, the optimal discriminator of GAN on the above standard formulation takes the form
[Equation]
f [Equation] x )) , where f ∗ ( x ) = log
q data ( x ) − log p G [Equation] x )
[Equation]
3) and its derivative ∇
[Equation]
x
[Equation]
can be unbounded or even incomputable.
This prompts us to introduce some regularity condition to the derivative of f ( x ) .
A particularly successful works in this array are (Qi, 2017; Gulrajani et al., 2017), which proposed methods to control the Lipschitz constant of the discriminator by adding regularization terms deﬁned on input examples x .
We would follow their footstep and search for the discriminator D from the set of K -Lipschitz continuous functions, that is,
[Equation]
where we mean by (cid:107) f (cid:107) Lip the smallest value M such that (cid:107) f
( x ) − f ( x (cid:48) )
(cid:107) / (cid:107) x − x (cid:48) (cid:107) ≤ M for any x , x (cid:48) , with the norm being the (cid:96) 2 norm.
While input based regularizations allow for relatively easy formulations based on samples, they also suffer from the fact that, they cannot impose regularization on the space outside of the supports of the generator and data distributions without introducing somewhat heuristic means.
A method we would introduce in this paper, called spectral normalization , is a method that aims to skirt this issue by normalizing the weight matrices using the technique devised by Yoshida & Miyato (2017).
2. S PECTRAL N ORMALIZATION
Our spectral normalization controls the Lipschitz constant of the discriminator function f by literally constraining the spectral norm of each layer g : h in (cid:55)→ h out .
By deﬁnition, Lipschitz norm (cid:107) g (cid:107) Lip is equal to sup h σ ( ∇ g ( h )) , where σ ( A ) is the spectral norm of the matrix
A ( L 2 matrix norm of A )
[Equation]
[Equation]
which is equivalent to the largest singular value of A .
Therefore, for a linear layer g ( h ) =
W h , the norm is given by (cid:107) g (cid:107) Lip = sup h σ ( ∇ g ( h ))
= sup h σ ( W ) =
σ ( W ) .
If the Lipschitz norm of the activation function (cid:107) a l (cid:107) Lip is equal to 1 1 , we can use the inequality (cid:107) g 1 ◦ g 2 (cid:107) Lip ≤ (cid:107) g 1 (cid:107) Lip ·(cid:107) g 2 (cid:107) Lip to observe the following bound on (cid:107) f (cid:107) Lip :
[Equation]
Our spectral normalization normalizes the spectral norm of the weight matrix W so that it satisﬁes the Lipschitz constraint σ ( W ) = 1 :
[Equation]
If we normalize each W l using (8), we can appeal to the inequality (7) and the fact that σ (cid:0) ¯ W SN ( W ) (cid:1) = 1 to see that (cid:107) f (cid:107) Lip is bounded from above by 1 .
2. F AST A PPROXIMATION OF THE S PECTRAL N ORM σ ( W )
As we mentioned above, the spectral norm σ ( W ) that we use to regularize each layer of the discriminator is the largest singular value of W .
If we naively apply singular value decomposition to compute the σ ( W ) at each round of the algorithm, the algorithm can become computationally heavy.
Instead, we can use the power iteration method to estimate σ ( W ) (Yoshida & Miyato, 2017).
With power iteration method, we can estimate the spectral norm with very small additional computational time relative to the full computational cost of the vanilla GANs.
Please see Appendix A for the detail method and Algorithm 1 for the summary of the actual spectral normalization algorithm.
2. G RADIENT ANALYSIS OF THE SPECTRALLY NORMALIZED WEIGHTS
The gradient of ¯ W SN ( W ) with respect to W ij is:
[Equation]
where E ij is the matrix whose ( i, j ) -th entry is 1 and zero everywhere else, and u 1 and v 1 are respectively the ﬁrst left and right singular vectors of W .
If h is the hidden node in the network to be transformed by ¯ W SN , the derivative of the V ( G, D ) calculated over the mini-batch with respect to W of the discriminator D is given by:
[Equation]
where δ := (cid:0) ∂V ( G, D ) /∂
(cid:0) ¯ W SN h (cid:1)(cid:1) T , λ := ˆE (cid:2) δ T (cid:0) ¯ W SN h (cid:1)(cid:3) , and ˆE[ · ] represents empirical expectation over the mini-batch.
∂V∂W
=
when ˆE[ δh T ] =
k u 1 v T 1 for some k ∈ R .
We would like to comment on the implication of (12).
The ﬁrst term ˆE (cid:2) δh T (cid:3) is the same as the derivative of the weights without normalization.
In this light, the second term in the expression can be seen as the regularization term penalizing the ﬁrst singular components with the adaptive regularization coefﬁcient λ .
λ is positive when δ and ¯ W SN h are pointing in similar direction, and this prevents the column space of W from concentrating into one particular direction in the course of the training.
In other words, spectral normalization prevents the transformation of each layer from becoming to sensitive in one direction.
We can also use spectral normalization to devise a new parametrization for the model.
Namely, we can split the layer map into two separate trainable components: spectrally normalized map and the spectral norm constant.
As it turns out, this parametrization has its merit on its own and promotes the performance of GANs (See Appendix E).
S PECTRAL N ORMALIZATION VS O THER R EGULARIZATION T ECHNIQUES
The weight normalization introduced by Salimans & Kingma (2016) is a method that normalizes the (cid:96) 2 norm of each row vector in the weight matrix.
Mathematically, this is equivalent to requiring the weight by the weight normalization ¯ W WN :
[Equation]
, where T = min( d i , d
[Equation]
[Equation]
(13) where σ t ( A ) is a t -th singular value of matrix A .
Therefore, up to a scaler, this is same as the
Frobenius normalization, which requires the sum of the squared singular values to be 1.
These normalizations, however, inadvertently impose much stronger constraint on the matrix than intended.
If ¯ W WN is the weight normalized matrix of dimension d i × d o , the norm (cid:107) ¯ W WN h (cid:107) 2 for a ﬁxed unit vector h is maximized at (cid:107) ¯ W WN
[Equation]
2 = √
[Equation]
o when σ 1 ( ¯ W WN ) = √ d o and σ t ( ¯ W WN )
=
for t = 2 , . . .
, T , which means that ¯ W WN is of rank one.
Similar thing can be said to the Frobenius normalization (See the appendix for more details).
Using such ¯ W WN corresponds to using only one feature to discriminate the model probability distribution from the target.
In order to retain as much norm of the input as possible and hence to make the discriminator more sensitive, one would hope to make the norm of ¯ W WN h large.
For weight normalization, however, this comes at the cost of reducing the rank and hence the number of features to be used for the discriminator.
Thus, there is a conﬂict of interests between weight normalization and our desire to use as many features as possible to distinguish the generator distribution from the target distribution.
The former interest often reigns over the other in many cases, inadvertently diminishing the number of features to be used by the discriminators.
Consequently, the algorithm would produce a rather arbitrary model distribution that matches the target distribution only at select few features.
Weight clipping (Arjovsky et al., 2017) also suffers from same pitfall.
Our spectral normalization, on the other hand, do not suffer from such a conﬂict in interest.
Note that the Lipschitz constant of a linear operator is determined only by the maximum singular value.
In other words, the spectral norm is independent of rank.
Thus, unlike the weight normalization, our spectral normalization allows the parameter matrix to use as many features as possible while satisfying local 1 -Lipschitz constraint.
Our spectral normalization leaves more freedom in choosing the number of singular components (features) to feed to the next layer of the discriminator.
Gulrajani et al.
(2017) used Gradient penalty method in combination with WGAN.
In their work, they placed K -Lipschitz constant on the discriminator by augmenting the objective function with the regularizer that rewards the function for having local 1-Lipschitz constant (i.e. (cid:107)∇ ˆ x f (cid:107) 2 = 1 ) at discrete sets of points of the form ˆ x := (cid:15)
˜ x
+ (1 − (cid:15) )
x generated by interpolating a sample ˜ x from generative distribution and a sample x from the data distribution.
While this rather straightforward approach does not suffer from the problems we mentioned above regarding the effective dimension of the feature space, the approach has an obvious weakness of being heavily dependent on the support of the current generative distribution.
As a matter of course, the generative distribution and its support gradually changes in the course of the training, and this can destabilize the effect of such regularization.
In fact, we empirically observed that a high learning rate can destabilize the performance of WGAN-GP.
On the contrary, our spectral normalization regularizes the function the operator space, and the effect of the regularization is more stable with respect to the choice of the batch.
Training with our spectral normalization does not falter with aggressive learning rate.
Moreover, WGAN-GP requires more computational cost than our spectral normalization with single-step power iteration, because the computation of (cid:107)∇ ˆ x f (cid:107) 2 requires one whole round of forward and backward propagation.
In the experiment section, we compare the computational cost of the two methods for the same number of updates.
E XPERIMENTS
In order to evaluate the efﬁcacy of our experiment and investigate the reason behind its efﬁcacy, we conducted a set of extensive experiments of unsupervised image generation on CIFAR-10 (Torralba et al., 2008) and STL-10 (Coates et al., 2011), and compared our method against other normalization techniques.
To see how our method fares against large dataset, we also applied our method on
ILSVRC2012 dataset (ImageNet) (Russakovsky et al., 2015) as well.
This section is structured as follows.
First, we will discuss the objective functions we used to train the architecture, and then we will describe the optimization settings we used in the experiments.
We will then explain two performance measures on the images to evaluate the images produced by the trained generators.
Finally, we will summarize our results on the CIFAR-10, STL-10, and on ImageNet.
As for the architecture of the discriminator and generator, we used convolutional neural networks.
Also, for the evaluation of the spectral norm for the convolutional weight W ∈ R d out × d in × h × w , we treated the operator as a square matrix of dimension d out × ( d in hw ) 2 .
We trained the parameters of the generator with batch normalization (Ioffe & Szegedy, 2015).
We refer the readers to Table 3 in the appendix section for more details of the architectures.
For all methods other than WGAN-GP, we used the following standard objective function for the adversarial loss:
[Equation]
where
z ∈ R
d
z is a latent variable, p ( z ) is the standard normal distribution N (0 , I ) , and G : R d z → R dis a deterministic generator function.
We set d z to 128 for all of our experiments.
For the updates of G , we used the alternate cost proposed by Goodfellow et al.
− E z ∼ p ( z ) [log( D ( G ( z )))] as used in Goodfellow et al.
(2014); Warde-Farley & Bengio (2017).
For the updates of D , we used the original cost deﬁned in 14.
We also tested the performance of the algorithm with the so-called hinge loss, which is given by
[Equation]
respectively for the discriminator and for the generator.
Optimizing these objectives is equivalent to minimizing the so-called reverse KL divergence KL[ p g || q data ] .
This types of loss has been already proposed and used in (Lim & Ye, 2017; Tran et al., 2017).
The algorithm based on the hinge loss also showed good performance when evaluated with inception score and FID.
For Wasserstein
GANs with gradient penalty (WGAN-GP) (Gulrajani et al., 2017), we used the following objective function: V ( G, D ) :=
E x ∼
[Equation]
data [ D ( x )] − E z ∼ p
( z )
[ D ( G ( z ))] − λ E ˆ x ∼ p ˆ x [( (cid:107)∇ ˆ x D (ˆ x ) (cid:107) 2 − 1) 2 ] , where the regularization term is the one we introduced in the appendix section D.4.
For quantitative assessment of generated examples, we used inception score (Salimans et al., 2016) and Fr´echet inception distance (FID)
Heusel et al.
Please see Appendix B.1 for the details of each score.
4. R ESULTS ON CIFAR10 AND STL-
In this section, we report the accuracy of the spectral normalization (we use the abbreviation: SNGAN for the spectrally normalized GANs) during the training, and the dependence of the algorithm’s performance on the hyperparmeters of the optimizer.
We also compare the performance quality of the algorithm against those of other regularization/normalization techniques for the discriminator networks, including: WGAN-GP (Gulrajani et al., 2017), batch-normalization (BN) (Ioffe &
Szegedy, 2015), layer normalization (LN) (Ba et al., 2016) and weight normalization (WN) (Salimans & Kingma, 2016).
In order to evaluate the stand-alone efﬁcacy of the gradient penalty, we also applied the penalty term (28) to the standard adversarial loss of GANs (14). We would refer to this [Table] [Figure]
[Table]
[Figure] [Table] [Figure] [Table]
[Figure]
[Table]
[Figure] [Table] method as ‘GAN-GP’. For each method with gradient penalty, we set λ to 10, as suggested in Gulrajani et al.
For all comparative studies throughout, we excluded the multiplier parameter γ in the weight normalization method, as well as in batch normalization and layer normalization method.
This was done in order to prevent the methods from overtly violating the Lipschitz condition.
When we experimented with the multiplier parameter, we were in fact not able to achieve any improvement.
For optimization, we used the Adam optimizer Kingma & Ba (2015) in all of our experiments.
We tested with 6 settings for (1) n dis , the number of updates of the discriminator per one update of the generator and (2) β 1 , β 2 , the ﬁrst and second order momentum of the hyper-parameters on Adam (the learning rate α ).
We list the details of these settings in Table 1 in the appendix section.
Out of these 6 settings, A, B, and C are the settings used in previous representative works.
The purpose of the settings D, E, and F is to the evaluate the performance of the algorithms implemented with more aggressive learning rates.
For the details of the architectures of convolutional networks deployed in the generator and the discriminator, we refer the readers to Table 3 in the appendix section.
Number of updates for GAN generator were 100K for all experiments, unless otherwise noted.
Firstly, we inspected the spectral norms of each layer during the training to make sure that our spectral normalization procedure is indeed serving its purpose.
As we can see in the Figure 8 in the C.1, the spectral norms of these layers ﬂoats around 1 – 1 . 05 region throughout the training.
Please see Appendix C.1 for more details.
In Figures 1 and 2 we show the inception scores of each method with the settings A–F. We can see that spectral normalization is relatively robust to aggressive learning rates and momentum parameters.
WGAN-GP fails to train good GANs at high learning rates and high momentum parameters on both CIFAR-10 and STL-10.
Weight normalization is more robust than WGAN-GP on CIFAR-10 in this aspect.
The optimal performance of weight normalization was inferior to both WGAN-GP and spectral normalization on STL-10, which consists of more diverse examples than CIFAR-10.
Best scores of spectral normalization are better than all other methods on both CIFAR-10 and STL-10.
[Figure]
P
[Figure]
G A N
[Figure]
G P [Figure]
N
L N
W N
S N
[Figure]
C D
E F (a) CIFARW G
A N - G
P
L N
W N
S N 1010 2 F I D
A B
C D
E F (b) STL-
[Table]
In Tables 2, we show the inception scores of the different methods with optimal settings on CIFAR10 and STL-10 dataset. We see that SN-GANs performed better than all contemporaries on the optimal settings 3 . SN-GANs performed even better with hinge loss (16). In Figure 5 we show the images produced by the generators trained with WGAN-GP, weight normalization, and spectral normalization.
SN-GANs were consistently better than GANs with weight normalization in terms of the quality of generated images. To be more precise, as we mentioned in Section 3, the set of images generated by spectral normalization was clearer and more diverse than the images produced by the weight normalization. We can also see that WGAN-GP failed to train good GANs with high learning rates and high momentums (D,E and F). The generated images with GAN-GP, batch normalization and layer normalization is shown in Figure 11 in the appendix section.
We compared our algorithm against multiple benchmark methods in Table 2. We also tested the performance of our method on ResNet based GAN in Gulrajani et al.
For the details, please see Appendix section.
Please note that all methods listed thereof are all different in both optimization methods and the architecture of the model.
Our implementation of our algorithm was able to superior to all the predecessors in the performance.
4.1. A NALYSIS OF S PECTRALLY N ORMALIZED GAN S Singular
values analysis on the weights of the discriminator D
In Figure 4, we show the squared singular values of the weight matrices in the ﬁnal discriminator D produced by each method using the parameter that yielded the best inception score.
As we prophesied in Section 3, the singular values of the ﬁrst to ﬁfth layers trained with the weight normalization concentrate on a few components.
That is, the weight matrices of these layers tend to be rank deﬁcit.
On the other hand, the singular values of the weight matrices in those layers trained with spectral normalization is more broadly distributed.
When the goal is to distinguish a pair of probability distributions on the lowdimensional nonlinear data manifold embedded in a high dimensional space, rank deﬁciencies in lower layers can be especially fatal.
Outputs of lower layers have gone through only few sets of rectiﬁed linear transformations, which means that they tend to lie on the space that is linear in most parts.
Marginalizing out many features of the input distribution in such space can result in oversimpliﬁed discriminator.
We can actually conﬁrm the effect of this phenomenon on the generated images especially in Figure 5b.
The images generated with spectral normalization is more diverse and complex than those generated with weight normalization.
Training time On CIFAR-10, SN-GANs is a tad slower than weight normalization (about 110 ∼ 120% computational time), but signiﬁcantly faster than WGAN-GP. As we mentioned in Section 3,
WGAN-GP is slower than other methods because WGAN-GP needs to calculate the gradient of gradient norm (cid:107)∇ x D (cid:107) 2 .
For STL-10, the computational time of SN-GANs is almost the same as vanilla GANs, because the relative computational cost of the power iteration (17) is negligible when compared to the cost of forward and backward propagation on CIFAR-10 (images size of STL-10 is larger ( 48 × 48 )).
Please see Figure 9 in the appendix section for the actual computational time.
4. C LASS C ONDITIONAL I MAGE G ENERATION
[Figure] I MAGE N [Figure] To show that our method remains effective on large high dimensional dataset, we also applied our method to the training of class conditional GANs on ILRSVRC2012 dataset with 1000 classes, each consisting of approximately 1300 images, which we compressed to 128 × 128 pixels.
Regarding the adversarial loss for conditional GANs, we used practically the same formulation used in Mirza & Osindero (2014), except that we replaced the standard GANs loss with hinge loss (16).
Please see Appendix B.3 for the details of experimental settings.
As we can see in the learning curves in Figure 3, our SN-GANs is the only methods with successful training sequence among all other methods.
To our knowledge, our method is the ﬁrst of its kind in succeeding to produce decent images from ImageNet dataset with a single pair of a discriminator and a generator (Figure 6).
To measure the degree of mode-collapse, we followed the footstep of Odena et al.
and computed the intra MS-SSIM Odena et al.
(2017) for pairs of independently generated GAN images of each class. .
C ONCLUSION
This paper proposes spectral normalization as a stabilizer of training of GANs.
When we apply spectral normalization to the GANs on image generation tasks, the generated examples are more diverse than the conventional weight normalization and achieve better or comparative inception scores relative to previous studies.
The method imposes global regularization on the discriminator as opposed to local regularization introduced by WGAN-GP, and can possibly used in combinations.
In the future work, we would like to further investigate where our methods stand amongst other methods on more theoretical basis, and experiment our algorithm on larger and more complex datasets.
A T HE A LGORITHM OF S PECTRAL N ORMALIZATION
Let us describe this shortcut in more detail.
We begin with vectors ˜ u and ˜ v that are randomly initialized.
If there is no multiplicity in the dominant singular values and if ˜ u and ˜ v are not orthogonal to the ﬁrst singular vectors 4 , ˜ u and ˜ v converge to the ﬁrst left and right singular vectors u and v through the following updating rule.
[Equation]
We can then approximate the spectral norm of W with the pair of so-approximated singular vectors:
[Equation]
If we use SGD for updating W , the change in W at each update would be small, and hence the change in its largest singular value.
In our implementation, we took advantage of this fact and reused the ˜ u computed at each step of the algorithm as the initial vector in the subsequent step.
In fact, with this ‘recycle’ procedure, one round of power iteration was sufﬁcient in the actual experiment to achieve satisfactory performance.
Algorithm 1 in the appendix summarizes the computation of the spectrally normalized weight matrix ¯ W with this approximation.
Note that this procedure is very computationally cheap even in comparison to the calculation of the forward and backward propagations on neural networks.
Please see Figure 9 in the experiment section for actual computational time with and without spectral normalization.
• Initialize ˜
u l ∈ R d l for l = 1 , . . .
, L with a random vector (sampled from isotropic distribution).
• For each update and each layer l : 1.
Apply power iteration method to a unnormalized weight W l
[Equation]
Calculate ¯ W SN with the spectral norm:
[Equation]
l /σ ( W l ) , where σ ( W l ) =
˜ u T l W l ˜ v l
[Equation]
[Equation]
Update W l with SGD on mini-batch dataset D M with a learning rate α :
[Equation]
B E XPERIMENTAL S ETTINGS B. P ERFORMANCE MEASURES
Inception score is introduced originally by Salimans et al.
I ( { x n } Nn =1 ) := exp(E[ D KL [ p ( y | x ) || p
( y )]]) , where p ( y ) is approximated by 1 N (cid:80)
Nn =1 p ( y | x n ) and p ( y | x ) is the trained Inception convolutional neural network (Szegedy et al.
(2015)), which we would refer to
Inception model for short.
In their work, Salimans et al.
(2016) reported that this score is strongly correlated with subjective human judgment of image quality.
Following the procedure in Salimans et al.
(2016); Warde-Farley & Bengio (2017)
, we calculated the score for randomly generatedexamples from each trained generator to evaluate its ability to generate natural images.
We repeated each experiment 10 times and reported the average and the standard deviation of the inception scores.
Fr´echet inception distance (Heusel et al., 2017) is another measure for the quality of the generated examples that uses 2nd order information of the ﬁnal layer of the inception model applied to the examples.
On its own, the Fre´chet distance Dowson & Landau (1982) is 2-Wasserstein distance between two distribution p 1 and p 2 assuming they are both multivariate Gaussian distributions:
[Equation]
where { µ p 1 , C p 1 } , { µ p 2 , C p 2 } are the mean and covariance of samples from q and p , respectively.
If f (cid:9) is the output of the ﬁnal layer of the inception model before the softmax, the Fr´echet inception distance (FID) between two distributions p 1 and p 2 on the images is the distance between f (cid:9) ◦ pand f (cid:9) ◦ p 2 .
We computed the Fr´echet inception distance between the true distribution and the generated distribution empirically over 5000 sample and reported the results.
Multiple repetition of the experiments did not exhibit any notable variations on this score.
B. I MAGE G ENERATION ON CIFAR-10 AND STL-
For the comparative study, we experimented with the recent ResNet architecture of Gulrajani et al.
(2017) as well as the standard CNN.
For this additional set of experiments, we used Adam again for the optimization and used the very hyper parameter used in Gulrajani et al.
( α = 0 .
0002 , β 1 = 0 , β 2 = 0 .
9 , n dis = 5 ).
For our SN-GANs, we doubled the feature map in the generator from the original, because this modiﬁcation achieved better results.
Note that when we doubled the dimension of the feature map for the WGAN-GP experiment, however, the performance deteriorated.
B. C ONDITIONAL I MAGE G ENERATION ON I MAGE N ET
The images used in this set of experiments were resized to 128 × 128 pixels.
We used conditional batch normalization (CBN) Dumoulin et al.
(2017) for the generator network.
Namely we replaced the standard batch normalization layer with the CBN conditional to the label information y ∈ { 1 , . .
The details of the architecture are given in Table 6.
For the optimization, we used Adam with the same hyperparameters we used for ResNet on CIFAR-10 and STL-10 dataset.
We trained the networks with 450K generator updates.
B. N ETWORK A RCHITECTURES
[Table]
[Table]
C A PPENDIX R ESULTS C. A CCURACY OF SPECTRAL NORMALIZATION
Figure 8 shows the spectral norm of each layer in the generator over the course of the training.
The setting of optimizers is C in Table 1 throughout the training.
In fact, they do not deviate by more than
[Figure]
[Table] [Table]
[Table] [Table]
[Figure]
[Table]
ResBlock down
ResBlock ResBlock
Global sum pooling
[Table] [Table]
[Figure] [Figure] [Table]
[Figure]
[Table]
ResBlock up
[Figure]
[Table]
ResBlock down
ResBlock down ResBlock down
ResBlock
[Table]
[Figure]
0.05 for the most part.
As an exception, 6 and 7-th convolutional layers with largest rank deviate by more than 0.1 in the beginning of the training, but the norm of this layer too stabilizes around 1 after some iterations.
[Figure] [Figure]
up
[Figure]
[Table]
ResBlock
[Table]
ResBlock down
Concat(embed( y ), h )
ResBlock
[Table]
Global
[Table]
ReLU,
[Table]
[Table]
C. T RAINING T IME
[Figure]
T HE EFFECT OF n dis ON SPECTRAL NORMALIZATION AND WEIGHT NORMALIZATION
Figure 10 shows the effect of n dis on the performance of weight normalization and spectral normalization.
All results shown in Figure 10 follows setting D, except for the value of n dis .
For WN, the performance deteriorates with larger n dis , which amounts to computing minimax with better accuracy.
Our SN does not suffer from this unintended effect.
[Figure] [Figure] [Figure] n
[Figure] ep t i on [Figure]
Inception scores after 10000 generator updates
SN
WN
C. G ENERATED I MAGES ON CIFAR10 WITH GAN-GP, L AYER N ORMALIZATION AND B ATCH N ORMALIZATION
D S PECTRAL N ORMALIZATION VS O THER R EGULARIZATION T ECHNIQUES
This section is dedicated to the comparative study of spectral normalization and other regularization methods for discriminators.
In particular, we will show that contemporary regularizations including weight normalization and weight clipping implicitly impose constraints on weight matrices that places unnecessary restriction on the search space of the discriminator.
More speciﬁcally, we will show that weight normalization and weight clipping unwittingly favor low-rank weight matrices.
This can force the trained discriminator to be largely dependent on select few features, rendering the algorithm to be able to match the model distribution with the target distribution only on very low dimensional feature space.
D. W EIGHT N ORMALIZATION AND F ROBENIUS N ORMALIZATION
The weight normalization introduced by Salimans & Kingma (2016) is a method that normalizes the (cid:96) 2 norm of each row vector in the weight matrix 5 :
[Equation]
where ¯ w i and w i are the i th row vector of ¯ W WN and W , respectively.
Still another technique to regularize the weight matrix is to use the Frobenius norm:
[Equation]
where (cid:107) W (cid:107) F := (cid:112) tr
( W T W ) =
(cid:113)(cid:80) i,j w 2 ij .
Originally, these regularization techniques were invented with the goal of improving the generalization performance of supervised training (Salimans & Kingma, 2016; Arpit et al., 2016).
However, recent works in the ﬁeld of GANs (Salimans et al., 2016; Xiang & Li, 2017) found their another raison d’etat as a regularizer of discriminators, and succeeded in improving the performance of the original.
These methods in fact can render the trained discriminator D to be K -Lipschitz for a some prescribed K and achieve the desired effect to a certain extent.
However, weight normalization (24) imposes the following implicit restriction on the choice of ¯
W WN :
[Equation]
T
[Equation] 26) where σ t ( A ) is a t -th singular value of matrix A .
The above equation holds because (cid:80) min( d i ,d o [Equation] =
[Equation]
2 = d o . Under this restriction, the norm (cid:107) ¯ W WN h (cid:107) 2 for a ﬁxed unit vector h is maximized at (cid:107) ¯ W WN h (cid:107) 2 = √
d
o when σ 1 ( ¯ W WN ) = √ d o and σ t ( ¯ W WN )
=
for t = 2 , . . .
, T , which means that ¯ W WN is of rank one.
Using such
W corresponds to using only one feature to discriminate the model probability distribution from the target.
Similarly, Frobenius normalization requires σ 1 ( ¯ W FN ) 2 + σ 2 ( ¯ W FN ) 2 + · · · + σ T ( ¯ W FN ) 2 = 1 , and the same argument as above follows.
Here, we see a critical problem in these two regularization methods.
In order to retain as much norm of the input as possible and hence to make the discriminator more sensitive, one would hope to make the norm of ¯ W WN h large.
For weight normalization, however, this comes at the cost of reducing the rank and hence the number of features to be used for the discriminator.
Thus, there is a conﬂict of interests between weight normalization and our desire to use as many features as possible to distinguish the generator distribution from the target distribution.
The former interest often reigns over the other in many cases, inadvertently diminishing the number of features to be used by the discriminators.
Consequently, the algorithm would produce a rather arbitrary model distribution that matches the target distribution only at select few features.
Our spectral normalization, on the other hand, do not suffer from such a conﬂict in interest.
Note that the Lipschitz constant of a linear operator is determined only by the maximum singular value.
In other words, the spectral norm is independent of rank.
Thus, unlike the weight normalization, our spectral normalization allows the parameter matrix to use as many features as possible while satisfying local 1 -Lipschitz constraint.
Our spectral normalization leaves more freedom in choosing the number of singular components (features) to feed to the next layer of the discriminator.
To see this more visually, we refer the reader to Figure (13).
Note that spectral normalization allows for a wider range of choices than weight normalization.
[Figure]
In summary, weight normalization and Frobenius normalization favor skewed distributions of singular values, making the column spaces of the weight matrices lie in (approximately) low dimensional vector spaces.
On the other hand, our spectral normalization does not compromise the number of feature dimensions used by the discriminator.
In fact, we will experimentally show that GANs trained with our spectral normalization can generate a synthetic dataset with wider variety and higher inception score than the GANs trained with other two regularization methods.
D. W EIGHT C LIPPING
Still another regularization technique is weight clipping introduced by Arjovsky et al.
(2017) in their training of Wasserstein GANs.
Weight clipping simply truncates each element of weight matrices so that its absolute value is bounded above by a prescribed constant c ∈ R + .
Unfortunately, weight clipping suffers from the same problem as weight normalization and Frobenius normalization.
With weight clipping with the truncation value c , the value (cid:107) W x (cid:107) 2 for a ﬁxed unit vector x is maximized when the rank of W is again one, and the training will again favor the discriminators that use only select few features.
Gulrajani et al.
(2017) refers to this problem as capacity underuse problem.
They also reported that the training of WGAN with weight clipping is slower than that of the original
DCGAN (Radford et al., 2016).
D. S INGULAR V ALUE C LIPPING AND S INGULAR V ALUE C ONSTRAINT
One direct and straightforward way of controlling the spectral norm is to clip the singular values (Saito et al., 2017), (Jia et al., 2017).
This approach, however, is computationally heavy because one needs to implement singular value decomposition in order to compute all the singular values.
A similar but less obvious approach is to parametrize W ∈ R d o × d i as follows from the get-go and train the discriminators with this constrained parametrization:
[Equation]
where U ∈ R d o × P , V ∈ R d i × P , and
S ∈ R P × P is a diagonal matrix.
However, it is not a simple task to train this model while remaining absolutely faithful to this parametrization constraint.
Our spectral normalization, on the other hand, can carry out the updates with relatively low computational cost without compromising the normalization constraint.
D. WGAN WITH G RADIENT P ENALTY (WGAN-GP)
Recently, Gulrajani et al.
(2017) introduced a technique to enhance the stability of the training of
Wasserstein GANs (Arjovsky et al., 2017).
In their work, they endeavored to place K -Lipschitz constraint (5) on the discriminator by augmenting the adversarial loss function with the following regularizer function:
[Equation]
where λ > 0 is a balancing coefﬁcient and ˆ x is:
[Equation]
29) where (cid:15) ∼ U [0 , 1] , x ∼ p data , ˜ x = G ( z ) , z ∼ p z .
Using this augmented objective function, Gulrajani et al.
(2017) succeeded in training a GAN based on ResNet (He et al., 2016) with an impressive performance.
The advantage of their method in comparison to spectral normalization is that they can impose local 1 -Lipschitz constraint directly on the discriminator function without a rather round-about layer-wise normalization.
This suggest that their method is less likely to underuse the capacity of the network structure.
At the same time, this type of method that penalizes the gradients at sample points ˆ x suffers from the obvious problem of not being able to regularize the function at the points outside of the support of the current generative distribution.
In fact, the generative distribution and its support gradually changes in the course of the training, and this can destabilize the effect of the regularization itself.
On the contrary, our spectral normalization regularizes the function itself, and the effect of the regularization is more stable with respect to the choice of the batch.
In fact, we observed in the experiment that a high learning rate can destabilize the performance of WGAN-GP.
Training with our spectral normalization does not falter with aggressive learning rate.
Moreover, WGAN-GP requires more computational cost than our spectral normalization with single-step power iteration, because the computation of (cid:107)∇ x D (cid:107) 2 requires one whole round of forward and backward propagation.
In the experiment section (Figure 9), we compare the computational cost of the two methods for the same number of updates.
Having said that, one shall not rule out the possibility that the gradient penalty can compliment spectral normalization and vice versa.
Because these two methods regularizes discriminators by completely different means, and in the experiment section, we actually conﬁrmed that combination of WGAN-GP and reparametrization with spectral normalization improves the quality of the generated examples over the baseline (WGAN-GP only).
E R EPARAMETRIZATION MOTIVATED BY THE SPECTRAL NORMALIZATION
We can take advantage of the regularization effect of the spectral normalization we saw above to develop another algorithm.
Let us consider another parametrization of the weight matrix of the discriminator given by:
[Equation]
[Equation]
where γ is a scalar variable to be learned.
This parametrization compromises the 1 -Lipschitz constraint at the layer of interest, but gives more freedom to the model while keeping the model from becoming degenerate.
For this reparametrization, we need to control the Lipschitz condition by other means, such as the gradient penalty (Gulrajani et al., 2017).
Indeed, we can think of analogous versions of reparametrization by replacing ¯ W SN in (31) with W normalized by other criterions.
The extension of this form is not new.
In Salimans & Kingma (2016), they originally introduced weight normalization in order to derive the reparametrization of the form (31) with ¯ W SN replaced (31) by
W [Table] [Table]
E. E XPERIMENTS : C OMPARISON OF R EPARAMETRIZATION WITH DIFFERENT NORMALIZATION METHODS
In this part of the addendum, we experimentally compare the reparametrizations derived from two different normalization methods (weight normalization and spectral normalization).
We tested the reprametrization methods for the training of the discriminator of WGAN-GP.
For the architecture of the network in WGAN-GP, we used the same CNN we used in the previous section.
For the
ResNet-based CNN
, we used the same architecture provided by (Gulrajani et al., 2017) 6 .
Tables 7, 8 summarize the result.
We see that our method signiﬁcantly improves the inception score from the baseline on the regular CNN, and slightly improves the score on the ResNet based CNN.
Figure 14 shows the learning curves of (a) critic losses, on train and validation sets and (b) the inception scores with different reparametrization methods.
We can see the beneﬁcial effect of spectral normalization in the learning curve of the discriminator as well.
We can verify in the ﬁgure 14a that the discriminator with spectral normalization overﬁts less to the training dataset than the discriminator without reparametrization and with weight normalization, The effect of overﬁtting can be observed on inception score as well, and the ﬁnal score with spectral normalization is better than the others.
As for the best inception score achieved in the course of the training, spectral normalization achieved 7.28, whereas the spectral normalization and vanilla normalization achieved 7.04 and 6.69, respectively.
F T HE GRADIENT OF G ENERAL N ORMALIZATION M ETHOD
Let us denote ¯ W := W/N ( W ) to be the normalized weight where N ( W ) to be a scalar normalized coefﬁcient (e.g. Spectral norm or Frobenius norm).
In general, we can write the derivative of loss
[Figure]
with respect to unnormalized weight W as follows:
[Equation]
where α := 1 /N ( W ) and λ := trace (cid:0) ( ∇ ¯ W V ) T ¯ W (cid:1) .
The gradient ∇ ¯ W V is calculated by ˆE (cid:2) δh T [Equation] where δ := (cid:0) ∂V ( G, D ) /∂
(cid:0) ¯ W h (cid:1)(cid:1) T , h is the hidden node in the network to be transformed by ¯ W and ˆE represents empirical expectation over the mini-batch.
When N ( W ) :
= (cid:107) W (cid:107) F , the derivative is:
[Equation]
35) and when N ( W ) :
= (cid:107) W
[Equation]
2 = σ ( W ) ,
[Equation]
We can represent the ﬁxed point(s) as follows (at least when n ( W ) :
= (cid:107) W (cid:107) F or n ( W ) :
= (cid:107) W (cid:107) 2 ):
[Equation]
37) where ∀ k ∈ R .