VAE-type Deep Generative Models

VAE-type Deep Generative
Models (Especially RNN + VAE)
Kenta Oono oono@preferred.jp
Preferred Networks Inc.
25th Jun. 2016
Tokyo Webmining @FreakOut
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Notations
• x: observable (visible) variables
• z: latent (hidden) variables
• D = {x1, x2, …, xN}: training dataset
• KL(q || p): KL divergence between two distributions q and p
• θ: parameters of generative model
• φ: parameters of inference model
• pθ: probability distribution modelled by generative model
• qφ: probability distribution modelled by inference model
• N(µ, σ2): Gaussian Distribution with mean µ and standard deviation σ
• Ber(p): Bernoulli Distribution with parameter p
• A := B, B =: A : Define A by B.
• Ex~p[ f (x)] : Expectation of f(x) with respect to x drawn from p. Namely, ∫ f(x) p(x) dx.
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Abbreviations
• NN: Neural Network
• RNN: Recurrent Neural Network
• CNN: Convolutional Neural Network
• ELBO: Evidence Lower BOund
• AE: Auto Encoder
• VAE: Variational Auto Encoder
• LSTM: Long Short-Term Memory
• NLL: Negative Log-Likelihood
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Agenda
• Mathematical formulation of generative models.
• Variational Auto Encoder (VAE)
• Variants of VAE
• RVAE, VRNN, DRAW, Convolutional DRAW, alignDRAW
• Chainer implementation of (Convolutional) DRAW
• Other VAE-like models
• Inverse DRAW, VAE + GAN
• Conclusion
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Generative models and discriminative
models
• Discriminative model
• Models p(z | x)
• e.g. SVM, Logistic Regression Naïve Bayes Classifier etc.
• Generative model ← Todayʼs Topic
• Models p(x, z) or p(x)
• e.g. RBM, HMM, VAE etc.
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Recent trend of generative models by NN
• Helmholtz machine type ← Todayʼs Topic
• Model p(x, z) as p(z) p(x | z)
• Prepare two NNs: Generative model and Inference model
• Use variational inference and train models to maximize ELBO
• e.g. VAE, ADGM, DRAW, IWAE, VRNN etc.
• Generative Adversarial Network (GAN) type
• Model p(x, z) as p(z) p(x | z)
• Prepare two NNs: Generator and Discriminator
• Train models by solving min-max problem
• e.g. GAN, DCGAN, LAPGAN, f-GAN, InfoGAN etc.
• Auto regressive type
• Model p(x) as Πi p(xi | x1, …, xi-1)
• e.g. Pixel RNN, MADE, NADE etc. 6/34

NN as a probabilistic model
• We assume p(x, z) are parameterized by NN whose
parameter (e.g. weights, biases) is θ and denote it by pθ(x, z).
• Training reduces to find θ that maximize some objective
function.
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NN as a probabilistic model (example)
• prior: pθ(z) = N(0, 1)
• generation: pθ(x | z) = N(x | µθ(z), σθ
2 (z))
• µθ and σθ are deteministic NNs which
takes z as a input and outputs scalar
value.
• Although pθ(x | z) is, simple, pθ(x) can
represent complex distribution.
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z
µ σ2
z ~ N(0, 1)
x x ~ N(x | µθ, σθ
2 )
deterministic NNs
sampling
pθ(x)
= ∫ pθ (x | z) pθ (z) dz
= ∫ N(x | µθ(z), σθ
2 (z)) pθ (z) dz
Generation pθ(x | z)

Difficulty of generative models
• Posterior pθ(z | x) is intractable.
9_34
z
x
pθ (x | z) is easy
to sample
×
pθ(z | x) is
intractable
pθ(z | x)
= pθ (x | z) pθ (z) / pθ (x) (Bayesʼ Thm.)
= pθ (x | z) pθ (z) / ∫ pθ (x, z’) dz’
= pθ (x | z) pθ (z) / ∫ pθ (x | z’) pθ (z’) dz’
• In typical situation, we cannot
calculate the integral analytically.
• When zʼ is high-dimensional, the
integral is difficult to estimate (e.g.
MCMC)

Variational inference
• Instead of posterior distribution pθ(z | x),
we consider the set of distributions
{qφ(z | x)}φ∈Φ .
• Φ is a some set of parameters.
• In addition to θ, we try to find φ that
approximates pθ(z | x) well in training.
• Choice of qφ(z | x)
• Easy to calculate or be sampled from.
• e.g. Mean field approximation
• e.g. VAE : NN with params. φ
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Note: To fully describe the distribution qφ, we
need to specify qφ(x). Typically we employ the
empirical distribution of training dataset.
z
x
×
z
x
approximate
Inference
model
qφ(z | x)
Generative
model
pθ (z | x)

Evidence Lower BOund (ELBO)
• Consider single training example x.
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L(x; θ)
L~(x; θ, φ)
difference
= KL(q(z | x) || p(z | x))
L(x; θ) := log pθ(x)
= log ∫ pθ(x, z)dz
= log ∫ qφ(z | x) pθ(x, z) / qφ(z | x) dz
≧ ∫ qφ(z | x) log pθ(x, z) / qφ(z | x) dz (Jensen)
=: L~(x; θ, φ)
• Instead of L(x; θ), we maximize L~(x; θ, φ)
with respect to θ and φ.
• We call L~ Evidence Lower BOund (ELBO).

Agenda
• Variants of VAE: RNN + VAE
• Conclusion
12_34

Variational AutoEncoder (VAE)
[Kingma+13]
• Use NN as an inference model.
• Training with backpropagation.
• How to calculate gradient?
• REINFORCE (a.k.a Likelihood Ratio (LR))
• Control Variate
• Reparameterization trick [Kingma+13]
(a.k.a Stochastic Gradient Variational
Bayes (SGVB) [Rezende+14])
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Kingma, D. P., & Welling, M. (2013). Auto-encoding variational bayes. arXiv preprint
arXiv:1312.6114.
Rezende, D. J., Mohamed, S., & Wierstra, D. (2014). Stochastic backpropagation and approximate
inference in deep generative models. arXiv preprint arXiv:1401.4082.
x
x’
Decoder
= Generative
model
Encoder
=Inference
model
z

Training Procedure
• ELBO L~(x; θ, φ) equals to Ez~q(z | x) [log p(x | z)] - KL(q(z | x) || p(z))
• 1st term: Reconstruction Loss
• 2nd term: Regularization Loss
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z
x
Inference
model
qφ
z
x’
Generative
model
pθ
2. Inference model tries to
make posterior close to the
prior of generation model
4. Generation model tries to
reconstruct the input data
Calculate Reconstruction loss
1. Input is fed to
inference model
3. Latent variable is pass
generation model.
Calculate regularized loss
NN +
sampling
NN +
sampling

Generation
• We can generate data points with trained generative models.
15/34
z
x’
Generative
model
pθ
NN +
sampling
1. sample from prior
~ pθ(z) (e.g. N(0, 1))
2. propagate down

Agenda
• Misc.
• Conclusion
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Variational Recurrent AutoEncoder (VRAE)
[Fabius+14]
• The modification of VAE where two models (inference model
and generative model) are replaced with RNNs.
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Fabius, O., & van Amersfoort, J. R. (2014). Variational recurrent auto-
encoders. arXiv preprint arXiv:1412.6581.
ht ht+1 hT
z h0
x1’
xt-1 xt xT-1
ht
xt+1’
Encoder
Decoder
RNN
RNNht-1
xt’

Variational RNN (VRNN) [Chung+15]
• Inference and generative
models share the hidden
state h and update it
throughout time. Latent
variable z is sampled from
the state.
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Chung, J., Kastner, K., Dinh, L., Goel, K., Courville, A. C., & Bengio, Y. (2015). A recurrent latent variable
model for sequential data. In Advances in neural information processing systems (pp. 2980-2988).
ht-1 ht ht+1
xt xt+1
ht-1 ht-1
xt’ xt+1
’
zt’ zt+1’
Encoder
Decoder
zt zt+1
RNN RNN

DRAW [Gregor+15]
• “Generative model of natural images that operates by
making a large number of small contributions to an additive
canvas using an attention model”.
• Inference and generative models are independent RNNs.
19/34
Gregor, K., Danihelka, I., Graves, A., Rezende, D. J., & Wierstra, D. (2015). DRAW: A
recurrent neural network for image generation. arXiv preprint arXiv:1502.04623.

DRAW without attention [Gregor+15]
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x
ht
e
ht
d
Δct
+
x
ht+1
e
ht+1
d
Δct+1
ct +ct-1 ct+1
Encoder
Decoder
zt zt+1
cT
x’
σ
RNN
RNN
RNN
RNN
RNN
RNN

DRAW [Gregor+15]
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x
rt
ht
e
ht
d
Δct
+
x
rt+1
ht+1
e
ht+1
d
Δct+1
at at+1
at
ct +ct-1
at+1
ct+1
zt zt+1
cT
x’
σ
RNN
RNN
RNN
RNN
RNN
RNN
Encoder
Decoder

Convolutional DRAW [Gregor+16]
• The variant of DRAW with following modifications:
• Linear connections are replaced with convolutions (including
connections in LSTM).
• Read and write attention mechanisms are removed.
• Instead of sampling from Standard Gaussian distribution in DRAW,
prior of generative model depends on decoderʼs state.
• But details of the implementation is not fully described in the
paper ...
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Gregor, K., Besse, F., Rezende, D. J., Danihelka, I., & Wierstra, D. (2016).
Towards Conceptual Compression. arXiv preprint arXiv:1604.08772.

alignDRAW [Mansimov+15]
• Generate image from its caption.
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Mansimov, E., Parisotto, E., Ba, J. L., & Salakhutdinov, R. (2015). Generating images
from captions with attention. arXiv preprint arXiv:1511.02793.

Implemantation of convolutional DRAW
with Chainer
24
Reconstruction
Generation
Generation (linear connection)

My implementation of
convolutional DRAW
25/34
y
x
+
eembe
ht
e LSTM ht
e
ztembd
+ht
d LSTM ht
d
Δct
+ct ct+1
µt
d σt
d2
µt
e σt
e2
Convolution
Linear
Identity
Samplingct
-
xt+1
’
σ
NLL loss
Deconvolution
y
Encoder
Decoder

Agenda
• Conclusion
26/34

VAE + GAN [Larsen+15]
• Use generative model of VAE as
the generator of GAN.
27/34
Larsen, A. B. L., Sønderby, S. K., & Winther, O. (2015). Autoencoding beyond
pixels using a learned similarity metric. arXiv preprint arXiv:1512.09300.

Inverse DRAW
• a
28/34https://openai.com/requests-for-research/#inverse-draw

cf. InfoGAN[Chen+16]
• Make latent variables of GAN interpretable.
29/34
Chen, X., Duan, Y., Houthooft, R., Schulman, J., Sutskever, I., & Abbeel, P. (2016). InfoGAN:
Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets.
arXiv preprint arXiv:1606.03657.

Agenda
• Conclusion
30/34

Challenges of VAE-like generative models
• Compared to GAN, the images generated by VAE-like models
are said to be blurry.
• Difficulty of evaluation.
• The following common evaluation criteria are independent in some
situation [Theis+15].
• average log-likelihood
• Parzen window estimates
• visual fidelity of samples
• We can evaluate exactly only lower bound of log-likelihood.
• Generation of high dimensional images is still challenging.
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Theis, L., Oord, A. V. D., & Bethge, M. (2015). A note on the
evaluation of generative models. arXiv preprint arXiv:1511.01844.

Many many topics are not covered today.
• VAE + Gaussian Process
• VAE-DGP, Variational GP, Recurrent GP
• Tighter lower bound of log-likelihood
• Importance Weighted AE
• Generative model with more complex prior distribution
• Hierachical Variational Model, Auxiliary Deep Generative Model,
Hamiltonial Variational Inference, Normalizing Flow, Gradient Flow,
Inverse Autoregressive Flow,
• Automatic Variational Inference
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Related conferences, workshops and blogs
• NIPS 2015
• Advances in Approximate Bayesian Inference (AABI)
• http://approximateinference.org/accepted/
• Black Box Learning and Inference
• http://www.blackboxworkshop.org
• ICLR 2016
• http://www.iclr.cc/doku.php?id=iclr2016:main
• OpenAI
• Blog: Generative Models
• https://openai.com/blog/generative-models/
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Summary
• VAE is a generative model that parameterize the inference
and generative models with NNs and optimize them by
maximizing the ELBO of loglikelihood.
• Recently the variant of VAE is proposed including RVAE,
VRNN, and (Convolutional) DRAW.
• Introduced the implementation of generative model with
Chainer.
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