Latent Constraints: Learning to Generate Conditionally from Unconditional Generative Models

Introduction

Generative modeling of complicated data such as images and audio is a long-standing challenge in machine learning. While unconditional sampling is an interesting technical problem, it is arguably of limited practical interest in its own right: if one needs a non-specific image (or sound, song, document, etc.), one can simply pull something at random from the unfathomably vast media databases on the web. But that naive approach may not work for conditional sampling (i.e., generating data to match a set of user-specified attributes), since as more attributes are specified, it becomes exponentially less likely that a satisfactory example can be pulled from a database. One might also want to modify some attributes of an object while preserving its core identity. These are crucial tasks in creative applications, where the typical user desires fine-grained controls (Bernardo et al., 2017).

One can enforce user-specified constraints at training time, either by training on a curated subset of data or with conditioning variables. These approaches can be effective if there is enough labeled data available, but they require expensive model retraining for each new set of constraints and may not leverage commonalities between tasks. Deep latent-variable models, such as Generative Adversarial Networks (GANs; Goodfellow et al., 2014) and Variational Autoencoders (VAEs; Kingma & Welling, 2013; Rezende et al., 2014), learn to unconditionally generate realistic and varied outputs by sampling from a semantically structured latent space. One might hope to leverage that structure in creating new conditional controls for sampling and transformations (Brock et al., 2016).

Here, we show that new constraints can be enforced post-hoc on pre-trained unsupervised generative models. This approach removes the need to retrain the model for each new set of constraints, allowing users to more easily define custom behavior. We separate the problem into (1) creating an unsupervised model that learns how to reconstruct data from latent embeddings, and (2) leveraging the latent structure exposed in that embedding space as a source of prior knowledge, upon which we can impose behavioral constraints.

We show that it is possible to generate conditionally from an unconditional model, learning a critic function $D(z)$ in latent space and generating high-value samples with either gradient-based optimization or an amortized actor function $G(z)$, even with a non-differentiable decoder (e.g., discrete sequences).

Focusing on VAEs, we address the tradeoff between reconstruction quality and sample quality (without sacrificing diversity) by enforcing a universal “realism” constraint that requires samples in latent space to be indistinguishable from encoded data (rather than prior samples).

Because we start from a VAE that can reconstruct inputs well, we are able to apply identity-preserving transformations by making the minimal adjustment in latent space needed to satisfy the desired constraints. For example, when we adjust a person’s expression or hair, the result is still clearly identifiable as the same person (see Figure 5). This contrasts with pure GAN-based transformation approaches, which often fail to preserve identity.

Zero-shot conditional generation. Using samples from the VAE to generate exemplars, we can learn an actor-critic pair that satisfies user-specified rule-based constraints in the absence of any labeled data.

Background

Decoder-based deep generative models such as VAEs and GANs generate samples that approximate a population distribution $p^{\star}(x)$ by passing samples from some simple tractable distribution $p(z)$ (often $p(z)\triangleq\mathcal{N}(0,I)$) through a deep neural network. GANs are trained to fool an auxiliary classifier that tries to learn to distinguish between real and synthetic samples. VAEs are fit to data using a variational approximation to maximum-likelihood estimation:

GANs and VAEs have complementary strengths and weaknesses. GANs suffer from the “mode-collapse” problem, where the generator assigns mass to a small subset of the support of the population distribution—that is, it may generate realistic samples, but there are many more realistic samples that it cannot generate. This is particularly problematic if we want to use GANs to manipulate data rather than generate new data; even GAN variants that include some kind of inference machinery (e.g., Donahue et al., 2016; Dumoulin et al., 2016; Perarnau et al., 2016) to determine what $z$ best matches some $x$ tend to produce reconstructions that are reminiscent of the input but do not preserve its identity.

Conditional GANs (CGAN; Mirza & Osindero, 2014) and conditional VAEs (CVAE; Sohn et al., 2015) can generate samples conditioned on attribute information when available, but they must be trained with knowledge of the attribute labels for the whole training set, and it is not clear how to adapt them to new attributes without retraining from scratch. Furthermore, CGANs and CVAEs suffer from the same problems of mode-collapse and blurriness as their unconditional cousins.

We take a different approach to conditional generation and identity-preserving transformation. We begin by training an unconditional VAE with hyperparameters chosen to ensure good reconstruction (at the expense of sample quality). We then train a “realism” critic to predict whether a given $z$ maps to a high-quality sample. We also train critics to predict whether a given $z$ maps to a sample that manifests various attributes of interest. To generate samples that are both realistic and exhibit desired attributes, one option is to optimize random $z$ vectors until they satisfy both the realism and attribute critics. Alternately, we can amortize this cost by training an “actor” network to map a random set of $z$ vectors to a subregion of latent space that satisfies the constraints encoded by the critics. By encouraging these transformed $z$ vectors to remain as close as possible to where they started, we alleviate the mode-collapse problem common to GANs.

Our approach is summarized visually in Figure 1. The details follow in sections 3, 4, 5, and 6.

The “Realism” Constraint: Sharpening VAE Samples

We define the realism constraint implicitly as being satisfied by samples from the marginal posterior $q(z)\triangleq\frac{1}{N}\sum_{n}q(z\mid x_{n})$ and not those from $p(z)$. By enforcing this constraint, we can close the gap between reconstruction quality and sample quality (without sacrificing sample diversity).

Attribute Constraints: Conditional Generation

We want to generate samples that are realistic, but we also want to control what attributes they exhibit. Given binary attribute labels $y$ for a dataset, we can accomplish this by using a CGAN in the latent space, which amounts to replacing $D(z)$ and $G(z)$ with conditional versions $D(z,y)$ and $G(z,y)$ and concatenating $y$ to $z$ as input. If both the actor and critic see attribute information, $G$ must find points in latent space that could be samples from $q(z)$ with attributes $y$.

This procedure is computationally inexpensive relative to training a generative model from scratch. In most of our experiments, we use a relatively large CGAN actor-critic pair (4 fully connected ReLU layers of 2048 units each), which during training uses about $96\times$ fewer FLOPs/iteration than the unconditional VAE. We also trained a much smaller CGAN actor-critic pair (3 fully connected ReLU layers of 256 units), which uses about $2884\times$ fewer FLOPs/iteration than the VAE, and achieves only slightly worse results than the larger CGAN (supplemental Figure 14 and Table 1).

Figure 4 demonstrates the quality of conditional samples from a CGAN actor-critic pair and the effect of the distance penalty, which constrains generation to be closer to the prior sample, maintaining similarity between samples with different attributes. The regularized CGAN actor has less freedom to ignore modes by pushing many random $z$ vectors to the same area of the latent space, since it is penalized for moving samples from $p(z)$ too far. The increased diversity across rows of the regularized CGAN is evidence that this regularization does fight mode-collapse (additional qualitative evidence is in supplemental Figures 7 and 8). However, without a distance penalty, samples appear more a bit realistic with more prominent attributes. This is supported by Table 1, where we use a separately trained attribute classification model to quantitatively evaluate samples. The actor with no penalty generates samples that are more accurately classified than the actor with a penalty but also shifts the samples much farther in latent space.

Although we used a VAE as the base generative model, our approach could also be used to generate high-quality conditional samples from pretrained classical autoencoders. We show in supplemental Figure 15 that we obtain reasonably good conditional samples (albeit with high-frequency spatial artifacts) as $\sigma_{x}\rightarrow 0$ (equivalent to a classical autoencoder). Learning the decoder using VAE training encourages q(z) to fill up as much of the latent space as possible (without sacrificing reconstruction quality), which in turn encourages the decoder to map more of the latent space to reasonable-looking images. The prior $p(z)=\mathcal{N}(0,I)$ also imposes a natural scale on the latent variables.

Identity-Preserving Transformations

Figure 5 demonstrates transformations applied to samples from the held-out evaluation dataset. Note that since the reconstructions are close to the original images, the transformed images also maintain much of their structure. This contrasts with supplemental Figure 10, where a distance-penalty-free CGAN actor produces transformations that share attributes with the original but shift identity. We could preserve identity by introducing a distance penalty, but find that it is much easier to find the correct weighting of realism cost, attribute cost, and distance penalty through optimization, as each combination does not require retraining the network.

Rule-based Constraints: Zero-shot Conditional Generation

Note that the reward function and VAE decoder need not necessarily be differentiable, as the critic learns a value function to approximate the reward, which the actor uses for training. To highlight this, we demonstrate that the output of a recurrent VAE model can be constrained to satisfy hard-coded rule-based constraints.

We first train an LSTM VAE (details in the Appendix) on melodic fragments. Each melody, $m$, is represented as a sequence of categorical variables. In order to examine our ability to constrain the pitch classes and note density of the outputs, we define two reward functions, one that encourages notes from a set of pitches $\mathcal{P}$, and another for that encourages melodies to have at least $d$ notes:

Figure 6 gives an example of controlling the pitch class and note density of generated outputs, which is quantitatively supported by the results in Table 2. During training, the actor goes through several phases of exploration and exploitation, oscillating between expanding to find new modes with high reward and then contracting to find the nearest locations of those modes, eventually settling into high value states that require only small movements in the latent space (supplemental Figure 11).

Related Work

Conditional GANs (Mirza & Osindero, 2014) and VAEs (Sohn et al., 2015) introduce conditioning variables at training time. Sohn et al. (2015) allow these variables to affect the distribution in latent $z$ space, but still require that $p(z\mid y)$ be a tractable distribution. Perarnau et al. (2016) use CGANs to adjust images, but because CGANs cannot usually reconstruct arbitrary inputs accurately, they must resort to image-space processing techniques to transfer effects to the original input. White (2016) propose adding “attribute vectors” to samples from $p(z)$ as a simple and effective heuristic to perform transformations, which relies heavily on the linearity of the latent space.

Some recent work has focused on applying more expressive prior constraints to VAEs (Rezende et al., 2014; Sønderby et al., 2016; Chen et al., 2017; Tomczak & Welling, 2017). The prior that maximizes the ELBO is $p^{\star}(z)=q(z)$ (Hoffman & Johnson, 2016); one can interpret our realism constraint as trying to find an implicit distribution that is indistinguishable from $q(z)$. Like the adversarial autoencoder of Makhzani et al. (2016), our realism constraint relies on a discriminative model, but instead of trying to force $q(z)$ to equal some simple $p(z)$, we only weakly constrain $q(z)$ and then use a classifier to “clean up” our results.

Like this work, the recently proposed adversarially regularized autoencoder (Junbo et al., 2017) uses adversarial training to generate latent codes in a latent space discovered by an autoencoder; that work focuses on unconditional generation. Gómez-Bombarelli et al. (2016) train classifiers in the latent space of a VAE to predict what latent variables map to molecules with various properties, and then use iterative gradient-based optimization in the latent space to find molecules that have a desired set of properties. On molecule data, their procedure generates invalid molecules rarely enough that they can simply reject these samples, which are detected using off-the-shelf software. By contrast, the probability of generating realistic images under our pretrained VAE is astronomically small, and no simple criterion for detecting valid images exists.

Jaques et al. (2017) also use a classifier to constrain generation; they use a Deep Q-network as an auxiliary loss for training an LSTM. Closest to Section 6, Nguyen et al. (2016a; b) generate very high quality conditional images by optimizing a sample from the latent space of a generative network to create an image that maximizes the class activations of a pretrained ImageNet classifier. Our work differs in that we learn an amortized generator/discriminator directly in the latent space and we achieve diversity through regularizing by the natural scale of the latent space rather than through a modified Langevin sampling algorithm.

Discussion and Future Work

We have demonstrated a new approach to conditional generation by constraining the latent space of an unconditional generative model. This approach could be extended in a number of ways.

One possibility would be to plug in different architectures, including powerful autoregressive decoders or adversarial decoder costs, as we make no assumptions specific to independent likelihoods. While we have considered constraints based on implicit density estimation, we could also estimate the constrained distribution directly with an explicit autoregressive model or another variational autoencoder. The efficacy of autoregressive priors in VAEs is promising for this approach (Kingma et al., 2016). Conditional samples could then be obtained by ancestral sampling, and transformations by using gradient ascent to increase the likelihood under the model. Active or semisupervised learning approaches could reduce the sample complexity of learning constraints. Real-time constraint learning would also enable new applications; it might be fruitful to extend the reward approximation of Section 6 to incorporate user preferences as in (Christiano et al., 2017).

Many thanks to Jascha Sohl-Dickstein, Colin Raffel, and Doug Eck for their helpful brainstorming and encouragement.

References

Appendix

For images, we use the MNIST digits dataset (LeCun et al., 2010) and the Large-scale CelebFaces Attributes (CelebA) dataset (Liu et al., 2015). MNIST images are $28\times 28$ pixels and greyscale scaled to . For attributes, we use the number class label of each digit. CelebA images are center-cropped to $128\times 128$ pixels and then downsampled to $64\times 64$ RGB pixels and scaled to . We find that many of the attribute labels are not strongly correlated with changes in the images, so we narrow the original 40 attributes to the 10 most visually salient: blond hair, black hair, brown hair, bald, eyeglasses, facial hair, hat, smiling, gender, and age.

For melodies, we scraped the web to collect over 1.5 million publicly available MIDI files. We then extracted 16-bar melodies by sliding a window with a single bar stride over each non-percussion instrument with a $\frac{4}{4}$ time signature, keeping only the note with the highest pitch when multiple overlap. This produced over 3 million unique melodies. We represent each melody as a sequence of 256 (16 per bar) categorical variables taking one of 130 discrete states at each sixteenth note: 128 note-on pitches, a hold state, and a rest state.

2 Model Architectures

All encoders, decoders, and classifiers are trained with the Adam optimizer (Kingma & Ba, 2015), with learning rate $=$ 3e-4, $\beta_{1}=$ 0.9, and $\beta_{2}=$ 0.999.

To model the MNIST data, we use a deep feed-forward neural network (Figure 13a).

The encoder is a series of 3 linear layers with 1024 outputs, each followed by a ReLU, after which an additional linear layer is used to produce 2048 outputs. Half of the outputs are used as the $\mu$ and the softplus of the other half are used as the $\sigma$ to parameterize a 1024-dimension multivariate Gaussian distribution with a diagonal covariance matrix for $z$.

The decoder is a series of 3 linear layers with 1024 outputs, each followed by a ReLU, after which an additional linear layer is used to produce 28x28 outputs. These outputs are then passed through a sigmoid to generate the output image.

2.2 CelebA Convolutional VAE

To model the CelebA data, we use a deep convolutional neural network (Figure 13b).

The encoder is a series of 4 2D convolutional layers, each followed by a ReLU. The convolution kernels are of size $3\times 3$, $3\times 3$, $5\times 5$, and $5\times 5$, with 2048, 1024, 512, and 256 output channels, respectively. All convolutional layers have a stride of 2. After the final ReLU, a linear layer is used to produce 2048 outputs. Half of the outputs are used as the $\mu$ and the softplus of the other half are used as the $\sigma$ to parameterize a 1024-dimension multivariate Gaussian distribution with a diagonal covariance matrix for $z$.

The decoder passes the $z$ through a 4x4x2048 linear layer, and then a series of 4 2D transposed convolutional layers, all but the last of which are followed by a ReLU. The deconvolution kernels are of size $5\times 5$, $5\times 5$, $3\times 3$, and $3\times 3$, with 1024, 512, 256, and 3 output channels, respectively. All deconvolution layers have a stride of 2. The output from the final deconvolution is passed through a sigmoid to generate the output image.

The classifier that is trained to predict labels from images are identical to the VAE encoders except that they end with a sigmoid cross-entropy loss.

2.3 Melody Sequence VAE

Music is fundamentally sequential, so we use an LSTM-based sequence VAE for modelling monophonic melodies (Figure 13c).

The encoder is made up of a single-layer bidirectional LSTM, with 2048 units per cell. The final output in each direction is concatenated and passed through a linear layer to produce 1024 outputs. Half of the outputs are used as the $\mu$ and the softplus of the other half are used as a $\sigma$ to parameterize a 512-dimension multivariate Gaussian distribution with a diagonal covariance matrix for $z$.

Since musical sequences often have structure at the bar level, we use a hierarchical decoder to model long melodies. First, the $z$ goes through a linear layer to initialize the state of a 2-layer LSTM with 1024 units per layer, which outputs 16 embeddings of size 512 each, one per bar. Each of these embeddings are passed through a linear layer to produce 16 initial states for another 2-layer LSTM with 1024 units per layer. This bar-level LSTM autoregressively produces individual sixteenth note events, passing its output through a linear layer and softmax to create a distribution over the 130 classes. This categorical distribution is used to compute a cross-entropy loss during training or samples at inference time. In addition to generating the initial state at the start of each bar, the embedding for the current bar is concatenated with the previous output as the input at each time step.

2.4 Actor Feed-forward Network

For $G(z)$, we use a deep feed-forward neural network (Figure 12a) in all of our experiments.

The network is a series of 4 linear layers with 2048 outputs, each followed by a ReLU, after which an additional linear layer is used to produce $2*dim(z)$ outputs. Half of the outputs are used as the $\delta z$ and the sigmoid of the other half are used as $gates$. The transformed $z^{\prime}$ is the computed as $(1-gates)*z+gates*\delta z$. This aids in training as the network only has to then predict shifts in $z$.

When conditioning on attribute labels, $y$, to compute $G(z,y)$, the labels are passed through a linear layer producing 2048 outputs which are concatenated with $z$ as the model input.

2.5 Critic Feed-forward Network

For $D(z)$, we use a deep feed-forward neural network (Figure 12b) in all of our experiments.

The network is a series of 4 linear layers with 2048 outputs, each followed by a ReLU, after which an additional linear layer is used to produce a single output. This output is passed through a sigmoid to compute $D(z)$.

When conditioning on attribute labels, $y$, to compute $D(z,y)$, the labels are passed through a linear layer producing 2048 outputs which are concatenated with $z$ as the model input.

3 Supplemental Figures