Rethinking Semi-Supervised Learning in VAEs

06/17/2020 ∙ by Tom Joy, et al. ∙ University of Oxford 21

We present an alternative approach to semi-supervision in variational autoencoders(VAEs) that incorporates labels through auxiliary variables rather than directly through the latent variables. Prior work has generally conflated the meaning of labels, i.e. the associated characteristics of interest, with the actual label values themselves-learning latent variables that directly correspond to the label values. We argue that to learn meaningful representations, semi-supervision should instead try to capture these richer characteristics and that the construction of latent variables as label values is not just unnecessary, but actively harmful. To this end, we develop a novel VAE model, the reparameterized VAE (ReVAE), which "reparameterizes" supervision through auxiliary variables and a concomitant variational objective. Through judicious structuring of mappings between latent and auxiliary variables, we show that the ReVAE can effectively learn meaningful representations of data. In particular, we demonstrate that the ReVAE is able to match, and even improve on the classification accuracy of previous approaches, but more importantly, it also allows for more effective and more general interventions to be performed. We include a demo of ReVAE at https://github.com/thwjoy/revae-demo.

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1 Introduction

Learning the characteristic factors of perceptual observations has long been desired for effective machine intelligence brooks1991intelligence ; bengio2013 ; hinton2006reducing ; tenenbaum1998mapping . In particular, the ability to learn meaningful factors—capturing human-understandable characteristics from data—has been of interest from the perspective of human-like learning tenenbaum2000separating ; lake2015human and improving decision making and generalization across tasks bengio2013 ; tenenbaum2000separating .

At a fundamental level, learning meaningful representations of data allows one to a) make predictions and b) manipulate factors, for individual data points. Prediction provides a mechanism to interpret observations in terms of the different meaningful factors. Manipulation allows for the expression of causal effects between the meaning of factors and their corresponding realizations in the data. This can be further categorized into conditional generation—the ability to construct whole exemplar data instances with characteristics dictated by constraining relevant factors—and intervention—the ability to manipulate just particular factors for a given data point, and subsequently affect only the associated characteristics. Together, the prediction and manipulation tasks can be used to construct measures of fidelity and robustness of the learned representations, and of how meaningful they actually are.

A particularly flexible and powerful framework within which to explore the learning of meaningful representations are VAE, a class of deep generative latent-variable models, where representations of data are captured in the latent variables of the underlying generative model. Learning meaningful factors in this framework typically manifests in the form of constraints, or inductive biases, locatello2019challenging ; mathieu2019disentangling ; mao2019neuro ; kingma2014semi ; siddharth2017learning ; vedantam2018generative —effected via either the model, objective, data, or the learning algorithm, or some combination thereof.

One such constraint is semi-supervised learning, where labels are provided for some small subset of the observed data. In this setting, the expectation is for the learned factors to capture the denotation—the characteristic meaning—of the given labels, by establishing a correspondence between factors and labels. The label values themselves are simply a discrete set of prototypes rosch1973natural ; gardenfors2004conceptual within the category of characteristics denoted by the label, i.e. the label values are used to refer to the class meaning, and do not describe the class by themselves. For example, the label “female” can be seen as denoting a varied set of characteristics, whereas the label value itself is simply a prototypical element in that set, and not the entire set itself (cf. Figure 1).

Figure 1: Conceptualizing the distinction between a label’s value and its denotation.

This is particularly relevant in the VAE setting, as it directly affects the question of how latent variables capture meaningful representations from labels. While it might be tempting to directly define some set of latent variables as labels, as done by prior work kingma2014semi ; siddharth2017learning ; maaloe2016auxiliary , this conflates the denotation of the labels with its values—we want the representations to capture the characteristic meaning of the labels, not just the label values themselves.

Here, we develop a novel framework for semi-supervised learning in VAE that respects the distinction between a label’s denotation and its values. The framework, which we refer to as the ReVAE, employs a novel VAE formulation that “reparameterizes” latent variables corresponding to labels through the introduction of auxiliary variables. Along with a principled variational objective and careful structuring of the mappings between latent and auxiliary variables, we show that ReVAE successfully capture meaningful representations, while also enabling better performance on prediction and manipulation tasks. In particular, they permit certain manipulation tasks that cannot be performed with conventional approaches, such as manipulating denotations without changing the labels themselves and producing multiple distinct samples consistent with the desired intervention. We summarize our contributions as follows:

  1. showing how semi-supervision can be used to disentangle rich characteristic information through careful structuring of the latent space and the introduction of auxiliary variables;

  2. formulating ReVAE, a novel model class and objective for semi-supervised learning in VAE that respects the distinction between labels and their denotations;

  3. developing of a set of quantitative measures for manipulation, including both conditional generation and interventions, that highlight the required invariances; and

  4. demonstrating ReVAE’ ability to successfully learn meaningful representations in practice.

2 Background

VAE kingma2013auto ; rezende2014stochastic are a powerful and flexible class of model that combine the unsupervised representation-learning capabilities of deep autoencoders hinton1994autoencoders with generative latent-variable models—a popular tool to capture factored low-dimensional representations of higher-dimensional observations. In contrast to deep autoencoders, generative models capture representations of data not as distinct values corresponding to observations, but rather as distributions

of values. A generative model defines a joint distribution over observed data 

and latent variables  as . Given a model, learning representations of data can be viewed as performing inference—learning the posterior distribution  that constructs the distribution of latent values for a given observation.

VAE employ amortized VI wainwright2008graphical ; kingma2013auto using the encoder and decoder of an autoencoder to transform this setup by i) taking the model likelihood 

to be parameterized by a neural network using the

decoder, and ii) constructing an amortized variational approximation  to the (intractable) posterior  using the encoder.

The variational approximation of the posterior enables effective estimation of the objective—maximizing the marginal likelihood—through importance sampling, to derive the ELBO of the model as

(1)

Given observations

taken to be realizations of random variables generated from an unknown distribution

, the overall objective is .

SSVAE kingma2014semi consider the setting where a subset of data is assumed to also have corresponding labels . Denoting the rest of the (unlabeled) data as , the log-marginal likelihood can be decomposed as

In the most common SSVAE approach, known as M2 kingma2014semi , the supervised term is estimated by taking label  to be an observed variable, yielding

The unsupervised term is then estimated by treating the label  as an unobserved latent to marginalize over. This has a caveat however—when given supervised data where  is observed,

is the only latent variable under consideration, and hence an additional classifier term 

is needed:

(2)

Subsequent work siddharth2017learning extended this setting with multiple labels, capturing arbitrary dependencies within labels, and between labels and latent variables.

In addition to M2, a simpler generative classifier model, called M1, is built in the spirit of dimensionality reduction. This simply learns a VAE over all data, and then training a classifier from the learned latent space. They also further construct a combined M1M2 model that leverages both.

3 Rethinking Semi–Supervision

The de facto assumption for most approaches to semi-supervision in VAE is that the labels correspond directly to discrete latent variables . However, this can cause a number of significant issues if we want the latent space to encapsulate not just the labels themselves, but also their denotations—the underlying characteristics that relate a datapoint to its label. For example, encapsulating the masculine characteristics of a face, not just the fact that it is a man’s face.

The first issue is that the information represented by a denotation (which is typically continuous) is more than can be stored through a single discrete variable. That is not to say this denotational information is not present in approaches like M2, but here it is entangled within the continuous latent variables, , which simultaneously contain the denotations for all the labels, rather than having the information disentangled to distinct latents associated with each label. Relatedly, it can be difficult to ensure that the VAE actually makes use of , rather than just storing all information relevant to reconstruction in the (higher capacity)

. Overcoming this is challenging and generally requires additional heuristics and hyperparameters, such as the need to tune

in (2).

Second, we may wish to manipulate aspects of the denotation without fully changing the label itself. For example, making somebody look more or less feminine without fully changing their gender. Here we do not know how to manipulate to achieve this desired effect: we can only do the binary operation of changing the relevant variable in . Relatedly, we often wish to keep a level of diversity when carrying out conditional generation and, in particular, interventions. For example, if we change somebody’s gender then there is no single correct answer for how they would then look, but taking only allows for a single point estimate for the change.

Finally, taking the labels to be explicit latent variables can cause a mismatch between the VAE prior and the pushforward distribution of the data to the latent space . During training, latents are effectively generated according to , but once learned, is used to make generations; variations between the two effectively corresponds to a train-test mismatch. As there is a ground truth data distribution over the labels (which are typically not independent), taking as the labels themselves implies that there will be a ground truth . However, as this is not generally known a priori, we will inevitably end up with a mismatch.

What do we want from semi-supervision?

Given these issues, it is natural to ask whether is actually necessary? To answer this, we need to think about exactly what it is we are hoping to achieve through the semi-supervision itself. Along with uses of VAE more generally, the three most common tasks SSVAE are used for are: a) Classification, we try to predict the labels of inputs where these are not known a priori; b) Conditional Generation, we generate new examples conditioned on those examples conforming to certain desired labels; and c) Intervention, we manipulate certain desired characteristics of a data point before reconstructing it.

Previous work has often implicitly assumed that carrying out these tasks requires the labels to correspond directly to latents. However, on close inspection we see that this is not actually the case. For classification, we need only some classifier going from to . For conditional generation, we need a mechanism for sampling given . For interventions, we need to know which latent variables relate to each label and a mechanism for manipulating, or in some cases resampling, them.

4 Reparameterized Variational Autoencoders

To demonstrate how one might apply the insights of the last section, we first introduce a simplistic approach: jointly learning a classifier with the VAE. Specifically, reflecting the fact that label information only captures particular aspects of data, we partition the latent space into disjoint subsets: , to encapsulate the label denotations, and for the rest. We then construct the objective

(3)

where is a hyperparameter that controls the trade-off between the ELBO and the classifier. In this formulation, which we refer to as M3, will implicitly influence the latent embedding; we can interpret as a stochastic layer in the overall classification. Critically, as is also used in the reconstructions, it will also contain the information required for this, including the denotations.

However, in general, the denotations of different class labels will be entangled within : though it will contain the required information, the latents will typically not be interpretable, and it is unclear how we could perform conditional generation or interventions. To disentangle the denotations of different labels, we further partition the latent space, such that the classification of particular labels only has access to particular latents and thus .

This has the critical effect of forcing the denotational information needed to classify to be stored in the corresponding , providing a means to encapsulate the characteristic information of each label separately. We further see that it addresses many of the prior issues: there are no measure-theoretic issues as is not discrete, diversity in interventions is achieved by sampling different for a given label, can be manipulated while remaining within class decision boundaries, and a mismatch between and does not manifest as there is no longer ground truth for .

However, how to conditionally generate or intervene with M3 is not immediately obvious. Critically, the classifier implicitly contains the requisite information to do this via inference in an implied Bayesian model. For example, conditional generation needs samples from that classify to the desired labels, e.g. through rejection sampling. See Appendix A for further details.

4.1 The Reparameterized Variational Autoencoder

Figure 2: ReVAE graphical model.

One way to address the need for inference at test time is to introduce a conditional generative model , simultaneously learnt alongside existing components of M3, along with a prior that is either learnt or held fixed. This approach, which we term the ReVAE, allows the required sampling for conditional generations and interventions directly. Further, by persisting with the latent partitioning above, we can introduce a factorized set of generative models , enabling easy generation and manipulation of individually. This approach has the advantage of ensuring that the labels remain a part of the model for unlabelled datapoints, which transpires to be an important component for effective learning in practice.

To address the issue of learning and semi-supervision, we perform variational inference, treating as an observation in the supervised case, and an additional variable in the unsupervised case. The final graphical model associated with our variational model is illustrated in Figure 2. The ReVAE can be seen as a way of combining top-down and bottom-up information to obtain a structured latent representation. By enforcing different auxiliary variables to link to a single latent dimension, we are able to align the labelled generative factors with the axes in the latent space.

4.2 Model Objective

We now construct an objective function that encapsulates the model described above, by deriving a lower bound on the full model log-likelihood which factors over the supervised and unsupervised subsets as discussed in § 2. The supervised objective can be defined as

(4)

with . Here, we avoid directly modelling ; instead leveraging the conditional independence , along with Bayes rule, to give

Using this equivalence in eq. 4 yields (see § B.1 for a derivation and numerical details)

(5)

Note that unlike M2, M3, and similar models, a classifier term falls out naturally from the derivation. Reparametrising labels as auxiliary variables rather than directly as latent variables is crucial for this feature. When defining latents directly to be labels (as in M2), observing both   and   detaches the mapping between them, resulting in the parameters () not being learned—motivating addition of an explicit (weighted) classifier. Here however, observing both data  and label  does not detach any mapping, since they are always connected via an unobserved random variable , and hence do not need additional terms.

The unsupervised part of the objective, , derives as the standard (unsupervised) ELBO. However, it requires marginalising over labels as . This can be computed exactly, but doing so can be prohibitively expensive if the number of possible label combinations is large. Here, we apply Jensen’s inequality a second time to the expectation over to produce a looser, but cheaper to calculate, ELBO. See § B.2 for details.

Putting this together, we get the complete ReVAE objective as

(6)

which, unlike some prior approaches, is a valid lower bound on the evidence. It is interesting to note that explicitly modelling the connection between labels and their corresponding latent variables yields such a markedly different objective. As we shall see in (§ 6), this enables a range of capabilities and behaviors that encapsulate the distinction between a label’s value and its denotation.

5 Related Work

Beyond the immediate connections to prior work described in § 2, there are a number of other existing approaches that are related to our proposed framework. An auxiliary-variable approach maaloe2016auxiliary more related to the M2 model augments the encoding distribution with an additional, unobserved latent variable, that enables better semi-supervised classification accuracies. From a pure modeling perspective, there also exists prior work on hieararchical VAE ranganath2016hierarchical ; zhao2017learning that involve hierarchies of latent variables, exploring richer higher-order inference and issues with redundancy among latent variables in an unsupervised setting. Unlike our approach, these auxiliary or hierarchical variables do not have a direct interpretation, but exist merely to improve the flexibility of the encoder. Regarding the disparity between continuous and discrete latent variables in the typical semi-supervised VAE, dupont2018learning provide an approach to enable effective unsupervised learning in this setting.

From a more conceptual standpoint, there are two approaches that also incorporate the separation of labels and latent variables. The first of these mueller2017sequence introduces interventions (called revisions) on VAE for text data, regressing to auxiliary sentiment scores as a means of influencing the latent variables. This formulation is similar to M3 eq. 3 in spirit, although in practice they employ a range of additional factoring and regularizations particular to their domain of interest, in addition to training models in stages, involving different objective terms. Nonetheless, we share the desire to enforce meaningfulness in the latent representations through auxiliary supervision.

The other approach involves explicitly treating labels as another data modality vedantam2018generative ; JMVAE_suzuki_1611 ; MVAE_wu_2018 ; shi2019mmvae . This work is motivated by the need to learn latent representations that jointly encode data from different modalities. Looking back to eq. 4, by refactoring as , and taking , one derives multi-modal VAE, where can construct a product MVAE_wu_2018 or mixture shi2019mmvae of experts. Of these, the MVAE MVAE_wu_2018 is more closely related to our setup here, as it explicitly targets cases where alternate data modalities are labels—only ever containing information that is a subset of the information contained in the data. However, they differ in that the latent representations are not structured explicitly to map to distinct classifiers, and do not explore the question of explicitly capturing the denotations of the labels in the latent representations.

6 Experiments

Following our reasoning in § 3 we now showcase the efficacy of ReVAE for the three broad aims of (a) classification, (b) conditional generation and (c) intervention on the FashionMNIST and CelebA dataset (restricing ourselves to the 18 labels which are distinguishable in reconstructions for the former, see § C.1 for details). For our encoder and decoder we use MLPs for FashionMNIST and standard architectures higgins2016beta for CelebA. The label-predictive distribution is defined as with MLP  for FashionMNIST, and as with a diagonal transformation  enforcing for CelebA. The conditional prior is defined as , with appropriate factorization for CelebA, and has its parameters also derived through MLPs. For FashionMNIST we calculate analytically as . For CelebA this is not feasible so we employ the additional variational bounding described in Section 4.2. See § C.2 for further details.

Classification

We first inspect the predictive ability of ReVAE for classification across a range of supervision rates. Table 1 shows the classification accuracies of the label predictive distribution learned by M2, M3 and ReVAE. It can be observed that ReVAE generally obtains prediction accuracies equivalent or slightly superior to M2 (except in the case of very low supervision on FashionMNIST) and much better than M3. We emphasize here that ReVAE is not setup up to achieve better classification accuracies; we are simply checking that it does not harm them.

FashionMNIST CelebA
Model
ReVAE 0.680.02 0.820.01 0.850.01 0.800.00 0.880.00 0.880.00
M2 0.730.03 0.830.00 0.850.00 0.790.01 0.860.00 0.870.00
M3 0.220.07 0.340.14 0.310.06 0.520.02 0.640.02 0.730.00
Table 1: Classification accuracies. Boldface denotes the best performing model and models whose performance is not statistically significantly different to it according to a non-parametric Mann–Whitney test.

Conditional Generation

To asses conditional generation, we first train an independent classifier for both datasets. We then conditionally generate samples given labels and evaluate them using this pre-trained classifier. To further give an indication of how “real” the generations are, i.e. whether they are overall out-of-distribution, we also measure the mutual information (MI) between the parameters of the classifier and the labels as per smith2018understanding . This provides a measure the epistemic uncertainty of the classifier, with lower MI values being preferable. See § C.3 for more details. Results are shown in Table 2. We can see that in all cases for FashionMNIST ReVAE outperforms M2, indicating that having a diverse set of generations actually improves performance. For CelebA, ReVAE also outperforms M2 for the accuracy of the generations. To measure visual fidelity, rather than MI we instead report the FID heusel2017gans , both M2 and ReVAE perform comparably, unlike M3 which is significantly worse, possible due to more weight being applied to the classifier than the VAE.

Model
Acc MI Acc MI Acc MI
FMNIST ReVAE 0.610.02 0.030.00 0.720.02 0.030.00 0.760.00 0.020.00
M2 0.550.01 0.060.00 0.680.01 0.050.00 0.69 0.00 0.050.00
M3 0.190.06 0.040.01 0.270.11 0.040.01 0.210.04 0.080.02
Acc FID Acc FID Acc FID
CelebA ReVAE 0.490.00 130.02.10 0.550.04 122.61.10 0.590.00 121.01.60
M2 0.430.00 129.21.79 0.530.00 121.41.69 0.530.00 118.42.00
M3 0.490.01 190.013.5 0.490.01 159.313.2 0.490.01 301.368.7
Table 2: Pre-trained clasifier accuracies and MI for FashionMNIST (top), and pre-trained clasifier accuracies and FID for CelebA (bottom). Boldface denotes the best performing model and models whose performance is not statistically significantly different to it according to a non-parametric Mann–Whitney test.
Figure 3: From left to right: original, reconstruction, then interventions from switching on the following labels: arched eyebrows, bags under eyes, bangs, black hair, blond hair, brown hair, bushy eyebrows, chubby, eyeglasses, heavy makeup, male, no beard, pale skin, receding hairline, smiling, wavy hair, wearing necktie, young.
Figure 4:

Intervention confusion matrices for M2 (left) and ReVAE (right). Here we intervene on the label of a column an report the probability change for the class given by the row. Condition number is given in the title.

Single Interventions

To assess the fidelity of interventions, we first consider intervening on a single label. For ReVAE, this corresponds to sampling from in the dimension of the class we want to intervene on. We demonstrate the qualitative results for ReVAE in Figure 3

, which shows only a single attribute changing in each column. Equivalent plots for M2 and M3 are given in the Appendix. We further quantitatively assess these intervention by constructing a confusion matrix on how interventions affect the prediction probability of

each class, the result of which are given in Figure 4

. Here, perfect interventions should produce an identity matrix and we see that ReVAE outperforms M2 as reflected in its lower matrix condition number.

Full Interventions

We next consider making wholesale interventions through “denotation swaps.” Namely, we encode an image, replace its with the of another image, and then reconstruct the image while holding constant. For ReVAE, this has the effect of applying the denotations of the second image to the first. By comparison, M2 only allows for the labels to be transferred and not the more subtle denotation information. The results, shown in Figure 5, reveal that this is indeed the case. The differences are perhaps most obvious in how for ReVAE, but not M2, the bangs are preserved in the fifth row and the grinning nature of the smile is preserved in the third row. The figure further provides a quantitative measure for these interventions: we report the difference in log probabilities of the classes when evaluated on a pre-trained classifier of the reconstruction, averaging these values over a large number of such interventions. Here we see ReVAE outperforms M2 on every class, providing a quantitative confirmation of its superior intervention performance.

Figure 5: [Left, Middle] “Denotation swaps” for M2 and ReVAE respectively. Here, the denotations of the leftmost column are applied to the top image, such that rows should contain the same characteristics. [Right] quantitative results produced by averaging the differences in log probabilities of each class from performing randomly chosen such interventions for each datapoint over the whole dataset (lower is better).

Intervention Diversity

As outlined in Section 3, there are typically multiple different interventions that are consistent with manipulating a label. For example, if we add glasses to a person, there are multiple styles of glasses we might add. We demonstrate ReVAE’ ability to encapsulate this diversity in Figure 6 where we draw multiple sample interventions and look at the different reconstructions it produces. We see that ReVAE produces a diverse set of label specific realizations. By comparison, such a plot cannot be generated at all for M2 as it only allows single point estimate interventions.

Figure 6: Diverse interventions on smiling and eyeglasses for CelebA.
Figure 7: Continuous interventions through traversal of . From left to second from the right: CelebA latent walks between pale skin and young; smiling and necktie; male and beard

. Far right: interpolation on surface in the latent space between the four FashionMNIST classes:

t-shirt, pullover, coat and dress.

Latent Walk Interventions

By manipulating a particular , ReVAE is able to smoothly manipulate the characteristics of an image relating to a given label, without severely affecting others. This allows for a fine control during intervention, unlike M2 which can only make binary changes. To demonstrate this, we traverse two dimensions of the latent space and display the reconstructions in Figure 7. These examples indicate that ReVAE is able to smoothly manipulate characteristics through its encapsulation and disentangling of denotations; no such traversals are possible for M2. For example, in the leftmost set of images we are able to induce varying skin tones rather than have this be a binary intervention on pale skin. In the second set, we find that the associated with the necktie label has also managed to encapsulate information about whether someone is wearing a shirt or is completely bare-necked. In the third, we are able to separately encapsulate the length of beard and gender as continuous variables that have separate impacts on the image. Finally, in the last set, we see that we are able to smoothly interpolate between classes in FashionMNIST, e.g. going from a t-shirt to a dress involves a steady elongation of the torso, as one would expect.

7 Discussion

We have presented a novel mechanism for performing semi-supervised learning in deep generative models, the ReVAE, wherein we avoid a direct correspondence between labels and latents, and instead treat the labels as auxiliary variables. This has allowed us to encapsulate and disentangle the denotations associated with labels, rather than just the label values. We are able to do so without affecting the ability to perform the tasks one typically does in the semi-supervised setting—namely classification, conditional generation, and intervention. In particular, we have shown that, not only does this lead to more effective conventional label-switch interventions, it also allows for more fine-grained interventions to be performed, such as producing diverse sets of samples consistent with an intervened label value, or performing continuous traversals of the denotation space both within and across class labels.

Broader Impact

Manipulating generative factors of data comes with obvious advantage such as the ability to manipulate certain characteristics without affecting others, e.g. seeing what someone will look like with a different hair color, or when wearing glasses. However, the ability to do so on such personal and potentially sensitive features leads to serious thought into the ethical considerations about how such approaches should be used. Moreover with the ever pressing issue of Deep-Fakeskorshunov2018deepfakes undermining the confidence of images representing a true scene, this work has the potential to stoke this problem even further. While this concern is very real, it is purely application based and limited to photographs of people—a domain which ReVAE certainly is not exclusively tied to. Moreover, these issues are common to all of semi–supervised learning in deep generative models, rather than being specific to our work. The conceptual and methodological ideas presented in the paper draw attention to how representations of denotations should be stored in latent variable models, which is potentially very useful in certain domains. Therefore, though the aforementioned concerns should not be forgotten, we do not feel like they are a basis to avoid pursuing such avenues of research.

TJ, PHST, and NS were supported by the ERC grant ERC-2012-AdG 321162-HELIOS, EPSRC grant Seebibyte EP/M013774/1 and EPSRC/MURI grant EP/N019474/1. Toshiba Research Europe also support TJ. PHST would also like to acknowledge the Royal Academy of Engineering and FiveAI. SMS was partially supported by the Engineering and Physical Sciences Research Council (EPSRC) grant EP/K503113/1. TR’s research leading to these results has received funding from a Christ Church Oxford Junior Research Fellowship and from Tencent AI Labs.

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Appendix A Conditional Generation and Intervention for M3

For the M3 model to be usable, we must consider whether it can carry out the classification, conditional generation, and intervention tasks outlined in the last section. Of these, classification is obviously trivial, but it is less immediately apparent how the others could be performed. The key here is to realize that the classifier itself implicitly contains the information required to perform these tasks.

Consider first conditional generation and note that we still have access to the prior as per a standard VAE. One simple way of performing conditional generation would be to conduct a rejection sampling where we draw samples and then accept these if and only if they lead to the classifier predicting the desired labels up to a desired level of confidence, i.e.  where is some chosen confidence threshold. Though such an approach is likely to be highly inefficient for any general

due to the curse of dimensionality, in the standard setting where each dimension of

is independent, this rejection sampling can be performed separately for each , for which it which actually often be relatively painless. More generally, we have that conditional generation becomes an inference problem where we wish to draw samples from

Interventions can also be performed in an analogous manner. Namely, for a conventional intervention where we change one or more labels, we can simply resample the associated we those labels, thereby sampling new denotations to match the new labels. Further, unlike in previous approaches, there are alternative interventions we can perform as well. For example, we might attempt to find the closest to the original that leads to the class label changing; this can be done in a manner akin to how adversarial attacks are performed. Alternatively, we might look to manipulate the without actually changing the class itself to see what other denotations are consistent with the labels.

To summarize, this M4 model provides a mechanism of learning semi-supervised VAE that avoid the pitfalls of directly fixing the latents to correspond to labels. It still allows us to perform all the tasks usually associated with semi-supervised VAE and in fact allows a more general form of interventions to be performed. However, this comes at the cost of requiring inference to be performed to perform conditional generation or interventions. Further, as the auxiliary variables are not present when the labels are unobserved, there may be empirical complications with forcing all the denotational information to be encoded to the appropriate . In particular, we still have a hyperparameter that must be carefully tuned to ensure the appropriate balance between classification and reconstruction.

Appendix B Model Formulation

b.1 Variational Lower Bound

In this section we provide the mathematical details of our objective functions. We show how to derive it as a lower bound to the marginal model likelihood and show how we estimate the model components.

The variational lower bound for the generative model in Figure 2, is given as

The overall likelihood in the semi-supervised case is given as

To derive a lower bound for the overall objective, we need to obtain lower bounds on and . When the labels are unobserved the latent state will consist of and . Using the factorization according to the graph in Figure 2 yields

where . For supervised data points we consider a lower bound on the likelihood ,

in order to make sense of the term , which is usually different from we consider the inference model

Returning to the lower bound on we obtain

where denotes the Radon-Nikodym derivative of with respect to .

b.2 Alternative Derivation of Unsupervised Bound

The bound for the unsupervised case can alternatively be derived by applying Jensen’s inequality twice. First, use the standard (unsupervised) ELBO

Now, since calculating can be expensive we can apply Jensen’s inequality a second time to the expectation over to obtain

Substituting this bound into the unsupervised ELBO yields again our bound

(7)

Appendix C Implementation

c.1 CelebA

We chose to use only a subset of the labels present in CelebA. The reason for this is two-fold: firstly, not all attributes are visually distinguishable in the reconstructions e.g. (earrings); secondly, some of the attributes are potentially offensive e.g (attractive). As such we limited ourselves to the following labels: arched eyebrows, bags under eyes, bangs, black hair, blond hair, brown hair, bushy eyebrows, chubby, eyeglasses, heavy makeup, male, no beard, pale skin, receding hairline, smiling, wavy hair, wearing necktie, young. No images were omitted or cropped, the only modifications were keeping the aforementioned labels and resizing the images to be 64 64 in dimension.

c.2 Implementation Details

For our experiments we define the generative and inference networks as follows. The approximate posterior is represented as with and being an MLP for FashionMNIST and the architecture from higgins2016beta for CelebA. The generative model

is represented by a Bernoulli distribution and also parametrised by an MLP for FashionMNIST and a Laplace distribution, again parametrised using the architecture from

higgins2016beta for CelebA. The label predictive distribution is represented as with being an MLP for FashionMNIST, or as with being a diagonal transformation forcing the factorisation for CelebA. The conditional prior is given as (with the appropriate factorisation for CelebA) where the parameters are represented by an MLP. Finally, the prior placed on the portion of the latent space reserved for unlabelled latent variables is . For the latent space and , where with and for FashionMNIST and and for CelebA. The architectures are given in Table 3 and Table 4.

Encoder Decoder
Input 1 x 28 x 28 image Input
784

600 Linear layer & ReLU

600 600 Linear layer & ReLU
600 600 Linear layer & ReLU 600 600 Linear layer & ReLU
600 (2) Linear layer 600 784 Linear layer & Sigmoid
Classifier Conditional Prior
Input Input
50 50 Linear layer & ReLU 10 50 Linear layer & ReLU
50 10 Linear layer 50 Linear layer
Table 3: Architectures for FashionMNIST and MNIST.
Encoder Decoder
Input 32 x 32 x 3 channel image Input

Conv2d stride 2 & ReLU

(2) 256 Linear layer
Conv2d stride 2 & ReLU ConvTranspose2d stride 1 & ReLU
Conv2d stride 2 & ReLU ConvTranspose2d stride 2 & ReLU
Conv2d stride 2 & ReLU ConvTranspose2d stride 2 & ReLU
Conv2d stride 1 & ReLU ConvTranspose2d stride 2 & ReLU
256 (2) Linear layer ConvTranspose2d stride 2 & Sigmoid
(a) Fashion-MNIST dataset.
Classifier Conditional Prior
Input Input
Diagonal layer Diagonal layer
Table 4: Architectures for CelebA.

Optimization

To perform the optimization, we trained the models on a GeForce GTX Titan GPU. Training consumed Gb memory for FashionMNIST and

Gb for CelebA, taking around 20 minutes and 4 hours to complete 100 epochs respectively. Both models were optimized using Adam with a learning rate of

and for FashionMNIST and CelebA respectively.

c.3 Classifier Uncertainty and Mutual Information

We use classifier uncertainty as an out-of-distribution measure on generated or intervened data. In order to estimate the uncertainty, we transform a fixed pre-trained classifier into a Bayesian predictive classifier that integrates over the posterior distribution of parameters  as . The utility of classifier uncertainties for out-of-distribution detection has previously been explored smith2018understanding , where dropout is also used at test time to estimate the mutual information (MI) between the predicted label  and parameters  gal2016uncertainty ; smith2018understanding as.

However, the Monte-Carlo (MC) dropout approach has the disadvantage of requiring ensembling over multiple instances of the classifier for a robust estimate and repeated forward passes through the classifier to estimate MI. To mitigate this, we instead employ a sparse variational GP (with 200 inducing points) as a replacement for the last linear layer of the classifier, fitting just the GP to the data and labels while holding the rest of the classifier fixed. This, in our experience, provides a more robust and cheaper alternative to MC-dropout for estimating MI.

Appendix D Additional Results

d.1 Classification and Generation

For the case where classification and generation is the primary goal, we can improve the resulting accuracies by setting , that is, forcing the classifier to use the whole of the latent space . The results for classification and generation are given in Table 5 and Table 6, Subscript indicates the size of .

The results indicate that when the primary goal is purely to perform classification or conditional generation, by not splitting the latent space, ReVAE is able to obtain superior results, particularly for the case of conditional generation. We posit that this is due to M2 having to sample from the continuous and the discrete latent space, which as we showed in the main paper, entangles class specific denotations. As such, situations could arise where and potentially provide conflicting information. Conversely, ReVAE learns to structure the latent space such that certain regions correspond to certain classes which implicitly contain the style information.

FashionMNIST MNIST
Model
ReVAE20 0.750.02 0.830.01 0.840.01 0.930.01 0.970.00 0.970.02
ReVAE10 0.750.01 0.830.01 0.860.00 0.930.00 0.960.00 0.970.00
M2 0.730.03 0.830.00 0.850.00 0.900.01 0.940.00 0.960.00
Table 5: Additional classification accuracies.
Model
Acc MI Acc MI Acc MI
FMNIST ReVAE20 0.700.02 0.030.00 0.760.02 0.030.00 0.780.02 0.030.00
ReVAE10 0.710.02 0.030.00 0.760.04 0.030.00 0.800.01 0.020.00
M2 0.550.01 0.060.00 0.680.02 0.050.00 0.69 0.01 0.050.00
MNIST ReVAE20 0.900.01 0.020.00 0.940.00 0.020.00 0.940.03 0.020.00
ReVAE10 0.910.01 0.020.00 0.950.00 0.010.00 0.960.00 0.010.00
M2 0.790.02 0.060.00 0.890.01 0.040.00 0.900.00 0.030.00
Table 6: Pre-trained classifier accuracies and MI for FashionMNIST (top) and MNIST (bottom).

d.2 Supervision using only a single label

When performing semi-supervision, the question of ‘how low can we go?’ naturally arises. Here we show that we can drop the supervision rate to the lowest possible level, that is, only having one label for a single instance of each class. This is achieved through an additional term, which makes use of the fact we can sample from and evaluate the likelihood on .

A naïve way to increase the performance of the classifier, is to sample and then maximize using as the labels. Thus increasing the strength of the gradients to the classifier. A simple regularizer incorporating this objective yields:

However, there is no guarantee that a sample , will fall in the same decision boundary as , where is the label for , thus providing adverse gradients. This is particularly common early on in training where the encoder distribution and the conditional prior do not have significant overlap. To counter this, rather than using from to evaluate , we instead use , where is stored from the single supervised example at a very small cost. This effectively acts as a NN classifier to obtain the appropriate label, but removes the dependence on a match between and . With this approach, the final additional term is given as:

With , the resulting objective allows us to achieve 76.7% accuracy on MNIST where supervision is present for only a single random image for each class.

d.3 CelebA Interventions

We would like to highlight the addition of a demo application included in the supplementary (‘./demo/main.ipynb’). The demo provides a user interface to alter characteristics on a chosen image. The demo performs latent walk interventions along the latent dimension for the corresponding chosen characteristic, thus providing a way alter multiple attributes as apposed to a pair in main paper (Figure 5). A screenshot is given in Figure 8, where the original image has been altered to add a smile and sunglasses.

Figure 8: Screenshot of demo included in the supplementary.

We present additional qualitative results for single interventions on ReVAE (Figure 9) and M2 (Figure 10).

Figure 9: ReVAE. From left to right: original, reconstruction, then interventions from switching on the following labels: arched eyebrows, bags under eyes, bangs, black hair, blond hair, brown hair, bushy eyebrows, chubby, eyeglasses, heavy makeup, male, no beard, pale skin, receding hairline, smiling, wavy hair, wearing necktie, young.
Figure 10: M2. From left to right: original, reconstruction, then interventions from switching on the following labels: arched eyebrows, bags under eyes, bangs, black hair, blond hair, brown hair, bushy eyebrows, chubby, eyeglasses, heavy makeup, male, no beard, pale skin, receding hairline, smiling, wavy hair, wearing necktie, young.