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Hierarchical Bayesian methods have the potential to unify many related tasks (e.g. k-shot classification, conditional, and unconditional generation) by framing each as inference within a single generative model.
We show that existing approaches for learning such models can fail on expressive generative networks such as PixelCNNs, by describing the global distribution with little reliance on latent variables.
To address this, we develop a modification of the Variational Autoencoder in which encoded observations are decoded to new elements from the same class; the result, which we call a Variational Homoencoder (VHE), may be understood as training a hierarchical latent variable model which better utilises latent variables in these cases.
Using this framework enables us to train a hierarchical PixelCNN for the Omniglot dataset, outperforming all existing models on test set likelihood.
With a single model we achieve both strong one-shot generation and near human-level classification, competitive with state-of-the-art discriminative classifiers.
The VHE objective extends naturally to richer dataset structures such as factorial or hierarchical categories, as we illustrate by training models to separate character content from simple variations in drawing style, and to generalise the style of an alphabet to new characters.
Learning from few examples is possible only with strong inductive biases.
In machine learning such biases can come from hand design, as in the parametrisation of a model, or can be the result of a meta-learning algorithm.
Furthermore they may be task-specific, as in discriminative modelling, or may describe the world causally so as to be naturally reused across many tasks.
Recent work has approached one-and few-shot learning from all of these perspectives.
Siamese Networks BID15 , Matching Networks , Prototypical Networks BID23 and MANNs BID22 ) are all models discriminatively trained for few-shot classification.
Such models can achieve state-of-the-art performance at the task they were trained for, but provide no principled means for transferring knowledge to other tasks.
Other work such as has developed conditional generative models, which take one or a few observations from a class as input, and return a distribution over new elements p(x|D).
These models may be used as classifiers despite not being explicitly trained for this purpose, by comparing conditional likelihoods.
They may also be used to generate full sets incrementally as p(X) = i p(x i |x 1 , . . .
, x i−1 ), as discussed in Generative Matching Networks BID0 .
However, such models are a more natural fit to sequences than sets as they typically lack exchangeability, and furthermore they do not expose any latent representation of shared structure within a set.
Finally are hierarchical approaches that model shared structure through latent variables, as p(X) = A VAE treats all datapoints as independent, so only a single random element need be encoded and decoded each step.
A Neural Statistician instead feeds a full set of elements X through both encoder and decoder networks, in order to share a latent variable c. In a VHE, we bound the full likelihood p(X) using only random subsamples D and x for encoding/decoding.
Optionally, p(x|c) may be defined through a local latent variable z.
In this work we propose the Variational Homoencoder (VHE), aiming to combine several advantages of the models described above:1.
Like conditional generative approaches, we train on a few-shot generation objective which matches how our model may be used at test time.
However, by introducing an encoding cost, we simultaneously optimise a likelihood lower bound for a hierarchical generative model, in which structure shared across elements is made explicit by shared latent variables.
2.
Previous work BID6 has learned hierarchical Bayesian models by applying Variational Autoencoders to sets, such as classes of images.
However, their approach requires feeding a full set through the model per gradient step FIG1 ), rendering it intractable to train on very large sets.
In practice, they avoid computational limits by sampling smaller subset as training data.
In a VHE, we instead optimise a likelihood bound for the complete dataset, while constructing this bound by subsampling.
This approach can not only improve generalisation, but also departs from previous work by extending to models with richer latent structure, for which the joint likelihood cannot be factorised.
3.
As with a VAE, the VHE objective includes both an encoding-and reconstruction-cost.
However, by sharing latent variables across a large set of elements, the encoding cost per element is reduced significantly.
This facilitates use of powerful autoregressive decoders, which otherwise often suffer from ignoring latent variables BID3 .
We demonstrate the significance of this by applying a VHE to the Omniglot dataset.
Using a Pixel-CNN decoder (Oord et al., 2016), our generative model is arguably the first with a general purpose architecture to both attain near human-level one-shot classification performance and produce high quality samples in one-shot generation.
When dealing with latent variable models of the form p(x) = z p(z)p(x|z)dz, the integration is necessary for both learning and inference but is often intractable to compute in closed form.
Variational Autoencoders (VAEs, Kingma & Welling (2013)) provide a method for learning such models by utilising neural-network based approximate posterior inference.
Specifically, a VAE comprises a generative network p(z)p(x|z) parametrised by θ, alongside a separate inference network q(z; x) parameterised by φ.
These are trained jointly to maximise a single objective: DISPLAYFORM0 DISPLAYFORM1 As can be seen from Equation 1, this objective L X is a lower bound on the total log likelihood of the dataset x∈X log p(x), while q(z; x) is trained to approximate the true posterior p(z|x) as accurately as possible.
If it could match this distribution exactly then the bound would be tight so that the VAE objective equals the true log likelihood of the data.
In practice, the resulting model is typically a compromise between two goals: pulling p towards a distribution that assigns high likelihood to the data, but also towards one which allows accurate inference by q. Equation 2 provides a formulation for the same objective which can be optimised stochastically, using Monte-Carlo integration to approximate the expectation.
The Neural Statistician BID6 ) is a Variational Autoencoder in which each item to be encoded is itself a set, such as the set X (i) of all images with a particular class label i: DISPLAYFORM0 The generative model for sets, p(X), is described by introduction of a corresponding latent variable c. Given c, individual x ∈ X are conditionally independent: DISPLAYFORM1 This functional form is justified by de Finetti's theorem under the assumption that elements within in each set X are exchangeable.
The likelihood is again intractable to compute, but it can be bounded below via: DISPLAYFORM2
Unfortunately, calculating the variational lower bound for each set X requires evaluating both q(c; X) and p(X|c), meaning that the entire set must be passed through both networks for each gradient update.
This can easily become intractable for classes with hundreds of examples.
Indeed, previous work BID6 ensures that sets used for training are always of small size by instead maximising a likelihood lower-bound for randomly sampled subsets.
In this work we instead replace the variational lower-bound in Equation 5 with a new objective, itself constructed via sub-sampled datasets of reduced size.
We use a constrained variational distribution q(c; D), D ⊆ X for posterior inference and an unbiased stochastic approximation log p(x|c), x ∈ X for the likelihood.
In the following section we show that the resulting objective can be interpreted as a lower-bound on the log-likelihood of the data.
This bound will typically be loose due to stochasticity in sampling D, and we view this as a regularization strategy: we aim to learn latent representations that are quickly inferable from a small number of instances, and the VHE objective is tailored for this purpose.
We would like to learn a generative model for sets X of the form DISPLAYFORM0 We will refer our full dataset as a union of disjoint sets X = X 1 X 2 . . .
X n , and use X (x) to refer to the set X i x. Using the standard consequent of Jensen's inequality, we can lower bound the log-likelihood of each set X using an arbitrary distribution q. In particular, we give q as a fixed function of arbitrary data.
DISPLAYFORM1 DISPLAYFORM2 where DISPLAYFORM3 Reparametrization gradient estimate using c, z (θ, φ) ← (θ, φ) + λg Gradient step, e.g SGD until convergence of (θ, φ)Splitting up individual likelihoods, we may rewrite DISPLAYFORM4 Finally, we can replace the universal quantification with an expectation under any distribution of D (e.g. uniform sampling from X without replacement): DISPLAYFORM5 This formulation suggests a simple modification to the VAE training procedure, as shown in Algorithm 1.
At each iteration we select an element x, use resampled elements D ⊂ X (x) to construct the approximate posterior q(c; D), and rescale the encoding cost appropriately.
If the generative model p(x|c) also describes a separate latent variable z for each element, we may simply introduce a second inference network q(z; c, x) in order to replace the exact reconstruction error of Equation 9 by a conditional VAE bound: DISPLAYFORM6
The above derivation applies to a dataset partitioned into disjoint subsets X = X 1 X 2 . . .
X n , each with a corresponding latent variable c i .
However, many datasets offer a richer organisational structure, such as the hierarchical grouping of characters into alphabets (Lake et al., 2015) or the factorial categorisation of rendered faces by identity, pose and lighting BID16 .Provided that such organisational structure is known in advance, we may generalise the training objective in FIG1 to include a separate latent variable c i for each group X i within the dataset, even when these groups overlap.
To do this we first rewrite this bound in its most general form, where c collects all latent variables: DISPLAYFORM0 As shown in Figure 2 , a separate D i ⊂ X i may be subsampled for inference of each latent variable DISPLAYFORM1 This leads to an analogous training objective (Equation 15), which Figure 2 : Application of VHE framework to hierarchical (left) and factorial (right) models.
Given an element x such that x ∈ X 1 and x ∈ X 2 , an approximate posterior is constructed for the corresponding shared latent variables c 1 , c 2 using subsampled sets DISPLAYFORM2 may be applied to data with factorial or hierarchical category structure.
For the hierarchical case, this objective may be further modified to infer layers sequentially, as in Supplementary Material 6.2.
DISPLAYFORM3 3.3 DISCUSSION
As evident in Equation 9, the VHE objective provides a formal motivation for KL rescaling in the variational objective (a common technique to increase use of latent variables in VAEs) by sharing these variables across many elements.
This is of particular importance when using autoregressive decoder models, for which a common failure mode is to learn a decoder p(x|z) with no dependence on the latent space, thus avoiding the encoding cost.
In the context of VAEs, this particular issue has been discussed by BID3 who suggest crippling the decoder as a potential remedy.
The same failure mode can occur when training a VAE for sets if the inference network q is not able to reduce its approximation error D KL [q(c; D) p(c|D)] to below the total correlation of D, either because |D| is too small, or the inference network q is too weak.
Variational Homoencoders suggest a potential remedy to this, encouraging use of the latent space by reusing the same latent variables across a large set X. This allows a VHE to learn useful representations even with |D| = 1, while at the same time utilising a powerful decoder model to achieve highly accurate density estimation.
In a VAE, use of a recognition network encourages learning of generative models whose structure permits accurate amortised inference.
In a VHE, this recognition network takes only a small subsample as input, which additionally encourages that the true posterior p(c|X) can be well approximated from only a few examples of X. For a subsample D ⊂ X, q(c; D) is implicitly trained to minimise the KL divergence from this posterior in expectation over possible sets X consistent with D. For a data distribution p d we may equivalently describe the VHE objective (Equation 12) as DISPLAYFORM0 Note that the variational gap on the right side of this equation is itself bounded by: DISPLAYFORM1 The left inequality is tightest when p(c|X) matches p(c|D) well across all X consistent with D, and exact only when these are equal.
We view this aspect of the VHE loss as regulariser for constrained posterior approximation, encouraging models for which the posterior p(c|X) can be well determined by sampled subsets D ⊂ X. This reflects how we expect the model to be used at test time, and in practice we have found this 'loose' bound to perform well in our experiments.
In principle, the bound may also be tightened by introducing an auxiliary inference network (see Supplementary Material 6.1) which we leave as a direction for future research.
With a Neural Statistician model, under-utilisation of latent variables is expected to pose the greatest difficulty either when |D| is too small, or the inference network q is insufficiently expressive.
We demonstrate on simple 1D distributions that a Variational Homoencoder can bring improvements under these circumstances.
For this we created five datasets as follows, each containing 100 classes from a particular parametric family, and with 100 elements sampled from each class.1.
Gaussian: Each class is Gaussian with µ drawn from a Gaussian hyperprior (fixed σ 2 ).2.
Mixture of Gaussians:
Each class is an even mixture of two Gaussian distributions with location drawn from a Gaussian hyperprior (fixed σ 2 and separation).
Our results show that, when |D| is small, the Neural Statistician often places little to no information in q(c; D) ( FIG2 , top row).
Our careful training suggests that this is not an optimisation difficulty, but is core to the objective as in BID3 .
In these cases a VHE better utilises the latent space, leading to improvements in both few-shot generation (by conditional NLL) and classification.
Importantly, this is achieved while retaining good likelihood of test-set classes, typically matching or improving upon that achieved by a Neural Statistician (including a standard VAE, corresponding to |D| = 1).
Please see Supplement 6.3 for further comparison to alternative objectives.
FORMULA0 ).
Secondly, since the VHE introduces both data-resampling and KL-rescaling as modifications to this baseline, we separate the contributions of each using two intermediate objectives: DISPLAYFORM0 Rescale only: DISPLAYFORM1 All models were trained on a random sample of 1200 Omniglot classes using images scaled to 28x28 pixels, dynamically binarised, and augmented by 8 rotations/reflections to produce new classes.
We additionally used 20 small random affine transformations to create new instances within each class.
Models were optimised using Adam (Kingma & Welling, 2013), and we used training error to select the best parameters from 5 independent training runs.
This was necessary to ensure a fair comparison with the Neural Statistician objective, which otherwise converges to a local optimum with q(c; D) = p(c).
We additionally implemented the 'sample dropout' trick of Edwards & Storkey (2016), but found that this did not have any effect on model performance.
Discriminative models, log q(y|x, X, Y )Siamese Networks BID15 88.1% 97.0% Matching Networks BID25 93.8% 98.7% Convnet with memory module BID13 95.0% 98.6% mAP-DLM BID24 95.4% 98.6% Model-Agnostic Meta-learning BID7 95.8% 98.9% Prototypical Networks BID23 96.0% 98.9%*Uses train/test split from Lake et al. (2015) TAB0 collects classification results of models trained using each of the four alternative training objectives, for both architectures.
When using a standard deconvolutional architecture, we find little difference in classification performance between all four training objectives, with the Neural Statistician and VHE models achieving equally high accuracy.
For the hierarchical PixelCNN architecture, however, significant differences arise between training objectives.
In this case, a Neural Statistician learns an strong global distribution over images but makes only minimal use of latent variables c. This means that, despite the use of a higher capacity model, classification accuracy is much poorer (66%) than that achieved using a deconvolutional architecture.
For the same reason, conditional samples display an improved sharpness but are no longer identifiable to the cue images on which they were conditioned ( FIG6 ).
Our careful training suggests that this is not an optimisation difficulty but is core to the objective, as discussed in BID3 .By contrast, a VHE is able to gain a large benefit from the hierarchical PixelCNN architecture, with a 3-fold reduction in classification error (5-shot accuracy 98.8%) and conditional samples which are simultaneously sharp and identifiable ( FIG6 ).
This improvement is in part achieved by increased utilisation of the latent space, due to rescaling of the KL divergence term in the objective.
However, our results show that this common technique is insufficient when used alone, leading to overfitting to cue images with an equally severe impairment of classification performance (accuracy 62.8%).
Rather, we find that KL-rescaling and data resampling must be used together in order to for the benefit of the powerful PixelCNN architecture to be realised.
TAB3 lists the classification accuracy achieved by VHEs with both |D| = 1 and |D| = 5, as compared to existing deep learning approaches.
We find that both networks are not only state-of-theart amongst deep generative models, but are also competitive against the best discriminative models trained directly for few-shot classification.
Unlike these discriminative models, a VHE is also able to generate new images of a character in one-shot, producing samples which are simultaneously realistic and faithful to the class of the cue image ( FIG5 ).
As our goal is to model shared structure across images, we evaluate generative performance using joint log likelihood of the entire Omniglot test set (rather than separately across images).
From this perspective, a single element VAE will perform poorly as it treats all datapoints as independent, optimising a sum over log likelihoods for each element.
By sharing latent variables across all elements of the same class, a VHE can improve upon this considerably.
Previous work which evaluates likelihood typically uses the train/test split of BID2 .
However, our most appropriate comparison is with Generative Matching Networks BID0 ) as they also model dependencies within a class; thus, we trained models under the same conditions as them, using the harder test split from Lake et al. FORMULA0 with no data augmentation.
We evaluate the joint log likelihood of full character classes from the test set, normalised by the number of elements, using importance weighting with k=500 samples from q(c; X).
As can be seen in TAB4 , our hierarchical PixelCNN architecture is able to achieve state-of-the-art log likelihood results only when trained using the full Variational Homoencoder objective.
1 n log i p(x i ) DRAW BID8 < 96.5 nats Conv DRAW BID9 < 91.0 nats VLAE BID3 89.83 nats
1 n log i p(x i |x 1:i−1 ) Variational Memory Addressing BID1 > 73.9 nats Generative Matching Networks BID0 62.42 nats 1Shared-latent models DISPLAYFORM0 Homoencoder 61.22 nats
To demonstrate how the VHE framework may apply to models with richer category structure, we built both a hierarchical and a factorial VHE (Figure 2 ) using simple modifications to the above architectures.
For the hierarchical VHE, we extended the deconvolutional model with an extra latent layer a using the same encoder and decoder architecture as c. This was used to encode alphabet level structure for the Omniglot dataset, learning a generative model for alphabets of the form DISPLAYFORM0 Again, we trained this model using a single objective, using separately resampled subsets D a and D c to infer each latent variable (Supplement 6.2).
We then tested our model at both one-shot character generation and 5-shot alphabet generation, using samples from previously unseen alphabets.
As shown in FIG7 , our single trained model is able to learn structure at both layers of abstraction.
For the factorial VHE, we extended the Omniglot dataset by assigning each image to one of 30 randomly generated styles (independent of its character class), modifying both the colour and pen stroke of each image.
We then extended the PixelCNN model to include a 6-dimensional latent variable s to represent the style of an image, alongside the existing c to represent the character.
We used a CNN for style encoder q(s|D s ), and for each image location we condition the PixelCNN decoder using the outer product s ⊗ c ij .We then test this model on a style transfer task by feeding separate images into the character encoder q(c|D c ) and style encoder q(s|D s ), then rendering a new image from the inferred (c, s) pair.
We find that synthesised samples are faithful to the respective character and style of both cue images FIG8 , demonstrating ability of a factorial VHE to successfully disentangle these two image factors using separate latent variables.
We introduced the Variational Homoencoder: a deep hierarchical generative model learned by a novel training procedure which resembles few-shot generation.
This framework allows latent variables to be shared across a large number of elements in a dataset, encouraging them to be well utilised even alongside highly expressive decoder networks.
We demonstrate this by training a hierarchical PixelCNN model on Omniglot dataset, and show that our novel training objective is responsible for the state-of-the-art results it achieves.
This model is arguably the first which uses a general purpose architecture to both attain near human-level one-shot classification performance and produce high quality samples in one-shot generation.
The VHE framework extends naturally to models with richer latent structure, as we demonstrate with two examples: a hierarchical model which generalises the style of an alphabet to produce new characters, and a factorial model which separates the content and drawing style of coloured character images.
In addition to these modelling extensions, our variational bound may also be tightened by learning a subsampling procedure q(D; X), or by introducing an auxiliary inference network r(D; c, X) as discussed in Supplementary Material 6.1.
While such modifications were unnecessary for our experiments on Omniglot character classes, we expect that they may yield improvements on other datasets with greater intra-class diversity.6 SUPPLEMENTARY MATERIAL
The likelihood lower bound in the VHE objective may also be tightened by introduction of an auxiliary network r(D; c, X), trained to infer which subset D ⊂ X was used in q. This meta-inference approach was introduced in BID21 to develop stochastic variational posteriors using MCMC inference, and has recently been applied to approximate inference evaluation (CusumanoTowner & Mansinghka, 2017).
Applied to Equation 12, this yields a modified bound for the VHE objective DISPLAYFORM0 where q (D; X) describes the stochastic sampling procedure for sampling D ⊂ X, which indeed may itself be learned using policy gradients.
We have conducted preliminary experiments using fixed q and a simple functional form r(D; c, X) = i r(d i ; c, X) ∝ i f ψ (c) · ξ di , learning parameters ψ and embeddings {ξ d : d ∈ X }; however, on the Omniglot dataset we found no additional benefit over the strictly loose bound (Equation 12).
We attribute this to the already high similarity between elements of the same Omniglot character class, allowing the approximate posterior q(c; D) to be relatively robust to different choices of D. However, we expect that the gain from using such a tightened objective may be much greater for domains with lower intra-class similarity (e.g. natural images), and thus suggest the tightened bound of Equation 21 as a direction for future research.
The resampling trick may be applied iteratively, to construct likelihood bounds over hierarchically organised data.
Expanding on Equation 12 , suppose that we have collection of datasets DISPLAYFORM0 For example, each X might be a different alphabet whose latent description a generates many character classes X i , and for each of these a corresponding latent c i is used to generate many images x ij .
From this perspective, we would like to learn a generative model for alphabets X of the form DISPLAYFORM1 Reapplying the same trick as before yields a bound taken over all elements x: DISPLAYFORM2 This suggests an analogous hierachical resampling procedure: Summing over every element x, we can bound the log likelihood of the full hierarchy by resampling subsets D c , D a , etc.
at each level to construct an approximate posterior.
All networks are trained together by this single objective, sampling x, D a and D c for each gradient step.
Note that this procedure need only require passing sampled elements, rather than full classes, into the upper-level encoder q a .
Our architecture uses a 8x28x28 latent variable c, with a full architecture detailed below.
For our classification experiments, we trained 5 models on each of the objectives (VHE, Rescale only, Resample only, NS).
Occasionally we found instability in optimisation, causing sudden large increases in the training objective.
When this happened, we halted and restarted training.
All models were trained for 100 epochs on 1000 characters from the training set (the remaining 200 have been used as validation data for model selection).
Finally, for each objective we selected the parameters achieving the best training error.
Note that we did not optimise or select models based on classification performance, other than through our development of our model's architecture.
However, we find that classification performance is well correlated the generative training objective, as can be seen in the full table of results.
We perform classification by calculating the expected conditional likelihood under the variational posterior: Eq(c;D) p(x|c).
This is approximated using 20 samples for the outer expectation, and importance sampling with k = 10 for the inner integral p(x|c) = Eq(t|x) DISPLAYFORM3 To evaluate and compare log likelihood, we trained 5 more models with the same architecture, this time on the canonical 30-20 alphabet split of Lake et al. We did not augment our training data.
Again, we split the background set into training data (25 alphabets) and validation data (5) but do not use the validation set in training or evaluation for our final results.
We estimate the total class log likelihood by importance weighting, using k=20 importance samples of the class latent c and k=10 importance samples of the transformation latent t for each instance.
[d] denotes a dimension d tensor.
{t} denotes a set with elements of type t. Posteriors q are Gaussian.
A PixelCNN with autoregressive weights along only the spatial (not depth) dimensions of c. We use 2 layers of masked 64x3x3 convolutions, followed by a ReLU and two 8x1x1 convolutions corresponding to the mean and log variance of a Gaussian posterior for the following pixel.
We extend the same architecture described in Appendix B of BID6 , with only a simple modification: we introduce a new latent layer containing a 64-dimensional variable a, with a Gaussian prior.
We give p(c|a) the same functional form as p(z|c), and give q(a|D a ) the same functional form as q(c; D c ) using the shared encoder.
We created a VHE using the same deconvolutional architecture as applied to omniglot, and trained it on the Caltech-101 Silhouettes dataset.
10 object classes were held out as test data, which we use to generate both 1-shot and 5-shot conditional samples.

assistant: text: Technique for learning deep generative models with shared latent variables, applied to Omniglot with a PixelCNN decoder.