
In my previous blog post, I described how simple distributions like Gaussians can be “deformed” to fit complex data distributions using normalizing flows. We implemented a simple flow by chaining 2D Affine Bijectors with PreLU nonlinearities to build a small invertible neural net.
However, this MLP flow is pretty weak: there are only 2 units per hidden layer. Furthermore, the nonlinearity is monotonic and piecewise linear, so all it does is slightly warp the data manifold around the origin. This flow completely fails to implement more complex transformations like separating an isotropic Gaussian into two modes when trying to learn the “Two Moons” dataset below:
Fortunately, there are several more powerful normalizing flows that have been introduced in recent Machine Learning literature. We will explore several of these techniques in this tutorial.
Autoregressive Models are Normalizing Flows
$$p(x) = \prod_i{p(x_i \,\, x_{1:i1})}$$
The conditional densities usually have learnable parameters. For example, a common choice is an autoregressive density $p(x_{1:D})$ whose conditional density is a univariate Gaussian, whose mean and standard deviations are computed by neural networks that depend on the previous $x_{1:i1}$.
$$p(x_i \,\, x_{1:i1}) = \mathcal{N}(x_i \,\,\mu_i, (\exp\alpha_i)^2)$$
$$p(x_i \,\, x_{1:i1}) = \mathcal{N}(x_i \,\,\mu_i, (\exp\alpha_i)^2)$$
$$\mu_i = f_{\mu_i}(x_{1:i1})$$
$$\alpha_i = f_{\alpha_i}(x_{1:i1})$$
Learning data with autoregressive density estimation makes the rather bold inductive bias that the ordering of variables are such that your earlier variables don’t depend on later variables. Intuitively, this shouldn’t be true at all for natural data (the top row of pixels in an image does have a causal, conditional dependency on the bottom of the image). However it’s still possible to generate plausible images in this manner (to the surprise of many researchers!).
To sample from this distribution, we compute $D$ “noise variates” $u_{1:D}$ from the standard Normal, $N(0,1)$, then apply the following recursion to get $x_{1:D}$.
$$x_i = u_i\exp{\alpha_i} + \mu_i$$
$$u_i \sim \mathcal{N}(0, 1)$$
The procedure of autoregressive sampling is a deterministic transformation of the underlying noise variates (sampled from $\mathcal{N}(0, \mathbb{I})$) into a new distribution, so autoregressive samples can actually be interpreted as a TransformedDistribution of the standard Normal!
Armed with this insight, we can stack multiple autoregressive transformations into a normalizing flow. The advantage of doing this is that we can change the ordering of variables $x_1,...x_D$ for each bijector in the flow, so that if one autoregressive factorization cannot model a distribution well (due to a poor choice of variable ordering), a subsequent layer might be able to do it.
The Masked Autoregressive Flow (MAF) bijector implements such a conditionalGaussian autoregressive model. Here is a schematic of the forward pass for a single entry in a sample of the transformed distribution, $x_i$:
The gray unit $x_i$ is the unit we are trying to compute, and the blue units are the values it depends on. $\alpha_i$ and $\mu_i$ are scalars that are computed by passing $x_{1:i1}$ through neural networks (magenta, orange circles). Even though the transformation is a mere scaleandshift, the scale and shift can have complex dependencies on previous variables. For the first unit $x_1$, $\mu$ and $\alpha$ are usually set to learnable scalar variables that don’t depend on any $x$ or $u$.
More importantly, the transformation is designed this way so that computing the inverse $u = f^{1}(x)$ does not require us to invert $f_\alpha$ or $f_\mu$. Because the transformation is parameterized as a scaleandshift, we can recover the original noise variates by reversing the shift and scale: $u = (xf_\mu(x))/\exp(f_\alpha(x))$. The forward and inverse pass of the bijector only depend on the forward evaluation of $f_\alpha(x)$ and $f_\mu(x)$, allowing us to use noninvertible functions like ReLU and nonsquare matrix multiplication in the neural networks $f_\mu$ and $f_\alpha$.
The inverse pass of the MAF model is used to evaluate density:
distribution.log_prob(bijector.inverse(x)) + bijector.inverse_log_det_jacobian(x))
Runtime Complexity and MADE
Autoregressive models and MAF can be trained “quickly” because all conditional likelihoods $p(x_1), p(x_2\,\, x_1), ... p(x_D\,\, x_{1:D1}))$ can be evaluated simultaneously in a single pass of D threads, leveraging the batch parallelism of modern GPUs. We are operating under the assumption that parallelism, such as SIMD vectorization on CPUs/GPUs, has zero runtime overhead.
On the other hand, sampling autoregressive models is slow because you must wait for all previous $x_{1:i1}$ to be computed before computing new $x_i$. The runtime complexity of generating a single sample is D sequential passes of a single thread, which fails to exploit processor parallelism.
Another issue: in the parallelizable inverse pass, should we use separate neural nets (with differentlysized inputs) for computing each $\alpha_i$ and $\mu_i$? That's inefficient, especially if we consider that learned representations between these D networks should be shared (as long as the autoregressive dependency is not violated). In the Masked Autoencoder for Distribution Estimation (MADE) paper, the authors propose a very nice solution: use a single neural net to output all values of $\alpha$ and $\mu$ simultaneously, but mask the weights so that the autoregressive property is preserved.
This trick makes it possible to recover all values of $u$ from all values of $x$ with a single pass through a single neural network (D inputs, D outputs). This is far more efficient than processing D neural networks simultaneously (D(D+1)/2 inputs, D outputs).
To summarize, MAF uses the MADE architecture as an efficiency trick for computing nonlinear parameters of shiftandscale autoregressive transformations, and casts these efficient autoregressive models into the normalizing flows framework.
Inverse Autoregressive Flow (IAF)
In Inverse Autoregressive Flow, the nonlinear shift/scale statistics are computed using the previous noise variates $u_{1:i1}$, instead of the data samples:
$$x_i = u_i\exp{\alpha_i} + \mu_i$$
$$\mu_i = f_{\mu_i}(u_{1:i1})$$
$$\alpha_i = f_{\alpha_i}(u_{1:i1})$$
The forward (sampling) pass of IAF is fast: all the $x_i$ can be computed in a single pass of $D$ threads working in parallel. IAF also uses MADE networks to implement this parallelism efficiently.
However, if we are given a new data point and asked to evaluate the density, we need to recover $u$ and this process is slow: first we recover $u_1 = (x\mu_1) * \exp(\alpha_1)$, then $u_i = (x\mu_i(u_{1:i1})) * \exp(\alpha_i(u_{1:i1}))$ sequentially. On the other hand, it’s trivial to track the (log) probability of samples generated by IAF, since we already know all of the $u$ values to begin with without having to invert from $x$.
The astute reader will notice that if you relabel the bottom row as x_1, .. x_D, and the top row as u_1, … u_D, this is exactly equivalent to the Inverse Pass of the MAF bijector! Likewise, the inverse of IAF is nothing more than the forward pass of MAF (with $x$ and $u$ swapped). Therefore in TensorFlow Distributions, MAF and IAF are actually implemented using the exact same Bijector class, and there is a convenient “Invert” feature for inverting Bijectors to swap their inverse and forward passes.
iaf_bijector = tfb.Invert(maf_bijector)
IAF and MAF make opposite computational tradeoffs  MAF trains quickly but samples slowly, while IAF trains slowly but samples quickly. For training neural networks, we usually demand way more throughput with density evaluation than sampling, so MAF is usually a more appropriate choice when learning distributions.
Parallel Wavenet
An obvious followup question is whether these two approaches can be combined to get the best of both worlds, i.e. fast training and sampling.
The answer is yes! The muchpublicized Parallel Wavenet by DeepMind does exactly this: an autoregressive model (MAF) is used to train a generative model efficiently, then an IAF model is trained to maximize the likelihood of its own samples under this teacher. Recall that with IAF, it is costly to compute density of external data points (such as those from the training set), but it can cheaply compute density of its own samples by caching the noise variates $u_{1:D}$, thereby circumventing the need to call the inverse pass. Thus, we can train the “student” IAF model by minimizing the divergence between the student and teacher distributions.
This is an incredibly impactful application of normalizing flows research  the end result is a realtime audio synthesis model that is 20 times faster to sample, and is already deployed in realworld products like the Google Assistant.
NICE and RealNVP
Finally, we consider is RealNVP, which can be thought of as a special case of the IAF bijector.
In a NVP “coupling layer”, we fix an integer $0 < d < D$. Like IAF, $x_{d+1}$ is a shiftandscale that depends on previous $u_{d}$ values. The difference is that we also force $x_{d+2}, x_{d+3}, … x_{D}$ to only depend on these $u_{d}$ values, so a single network pass can be used to produce $\alpha_{d+1:D}$ and $\mu_{d+1:D}$.
As for $x_1:d$ they are “passthrough” units that are set equivalently to $u_{1:d}$. Therefore, RealNVP is also a special case of the MAF bijector (since $\alpha(u_{1:d}) = \alpha(x_{1:d})$).
Because the shiftandscale statistics for the whole layer can be computed from either $x_{1:d}$ or $u_{1:d}$ in a single pass, NVP can perform forward and inverse computations in a single parallel pass (sampling and estimation are both fast). MADE is also not needed.
However, empirical studies suggest that RealNVP tends to underperform MAF and IAF and my experience has been that NVP tends to fit my toy 2D datasets (e.g. SIGGRAPH dataset) more poorly when using the same number of layers.
RealNVP was a followup work to the NICE bijector, which is a shiftonly variant that assumes $\alpha=0$. Because NICE does not scale the distribution, the ILDJ is actually constant!
Batch Normalization Bijector
In normalizing flows, batch norm is used in bijector.inverse during training, and the accumulated statistics are used to denormalize data at “test time” (bijector.forward). Concretely, BatchNorm Bijectors are typically implemented as follows:
Inverse pass:
 Compute the current mean and standard deviation of the data distribution $x$.
 Update running mean and standard deviation
 Batch normalize the data using current mean/std
 Use running mean and standard deviation to unnormalize the data distribution.
Thanks to TF Bijectors, this can be implemented with only a few lines of code:
The ILDJ can be derived easily by simply taking the log derivative of inverse function (consider the univariate case).
Code Example
Thanks to the efforts of Josh Dillon and the Google Bayesflow team, there is already a flexible implementation of MaskedAutoregressiveFlow Bijector that uses MADE networks to implement efficient recovery of $u$ for training.
I’ve created a complex 2D distribution, which is a point cloud in the shape of the letters “SIGGRAPH” using this blender script. We construct our dataset, bijector, and transformed distribution in a very similar fashion to the first tutorial, so I won’t repeat the code snippets here  you can find the Jupyter notebook here. This notebook can train a normalizing flow using MAF, IAF, RealNVP with/without BatchNorm, for both the "Two Moons" and "SIGGRAPH" datasets.
One detail that’s easy to miss / introduce bugs on is that this doesn’t work at all unless you permute the ordering of variable at each flow. Otherwise, none of the layers’ autoregressive factorization will be learn structure of $p(x1  x2)$. Fortunately, TensorFlow has a Permute bijector specially made for doing this.
Here’s the learned flow, along with the final result. It reminds me a lot of a taffy pulling machine.
Discussion
TensorFlow distributions makes normalizing flows easy to implement, and automatically accumulate all the Jacobians determinants in a chain for us in a way that is clean and highly readable. When deciding which Normalizing Flow to use, consider the design tradeoff between a fast forward pass and a fast inverse pass, as well as between an expressive flow and a speedy ILJD.
Although explicitdensity models like normalizing flows are amenable to training via maximum likelihood, this is not the only way they can be used and are complementary to VAEs and GANs. It’s possible to use normalizing flow as a dropin replacement for anywhere you would use a Gaussian, such as VAE priors and latent codes in GANs. For example, this paper use normalizing flows as flexible variational priors, and the TensorFlow distributions paper presents a VAE that uses a normalizing flow as a prior along with a PixelCNN decoder. Parallel Wavenet trains an IAF "student" model via KL divergence.
One of the most intriguing properties of normalizing flows is that they implement reversible computation (i.e. have a defined inverse of an expressive function). This means that if we want to perform a backprop pass, we can recompute the forward activation values without having to store them in memory during the forward pass (potentially expensive for large graphs). In a setting where credit assignment may take place over very long time scales, we can use reversible computation to “recover” past decision states while keeping memory usage bounded. In fact, this idea was utilized in the RevNets paper, and was actually inspired by the invertibility of the NICE bijector. I’m reminded of the main character from the film Memento who is unable to store memories, so he uses invertible compute to remember things.
Thank you for reading.
Acknowledgements
I’m grateful to Dustin Tran, Luke Metz, Jonathan Shen, Katherine Lee, and Samy Bengio for proofreading this post.
References and Further Reading
 The content and outline of this blog post was heavily influenced by the Masked Autoregressive Flow for Density Estimation paper, which is very wellwritten and is more or less my primary source for understanding this topic. Give it a read!
 Some earlier work on NFs: https://math.nyu.edu/faculty/tabak/publications/TabakTurner.pdf and https://arxiv.org/pdf/1302.5125.pdf and https://arxiv.org/abs/1505.05770
 Talk by Laurent Dinh & discussion by Twitter Cortex researchers. Some neat ideas and discussion here.
 Tutorial on Normalizing Flows using PyMC.
 There’s a body of work that I don’t fully understand yet, bridging Normalizing Flows to Langevin Flow and Hamiltonian Flow. As the number of Bijectors in a normalizing flow goes to infinity, one arrives at a ContinuousTime Flow, which apparently can express even richer transformations.
"so all it does is slightly the data manifold around the origin"
ReplyDeleteMissing a word at the beginning of the blog post... warp? pivot? transform?
Thank you! Fixed.
DeleteHow would you solve the issue that none of the layers’ autoregressive factorization will be learn the structure of p(x1x2) in a high dimensinal space? Permutation would become quite expensive.
ReplyDeletepermutation is expensive, but in practice this only needs to be done 45 times to get good results (e.g. fast wavenet).
DeleteThis comment has been removed by the author.
ReplyDeleteGreat post!
ReplyDeleteI just wanted to point out one passage which comes across as slightly inaccurate:
"Learning data with autoregressive density estimation makes the rather bold inductive bias that the ordering of variables are such that your earlier variables don’t depend on later variables"
As far as I can tell this assumption is not actually made: by the chain rule of probability we can write *any* joint probability density as a product of "telescopic" conditional densities, as in autoregressive models. The inductive bias comes from the fact that, for a fixed functional form of the conditional densities (e.g. Gaussian), not all orderings might be able to give rise to the desired joint distribution (see example in MAF paper).
Hope that makes sense.
In order to compute "the divergence between the student and teacher distributions", do we draw multiple samples from base distribution (noise) or from the output of the student?
ReplyDeleteCan IAF be used to transform the noise to a "mixture" of logistics distribution or it is only for single logistic distribution?
ReplyDelete