Posit AI Weblog: Moving into the stream: Bijectors in TensorFlow Chance

As of right now, deep studying’s best successes have taken place within the realm of supervised studying, requiring tons and many annotated coaching knowledge. Nevertheless, knowledge doesn’t (usually) include annotations or labels. Additionally, unsupervised studying is enticing due to the analogy to human cognition.

On this weblog to date, we’ve seen two main architectures for unsupervised studying: variational autoencoders and generative adversarial networks. Lesser identified, however interesting for conceptual in addition to for efficiency causes are normalizing flows (Jimenez Rezende and Mohamed 2015). On this and the subsequent publish, we’ll introduce flows, specializing in how you can implement them utilizing TensorFlow Chance (TFP).

In distinction to earlier posts involving TFP that accessed its performance utilizing low-level $-syntax, we now make use of tfprobability, an R wrapper within the fashion of keras, tensorflow and tfdatasets. A notice concerning this bundle: It’s nonetheless underneath heavy improvement and the API could change. As of this writing, wrappers don’t but exist for all TFP modules, however all TFP performance is accessible utilizing $-syntax if want be.

Density estimation and sampling

Again to unsupervised studying, and particularly considering of variational autoencoders, what are the primary issues they offer us? One factor that’s seldom lacking from papers on generative strategies are photos of super-real-looking faces (or mattress rooms, or animals …). So evidently sampling (or: era) is a crucial half. If we are able to pattern from a mannequin and procure real-seeming entities, this implies the mannequin has discovered one thing about how issues are distributed on the planet: it has discovered a distribution.
Within the case of variational autoencoders, there may be extra: The entities are alleged to be decided by a set of distinct, disentangled (hopefully!) latent elements. However this isn’t the idea within the case of normalizing flows, so we’re not going to elaborate on this right here.

As a recap, how can we pattern from a VAE? We draw from (z), the latent variable, and run the decoder community on it. The outcome ought to – we hope – appear to be it comes from the empirical knowledge distribution. It mustn’t, nevertheless, look precisely like several of the objects used to coach the VAE, or else we’ve not discovered something helpful.

The second factor we could get from a VAE is an evaluation of the plausibility of particular person knowledge, for use, for instance, in anomaly detection. Right here “plausibility” is imprecise on function: With VAE, we don’t have a method to compute an precise density underneath the posterior.

What if we would like, or want, each: era of samples in addition to density estimation? That is the place normalizing flows are available in.

Normalizing flows

A stream is a sequence of differentiable, invertible mappings from knowledge to a “good” distribution, one thing we are able to simply pattern from and use to calculate a density. Let’s take as instance the canonical technique to generate samples from some distribution, the exponential, say.

We begin by asking our random quantity generator for some quantity between 0 and 1:

This quantity we deal with as coming from a cumulative likelihood distribution (CDF) – from an exponential CDF, to be exact. Now that we’ve a worth from the CDF, all we have to do is map that “again” to a worth. That mapping CDF -> worth we’re in search of is simply the inverse of the CDF of an exponential distribution, the CDF being

[F(x) = 1 – e^{-lambda x}]

The inverse then is

[
F^{-1}(u) = -frac{1}{lambda} ln (1 – u)
]

which suggests we could get our exponential pattern doing

lambda <- 0.5 # decide some lambda
x <- -1/lambda * log(1-u)

We see the CDF is definitely a stream (or a constructing block thereof, if we image most flows as comprising a number of transformations), since

• It maps knowledge to a uniform distribution between 0 and 1, permitting to evaluate knowledge chance.
• Conversely, it maps a likelihood to an precise worth, thus permitting to generate samples.

From this instance, we see why a stream needs to be invertible, however we don’t but see why it needs to be differentiable. It will turn out to be clear shortly, however first let’s check out how flows can be found in tfprobability.

Bijectors

TFP comes with a treasure trove of transformations, known as bijectors, starting from easy computations like exponentiation to extra advanced ones just like the discrete cosine rework.

To get began, let’s use tfprobability to generate samples from the traditional distribution.
There’s a bijector tfb_normal_cdf() that takes enter knowledge to the interval ([0,1]). Its inverse rework then yields a random variable with the usual regular distribution:

Conversely, we are able to use this bijector to find out the (log) likelihood of a pattern from the traditional distribution. We’ll test towards a simple use of tfd_normal within the distributions module:

x <- 2.01
d_n <- tfd_normal(loc = 0, scale = 1) 

d_n %>% tfd_log_prob(x) %>% as.numeric() # -2.938989

To acquire that very same log likelihood from the bijector, we add two parts:

• Firstly, we run the pattern via the ahead transformation and compute log likelihood underneath the uniform distribution.
• Secondly, as we’re utilizing the uniform distribution to find out likelihood of a standard pattern, we have to monitor how likelihood modifications underneath this transformation. That is executed by calling tfb_forward_log_det_jacobian (to be additional elaborated on under).

b <- tfb_normal_cdf()
d_u <- tfd_uniform()

l <- d_u %>% tfd_log_prob(b %>% tfb_forward(x))
j <- b %>% tfb_forward_log_det_jacobian(x, event_ndims = 0)

(l + j) %>% as.numeric() # -2.938989

Why does this work? Let’s get some background.

Chance mass is conserved

Flows are based mostly on the precept that underneath transformation, likelihood mass is conserved. Say we’ve a stream from (x) to (z):
[z = f(x)]

Suppose we pattern from (z) after which, compute the inverse rework to acquire (x). We all know the likelihood of (z). What’s the likelihood that (x), the reworked pattern, lies between (x_0) and (x_0 + dx)?

This likelihood is (p(x) dx), the density occasions the size of the interval. This has to equal the likelihood that (z) lies between (f(x)) and (f(x + dx)). That new interval has size (f'(x) dx), so:

[p(x) dx = p(z) f'(x) dx]

Or equivalently

[p(x) = p(z) * dz/dx]

Thus, the pattern likelihood (p(x)) is set by the bottom likelihood (p(z)) of the reworked distribution, multiplied by how a lot the stream stretches area.

The identical goes in greater dimensions: Once more, the stream is concerning the change in likelihood quantity between the (z) and (y) areas:

[p(x) = p(z) frac{vol(dz)}{vol(dx)}]

In greater dimensions, the Jacobian replaces the by-product. Then, the change in quantity is captured by absolutely the worth of its determinant:

[p(mathbf{x}) = p(f(mathbf{x})) bigg|detfrac{partial f({mathbf{x})}}{partial{mathbf{x}}}bigg|]

In apply, we work with log possibilities, so

[log p(mathbf{x}) = log p(f(mathbf{x})) + log bigg|detfrac{partial f({mathbf{x})}}{partial{mathbf{x}}}bigg| ]

Let’s see this with one other bijector instance, tfb_affine_scalar. Beneath, we assemble a mini-flow that maps just a few arbitrary chosen (x) values to double their worth (scale = 2):

x <- c(0, 0.5, 1)
b <- tfb_affine_scalar(shift = 0, scale = 2)

To match densities underneath the stream, we select the traditional distribution, and have a look at the log densities:

d_n <- tfd_normal(loc = 0, scale = 1)
d_n %>% tfd_log_prob(x) %>% as.numeric() # -0.9189385 -1.0439385 -1.4189385

Now apply the stream and compute the brand new log densities as a sum of the log densities of the corresponding (x) values and the log determinant of the Jacobian:

z <- b %>% tfb_forward(x)

(d_n  %>% tfd_log_prob(b %>% tfb_inverse(z))) +
  (b %>% tfb_inverse_log_det_jacobian(z, event_ndims = 0)) %>%
  as.numeric() # -1.6120857 -1.7370857 -2.1120858

We see that because the values get stretched in area (we multiply by 2), the person log densities go down.
We are able to confirm the cumulative likelihood stays the identical utilizing tfd_transformed_distribution():

d_t <- tfd_transformed_distribution(distribution = d_n, bijector = b)
d_n %>% tfd_cdf(x) %>% as.numeric()  # 0.5000000 0.6914625 0.8413447

d_t %>% tfd_cdf(y) %>% as.numeric()  # 0.5000000 0.6914625 0.8413447

Thus far, the flows we noticed have been static – how does this match into the framework of neural networks?

Coaching a stream

On condition that flows are bidirectional, there are two methods to consider them. Above, we’ve largely confused the inverse mapping: We wish a easy distribution we are able to pattern from, and which we are able to use to compute a density. In that line, flows are generally known as “mappings from knowledge to noise” – noise largely being an isotropic Gaussian. Nevertheless in apply, we don’t have that “noise” but, we simply have knowledge.
So in apply, we’ve to study a stream that does such a mapping. We do that by utilizing bijectors with trainable parameters.
We’ll see a quite simple instance right here, and go away “actual world flows” to the subsequent publish.

The instance relies on half 1 of Eric Jang’s introduction to normalizing flows. The principle distinction (other than simplification to indicate the essential sample) is that we’re utilizing keen execution.

We begin from a two-dimensional, isotropic Gaussian, and we wish to mannequin knowledge that’s additionally regular, however with a imply of 1 and a variance of two (in each dimensions).

library(tensorflow)
library(tfprobability)

tfe_enable_eager_execution(device_policy = "silent")

library(tfdatasets)

# the place we begin from
base_dist <- tfd_multivariate_normal_diag(loc = c(0, 0))

# the place we wish to go
target_dist <- tfd_multivariate_normal_diag(loc = c(1, 1), scale_identity_multiplier = 2)

# create coaching knowledge from the goal distribution
target_samples <- target_dist %>% tfd_sample(1000) %>% tf$solid(tf$float32)

batch_size <- 100
dataset <- tensor_slices_dataset(target_samples) %>%
  dataset_shuffle(buffer_size = dim(target_samples)[1]) %>%
  dataset_batch(batch_size)

Now we’ll construct a tiny neural community, consisting of an affine transformation and a nonlinearity.
For the previous, we are able to make use of tfb_affine, the multi-dimensional relative of tfb_affine_scalar.
As to nonlinearities, presently TFP comes with tfb_sigmoid and tfb_tanh, however we are able to construct our personal parameterized ReLU utilizing tfb_inline:

# alpha is a learnable parameter
bijector_leaky_relu <- operate(alpha) {
  
  tfb_inline(
    # ahead rework leaves optimistic values untouched and scales unfavorable ones by alpha
    forward_fn = operate(x)
      tf$the place(tf$greater_equal(x, 0), x, alpha * x),
    # inverse rework leaves optimistic values untouched and scales unfavorable ones by 1/alpha
    inverse_fn = operate(y)
      tf$the place(tf$greater_equal(y, 0), y, 1/alpha * y),
    # quantity change is 0 when optimistic and 1/alpha when unfavorable
    inverse_log_det_jacobian_fn = operate(y) {
      I <- tf$ones_like(y)
      J_inv <- tf$the place(tf$greater_equal(y, 0), I, 1/alpha * I)
      log_abs_det_J_inv <- tf$log(tf$abs(J_inv))
      tf$reduce_sum(log_abs_det_J_inv, axis = 1L)
    },
    forward_min_event_ndims = 1
  )
}

Outline the learnable variables for the affine and the PReLU layers:

d <- 2 # dimensionality
r <- 2 # rank of replace

# shift of affine bijector
shift <- tf$get_variable("shift", d)
# scale of affine bijector
L <- tf$get_variable('L', c(d * (d + 1) / 2))
# rank-r replace
V <- tf$get_variable("V", c(d, r))

# scaling issue of parameterized relu
alpha <- tf$abs(tf$get_variable('alpha', listing())) + 0.01

With keen execution, the variables have for use contained in the loss operate, so that’s the place we outline the bijectors. Our little stream now could be a tfb_chain of bijectors, and we wrap it in a TransformedDistribution (tfd_transformed_distribution) that hyperlinks supply and goal distributions.

loss <- operate() {
  
 affine <- tfb_affine(
        scale_tril = tfb_fill_triangular() %>% tfb_forward(L),
        scale_perturb_factor = V,
        shift = shift
      )
 lrelu <- bijector_leaky_relu(alpha = alpha)  
 
 stream <- listing(lrelu, affine) %>% tfb_chain()
 
 dist <- tfd_transformed_distribution(distribution = base_dist,
                          bijector = stream)
  
 l <- -tf$reduce_mean(dist$log_prob(batch))
 # preserve monitor of progress
 print(spherical(as.numeric(l), 2))
 l
}

Now we are able to truly run the coaching!

optimizer <- tf$practice$AdamOptimizer(1e-4)

n_epochs <- 100
for (i in 1:n_epochs) {
  iter <- make_iterator_one_shot(dataset)
  until_out_of_range({
    batch <- iterator_get_next(iter)
    optimizer$reduce(loss)
  })
}

Outcomes will differ relying on random initialization, however it’s best to see a gentle (if gradual) progress. Utilizing bijectors, we’ve truly educated and outlined a bit of neural community.

Outlook

Undoubtedly, this stream is just too easy to mannequin advanced knowledge, however it’s instructive to have seen the essential rules earlier than delving into extra advanced flows. Within the subsequent publish, we’ll try autoregressive flows, once more utilizing TFP and tfprobability.