Posit AI Weblog: Stepping into the circulate: Bijectors in TensorFlow Likelihood

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

On this weblog up to now, we now have 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 following publish, we’ll introduce flows, specializing in the best way to implement them utilizing TensorFlow Likelihood (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 model of keras, tensorflow and tfdatasets. A observe concerning this bundle: It’s nonetheless below heavy growth 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 pondering of variational autoencoders, what are the primary issues they provide us? One factor that’s seldom lacking from papers on generative strategies are footage of super-real-looking faces (or mattress rooms, or animals …). So evidently sampling (or: technology) 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 earth: it has discovered a distribution.
Within the case of variational autoencoders, there may be extra: The entities are presupposed to be decided by a set of distinct, disentangled (hopefully!) latent components. 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 end result ought to – we hope – appear to be it comes from the empirical information distribution. It mustn’t, nonetheless, look precisely like several of the objects used to coach the VAE, or else we now have not discovered something helpful.

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

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

Normalizing flows

A circulate is a sequence of differentiable, invertible mappings from information 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 option 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 chance distribution (CDF) – from an exponential CDF, to be exact. Now that we now have a worth from the CDF, all we have to do is map that “again” to a worth. That mapping CDF -> worth we’re searching for 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 implies we could get our exponential pattern doing

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

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

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

From this instance, we see why a circulate must be invertible, however we don’t but see why it must be differentiable. This may 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, referred to as bijectors, starting from easy computations like exponentiation to extra advanced ones just like the discrete cosine remodel.

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

Conversely, we are able to use this bijector to find out the (log) chance of a pattern from the conventional distribution. We’ll test in opposition to an easy 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 chance from the bijector, we add two elements:

• Firstly, we run the pattern by way of the ahead transformation and compute log chance below the uniform distribution.
• Secondly, as we’re utilizing the uniform distribution to find out chance of a standard pattern, we have to observe how chance modifications below this transformation. That is completed 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.

Likelihood mass is conserved

Flows are primarily based on the precept that below transformation, chance mass is conserved. Say we now have a circulate from (x) to (z):
[z = f(x)]

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

This chance is (p(x) dx), the density instances the size of the interval. This has to equal the chance 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 chance (p(x)) is set by the bottom chance (p(z)) of the remodeled distribution, multiplied by how a lot the circulate stretches house.

The identical goes in greater dimensions: Once more, the circulate is in regards to the change in chance 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 chances, 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 a couple of 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 below the circulate, we select the conventional 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 circulate 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 house (we multiply by 2), the person log densities go down.
We will confirm the cumulative chance 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

Up to now, the flows we noticed have been static – how does this match into the framework of neural networks?

Coaching a circulate

On condition that flows are bidirectional, there are two methods to consider them. Above, we now have largely harassed 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 typically referred to as “mappings from information to noise” – noise largely being an isotropic Gaussian. Nevertheless in apply, we don’t have that “noise” but, we simply have information.
So in apply, we now have to study a circulate 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 following publish.

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

We begin from a two-dimensional, isotropic Gaussian, and we need to mannequin information 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 need to go
target_dist <- tfd_multivariate_normal_diag(loc = c(1, 1), scale_identity_multiplier = 2)

# create coaching information from the goal distribution
target_samples <- target_dist %>% tfd_sample(1000) %>% tf$forged(tf$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, at the moment 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 <- perform(alpha) {
  
  tfb_inline(
    # ahead remodel leaves optimistic values untouched and scales adverse ones by alpha
    forward_fn = perform(x)
      tf$the place(tf$greater_equal(x, 0), x, alpha * x),
    # inverse remodel leaves optimistic values untouched and scales adverse ones by 1/alpha
    inverse_fn = perform(y)
      tf$the place(tf$greater_equal(y, 0), y, 1/alpha * y),
    # quantity change is 0 when optimistic and 1/alpha when adverse
    inverse_log_det_jacobian_fn = perform(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', record())) + 0.01

With keen execution, the variables have for use contained in the loss perform, so that’s the place we outline the bijectors. Our little circulate 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 <- perform() {
  
 affine <- tfb_affine(
        scale_tril = tfb_fill_triangular() %>% tfb_forward(L),
        scale_perturb_factor = V,
        shift = shift
      )
 lrelu <- bijector_leaky_relu(alpha = alpha)  
 
 circulate <- record(lrelu, affine) %>% tfb_chain()
 
 dist <- tfd_transformed_distribution(distribution = base_dist,
                          bijector = circulate)
  
 l <- -tf$reduce_mean(dist$log_prob(batch))
 # preserve observe 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 is best to see a gentle (if sluggish) progress. Utilizing bijectors, we now have truly skilled and outlined somewhat neural community.

Outlook

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