Posit AI Weblog: Stepping into the movement: Bijectors in TensorFlow Chance

As of right this moment, 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 enticing due to the analogy to human cognition.

On this weblog thus far, now we 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 right way to 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 below heavy growth and the API might change. As of this writing, wrappers don’t but exist for all TFP modules, however all TFP performance is on the market utilizing $-syntax if want be.

Density estimation and sampling

Again to unsupervised studying, and particularly pondering of variational autoencoders, what are the principle 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 will 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’s extra: The entities are presupposed to be decided by a set of distinct, disentangled (hopefully!) latent elements. However this isn’t the belief within the case of normalizing flows, so we aren’t going to elaborate on this right here.

As a recap, how will 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 – seem like it comes from the empirical information distribution. It mustn’t, nevertheless, look precisely like several of the objects used to coach the VAE, or else now we have not discovered something helpful.

The second factor we might 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 way to compute an precise density below the posterior.

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

Normalizing flows

A movement is a sequence of differentiable, invertible mappings from information to a “good” distribution, one thing we will simply pattern from and use to calculate a density. Let’s take as instance the canonical approach 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 now we have a price from the CDF, all we have to do is map that “again” to a price. 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 implies we might get our exponential pattern doing

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

We see the CDF is definitely a movement (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 probability.
• Conversely, it maps a likelihood to an precise worth, thus permitting to generate samples.

From this instance, we see why a movement must be invertible, however we don’t but see why it must be differentiable. It will change into 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 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 information to the interval ([0,1]). Its inverse rework then yields a random variable with the usual regular distribution:

Conversely, we will use this bijector to find out the (log) likelihood of a pattern from the traditional distribution. We’ll test towards 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 likelihood from the bijector, we add two parts:

• Firstly, we run the pattern via the ahead transformation and compute log likelihood below the uniform distribution.
• Secondly, as we’re utilizing the uniform distribution to find out likelihood of a standard pattern, we have to observe how likelihood modifications below 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 primarily based on the precept that below transformation, likelihood mass is conserved. Say now we have a movement 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 remodeled pattern, lies between (x_0) and (x_0 + dx)?

This likelihood is (p(x) dx), the density instances 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 decided by the bottom likelihood (p(z)) of the remodeled distribution, multiplied by how a lot the movement stretches house.

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

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

In larger 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 follow, 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 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 check densities below the movement, we select the traditional distribution, and take 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 movement 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 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

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

Coaching a movement

On condition that flows are bidirectional, there are two methods to consider them. Above, now we have principally careworn the inverse mapping: We wish a easy distribution we will pattern from, and which we will use to compute a density. In that line, flows are generally referred to as “mappings from information to noise” – noise principally being an isotropic Gaussian. Nevertheless in follow, we don’t have that “noise” but, we simply have information.
So in follow, now we have to be taught a movement 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 is predicated on half 1 of Eric Jang’s introduction to normalizing flows. The primary distinction (aside from simplification to point out the essential sample) is that we’re utilizing keen execution.

We begin from a two-dimensional, isotropic Gaussian, and we wish 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 wish 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 will make use of tfb_affine, the multi-dimensional relative of tfb_affine_scalar.
As to nonlinearities, at present TFP comes with tfb_sigmoid and tfb_tanh, however we will construct our personal parameterized ReLU utilizing tfb_inline:

# alpha is a learnable parameter
bijector_leaky_relu <- perform(alpha) {
  
  tfb_inline(
    # ahead rework leaves optimistic values untouched and scales unfavorable ones by alpha
    forward_fn = perform(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 = perform(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 = 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', listing())) + 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 movement 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)  
 
 movement <- listing(lrelu, affine) %>% tfb_chain()
 
 dist <- tfd_transformed_distribution(distribution = base_dist,
                          bijector = movement)
  
 l <- -tf$reduce_mean(dist$log_prob(batch))
 # hold observe of progress
 print(spherical(as.numeric(l), 2))
 l
}

Now we will 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$decrease(loss)
  })
}

Outcomes will differ relying on random initialization, however it is best to see a gradual (if gradual) progress. Utilizing bijectors, now we have truly skilled and outlined a bit of neural community.

Outlook

Undoubtedly, this movement is simply too easy to mannequin advanced information, however it’s instructive to have seen the essential rules earlier than delving into extra advanced flows. Within the subsequent publish, we’ll take a look at autoregressive flows, once more utilizing TFP and tfprobability.