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

As of in the present day, deep studying’s best successes have taken place within the realm of supervised studying, requiring heaps and plenty of annotated coaching knowledge. Nonetheless, knowledge doesn’t (usually) include annotations or labels. Additionally, unsupervised studying is engaging due to the analogy to human cognition.

On this weblog to date, 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 subsequent submit, we’ll introduce flows, specializing in how one can implement them utilizing TensorFlow Likelihood (TFP).

In distinction to previous 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 word concerning this package deal: 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 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 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: technology) is a crucial half. If we are able to pattern from a mannequin and acquire 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 components. However this isn’t the idea 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 consequence ought to – we hope – appear to be it comes from the empirical knowledge 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 knowledge, 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 circulation 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 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 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 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 circulation (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 chance to an precise worth, thus permitting to generate samples.

From this instance, we see why a circulation needs to be invertible, however we don’t but see why it needs to be differentiable. This may 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 transform.

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) chance of a pattern from the traditional 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 parts:

Firstly, we run the pattern via 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 traditional pattern, we have to monitor how chance adjustments below this transformation. That is finished by calling tfb_forward_log_det_jacobian (to be additional elaborated on beneath).

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 based mostly on the precept that below transformation, chance mass is conserved. Say we now have a circulation 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 chance of (z). What’s the chance that (x), the reworked 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 reworked distribution, multiplied by how a lot the circulation stretches house.

The identical goes in greater dimensions: Once more, the circulation is concerning 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 follow, 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 number 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 circulation, 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 circulation 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 had been static – how does this match into the framework of neural networks?

Coaching a circulation

Provided that flows are bidirectional, there are two methods to consider them. Above, we now have largely pressured 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 knowledge to noise” – noise largely being an isotropic Gaussian. Nonetheless in follow, we don’t have that “noise” but, we simply have knowledge.
So in follow, we now have to study a circulation that does such a mapping. We do that through the use of bijectors with trainable parameters.
We’ll see a quite simple instance right here, and depart “actual world flows” to the subsequent submit.

The instance is predicated on half 1 of Eric Jang’s introduction to normalizing flows. The principle 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 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$forged(tf$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 <- operate(alpha) {
  
  tfb_inline(
    # ahead rework leaves optimistic values untouched and scales damaging 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 damaging 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 damaging
    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', list())) + 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 circulation 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)  
 
 circulation <- list(lrelu, affine) %>% tfb_chain()
 
 dist <- tfd_transformed_distribution(distribution = base_dist,
                          bijector = circulation)
  
 l <- -tf$reduce_mean(dist$log_prob(batch))
 # hold monitor of progress
 print(round(as.numeric(l), 2))
 l
}

Now we are able to really run the coaching!

optimizer <- tf$prepare$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 you must see a gentle (if gradual) progress. Utilizing bijectors, we now have really educated and outlined just a little neural community.

Outlook

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

Jimenez Rezende, Danilo, and Shakir Mohamed. 2015. “Variational Inference with Normalizing Flows.” arXiv e-Prints, Might, arXiv:1505.05770. https://arxiv.org/abs/1505.05770.