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

As of at present, deep studying’s best successes have taken place within the realm of supervised studying, requiring heaps 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 to this point, now we have seen two main architectures for unsupervised studying: variational autoencoders and generative adversarial networks. Lesser recognized, however interesting for conceptual in addition to for efficiency causes are normalizing flows (Jimenez Rezende and Mohamed 2015). On this and the following put up, we’ll introduce flows, specializing in how one can 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 type of keras, tensorflow and tfdatasets. A notice concerning this bundle: It’s nonetheless underneath 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 offered 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: technology) is a vital 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 earth: 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 components. However this isn’t the belief 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 information distribution. It mustn’t, nevertheless, look precisely like all of the gadgets 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 objective: With VAE, we don’t have a method to compute an precise density underneath the posterior.

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

Normalizing flows

A stream 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 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 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 on the lookout 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 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 stream (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 likelihood to an precise worth, thus permitting to generate samples.

From this instance, we see why a stream ought to be invertible, however we don’t but see why it ought to be differentiable. This may grow 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 bijectorsstarting 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 conventional 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 conventional distribution. We’ll verify 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 elements:

• 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 observe how likelihood modifications underneath this transformation. That is carried out 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 primarily based on the precept that underneath transformation, likelihood mass is conserved. Say now we have 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 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 reworked distribution, multiplied by how a lot the stream stretches area.

The identical goes in larger 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 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 observe, 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. Under, 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 underneath the stream, 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 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 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

Thus far, the flows we noticed had 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, now we have principally burdened the inverse mapping: We would like a easy distribution we will pattern from, and which we will use to compute a density. In that line, flows are typically referred to as “mappings from information to noise” – noise principally being an isotropic Gaussian. Nevertheless in observe, we don’t have that “noise” but, we simply have information.
So in observe, now we have to be taught 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 depart “actual world flows” to the following put up.

The instance relies 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 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 will make use of tfb_affinethe multi-dimensional relative of tfb_affine_scalar.
As to nonlinearities, at present TFP comes with tfb_sigmoid and tfb_tanhhowever 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 constructive values untouched and scales destructive ones by alpha
    forward_fn = perform(x)
      tf$the place(tf$greater_equal(x, 0), x, alpha * x),
    # inverse rework leaves constructive values untouched and scales destructive 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 constructive and 1/alpha when destructive
    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', checklist())) + 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 stream now’s 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)  
 
 stream <- checklist(lrelu, affine) %>% tfb_chain()
 
 dist <- tfd_transformed_distribution(distribution = base_dist,
                          bijector = stream)
  
 l <- -tf$reduce_mean(dist$log_prob(batch))
 # hold observe of progress
 print(spherical(as.numeric(l), 2))
 l
}

Now we will really 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 gradual (if gradual) progress. Utilizing bijectors, now we have really educated and outlined slightly neural community.

Outlook

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