Artificial Intelligence

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

August 18, 2024
roosho

As of right this moment, deep studying’s best successes have taken place within the realm of supervised studying, requiring heaps and plenty of 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, we’ve got 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 put up, we’ll introduce flows, specializing in find out how 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 fashion of keras, tensorflow and tfdatasets. A be aware relating to this package deal: 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 offered utilizing $-syntax if want be.

Density estimation and sampling

Again to unsupervised studying, and particularly considering 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 vital 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 earth: it has discovered a distribution.
Within the case of variational autoencoders, there’s extra: The entities are imagined 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 will 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 – seem like it comes from the empirical information distribution. It mustn’t, nonetheless, look precisely like every of the gadgets used to coach the VAE, or else we’ve got 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 function: With VAE, we don’t have a method to compute an precise density underneath 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.

Normalizing flows

A stream 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 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 chance distribution (CDF) – from an exponential CDF, to be exact. Now that we’ve got 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 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 information to a uniform distribution between 0 and 1, permitting to evaluate information probability.
  • Conversely, it maps a chance 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 develop 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 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 examine 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 chance from the bijector, we add two elements:

  • Firstly, we run the pattern by the ahead transformation and compute log chance underneath 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 underneath this transformation. That is achieved 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, chance mass is conserved. Say we’ve got a stream 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 reworked pattern, lies between (x_0) and (x_0 + dx)?

This chance is (p(x) dx), the density occasions 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 decided by the bottom chance (p(z)) of the reworked distribution, multiplied by how a lot the stream stretches house.

The identical goes in larger dimensions: Once more, the stream 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 larger dimensions, the Jacobian replaces the spinoff. 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 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. Under, 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 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 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

To date, 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 got principally harassed the inverse mapping: We would like 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 principally being an isotropic Gaussian. Nevertheless in observe, we don’t have that “noise” but, we simply have information.
So in observe, we’ve got 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 go away “actual world flows” to the following put up.

The instance is predicated on half 1 of Eric Jang’s introduction to normalizing flows. The principle distinction (other than 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, 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 <- perform(alpha) {
  
  tfb_inline(
    # ahead remodel leaves optimistic values untouched and scales unfavourable 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 unfavourable 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 unfavourable
    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 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)  
 
 stream <- checklist(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 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$decrease(loss)
  })
}

Outcomes will differ relying on random initialization, however it is best to see a gentle (if gradual) progress. Utilizing bijectors, we’ve got really educated and outlined just a little neural community.

Outlook

Undoubtedly, this stream is just too easy to mannequin advanced information, nevertheless it’s instructive to have seen the fundamental rules earlier than delving into extra advanced flows. Within the subsequent put up, we’ll take a look at 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.