With the abundance of nice libraries, in R, for statistical computing, why would you be involved in TensorFlow Chance (TFP, for brief)? Effectively – let’s have a look at an inventory of its elements:

• Distributions and bijectors (bijectors are reversible, composable maps)
• Probabilistic modeling (Edward2 and probabilistic community layers)
• Probabilistic inference (through MCMC or variational inference)

Now think about all these working seamlessly with the TensorFlow framework – core, Keras, contributed modules – and in addition, operating distributed and on GPU. The sector of doable functions is huge – and much too various to cowl as a complete in an introductory weblog put up.

As an alternative, our goal right here is to supply a primary introduction to TFP, specializing in direct applicability to and interoperability with deep studying.
We’ll rapidly present how you can get began with one of many fundamental constructing blocks: distributions. Then, we’ll construct a variational autoencoder just like that in Illustration studying with MMD-VAE. This time although, we’ll make use of TFP to pattern from the prior and approximate posterior distributions.

We’ll regard this put up as a “proof on idea” for utilizing TFP with Keras – from R – and plan to observe up with extra elaborate examples from the world of semi-supervised illustration studying.

To put in TFP along with TensorFlow, merely append tensorflow-probability to the default record of additional packages:

library(tensorflow)
install_tensorflow(
  extra_packages = c("keras", "tensorflow-hub", "tensorflow-probability"),
  model = "1.12"
)

Now to make use of TFP, all we have to do is import it and create some helpful handles.

And right here we go, sampling from a normal regular distribution.

n <- tfd$Regular(loc = 0, scale = 1)
n$pattern(6L)

tf.Tensor(
"Normal_1/pattern/Reshape:0", form=(6,), dtype=float32
)

Now that’s good, however it’s 2019, we don’t need to need to create a session to guage these tensors anymore. Within the variational autoencoder instance under, we’re going to see how TFP and TF keen execution are the right match, so why not begin utilizing it now.

To make use of keen execution, we now have to execute the next strains in a contemporary (R) session:

… and import TFP, similar as above.

tfp <- import("tensorflow_probability")
tfd <- tfp$distributions

Now let’s rapidly have a look at TFP distributions.

Utilizing distributions

Right here’s that customary regular once more.

n <- tfd$Regular(loc = 0, scale = 1)

Issues generally carried out with a distribution embody sampling:

# simply as in low-level tensorflow, we have to append L to point integer arguments
n$pattern(6L) 

tf.Tensor(
[-0.34403768 -0.14122334 -1.3832929   1.618252    1.364448   -1.1299014 ],
form=(6,),
dtype=float32
)

In addition to getting the log likelihood. Right here we do this concurrently for 3 values.

tf.Tensor(
[-1.4189385 -0.9189385 -1.4189385], form=(3,), dtype=float32
)

We are able to do the identical issues with plenty of different distributions, e.g., the Bernoulli:

b <- tfd$Bernoulli(0.9)
b$pattern(10L)

tf.Tensor(
[1 1 1 0 1 1 0 1 0 1], form=(10,), dtype=int32
)

tf.Tensor(
[-1.2411538 -0.3411539 -1.2411538 -1.2411538], form=(4,), dtype=float32
)

Be aware that within the final chunk, we’re asking for the log possibilities of 4 impartial attracts.

Batch shapes and occasion shapes

In TFP, we will do the next.

ns <- tfd$Regular(
  loc = c(1, 10, -200),
  scale = c(0.1, 0.1, 1)
)
ns

tfp.distributions.Regular(
"Regular/", batch_shape=(3,), event_shape=(), dtype=float32
)

Opposite to what it would seem like, this isn’t a multivariate regular. As indicated by batch_shape=(3,), this can be a “batch” of impartial univariate distributions. The truth that these are univariate is seen in event_shape=(): Every of them lives in one-dimensional occasion area.

If as an alternative we create a single, two-dimensional multivariate regular:

n <- tfd$MultivariateNormalDiag(loc = c(0, 10), scale_diag = c(1, 4))
n

tfp.distributions.MultivariateNormalDiag(
"MultivariateNormalDiag/", batch_shape=(), event_shape=(2,), dtype=float32
)

we see batch_shape=(), event_shape=(2,), as anticipated.

After all, we will mix each, creating batches of multivariate distributions:

nd_batch <- tfd$MultivariateNormalFullCovariance(
  loc = record(c(0., 0.), c(1., 1.), c(2., 2.)),
  covariance_matrix = record(
    matrix(c(1, .1, .1, 1), ncol = 2),
    matrix(c(1, .3, .3, 1), ncol = 2),
    matrix(c(1, .5, .5, 1), ncol = 2))
)

This instance defines a batch of three two-dimensional multivariate regular distributions.

Changing between batch shapes and occasion shapes

Unusual as it could sound, conditions come up the place we need to rework distribution shapes between these sorts – in actual fact, we’ll see such a case very quickly.

tfd$Impartial is used to transform dimensions in batch_shape to dimensions in event_shape.

Here’s a batch of three impartial Bernoulli distributions.

bs <- tfd$Bernoulli(probs=c(.3,.5,.7))
bs

tfp.distributions.Bernoulli(
"Bernoulli/", batch_shape=(3,), event_shape=(), dtype=int32
)

We are able to convert this to a digital “three-dimensional” Bernoulli like this:

b <- tfd$Impartial(bs, reinterpreted_batch_ndims = 1L)
b

tfp.distributions.Impartial(
"IndependentBernoulli/", batch_shape=(), event_shape=(3,), dtype=int32
)

Right here reinterpreted_batch_ndims tells TFP how lots of the batch dimensions are getting used for the occasion area, beginning to depend from the correct of the form record.

With this fundamental understanding of TFP distributions, we’re able to see them utilized in a VAE.

We’ll take the (not so) deep convolutional structure from Illustration studying with MMD-VAE and use distributions for sampling and computing possibilities. Optionally, our new VAE will be capable of be taught the prior distribution.

Concretely, the next exposition will encompass three elements.
First, we current frequent code relevant to each a VAE with a static prior, and one which learns the parameters of the prior distribution.
Then, we now have the coaching loop for the primary (static-prior) VAE. Lastly, we focus on the coaching loop and extra mannequin concerned within the second (prior-learning) VAE.

Presenting each variations one after the opposite results in code duplications, however avoids scattering complicated if-else branches all through the code.

The second VAE is out there as a part of the Keras examples so that you don’t have to repeat out code snippets. The code additionally comprises extra performance not mentioned and replicated right here, similar to for saving mannequin weights.

So, let’s begin with the frequent half.

On the danger of repeating ourselves, right here once more are the preparatory steps (together with a couple of extra library masses).

Dataset

For a change from MNIST and Trend-MNIST, we’ll use the model new Kuzushiji-MNIST(Clanuwat et al. 2018).

np <- import("numpy")

kuzushiji <- np$load("kmnist-train-imgs.npz")
kuzushiji <- kuzushiji$get("arr_0")
 
train_images <- kuzushiji %>%
  k_expand_dims() %>%
  k_cast(dtype = "float32")

train_images <- train_images %>% `/`(255)

As in that different put up, we stream the information through tfdatasets:

buffer_size <- 60000
batch_size <- 256
batches_per_epoch <- buffer_size / batch_size

train_dataset <- tensor_slices_dataset(train_images) %>%
  dataset_shuffle(buffer_size) %>%
  dataset_batch(batch_size)

Now let’s see what modifications within the encoder and decoder fashions.

Encoder

The encoder differs from what we had with out TFP in that it doesn’t return the approximate posterior means and variances instantly as tensors. As an alternative, it returns a batch of multivariate regular distributions:

# you may need to change this relying on the dataset
latent_dim <- 2

encoder_model <- perform(identify = NULL) {

  keras_model_custom(identify = identify, perform(self) {
  
    self$conv1 <-
      layer_conv_2d(
        filters = 32,
        kernel_size = 3,
        strides = 2,
        activation = "relu"
      )
    self$conv2 <-
      layer_conv_2d(
        filters = 64,
        kernel_size = 3,
        strides = 2,
        activation = "relu"
      )
    self$flatten <- layer_flatten()
    self$dense <- layer_dense(items = 2 * latent_dim)
    
    perform (x, masks = NULL) {
      x <- x %>%
        self$conv1() %>%
        self$conv2() %>%
        self$flatten() %>%
        self$dense()
        
      tfd$MultivariateNormalDiag(
        loc = x[, 1:latent_dim],
        scale_diag = tf$nn$softplus(x[, (latent_dim + 1):(2 * latent_dim)] + 1e-5)
      )
    }
  })
}

Let’s do this out.

encoder <- encoder_model()

iter <- make_iterator_one_shot(train_dataset)
x <-  iterator_get_next(iter)

approx_posterior <- encoder(x)
approx_posterior

tfp.distributions.MultivariateNormalDiag(
"MultivariateNormalDiag/", batch_shape=(256,), event_shape=(2,), dtype=float32
)

approx_posterior$pattern()

tf.Tensor(
[[ 5.77791929e-01 -1.64988488e-02]
 [ 7.93901443e-01 -1.00042784e+00]
 [-1.56279251e-01 -4.06365871e-01]
 ...
 ...
 [-6.47531569e-01  2.10889503e-02]], form=(256, 2), dtype=float32)

We don’t find out about you, however we nonetheless benefit from the ease of inspecting values with keen execution – rather a lot.

Now, on to the decoder, which too returns a distribution as an alternative of a tensor.

Decoder

Within the decoder, we see why transformations between batch form and occasion form are helpful.
The output of self$deconv3 is four-dimensional. What we’d like is an on-off-probability for each pixel.
Previously, this was achieved by feeding the tensor right into a dense layer and making use of a sigmoid activation.
Right here, we use tfd$Impartial to successfully tranform the tensor right into a likelihood distribution over three-dimensional pictures (width, peak, channel(s)).

decoder_model <- perform(identify = NULL) {
  
  keras_model_custom(identify = identify, perform(self) {
    
    self$dense <- layer_dense(items = 7 * 7 * 32, activation = "relu")
    self$reshape <- layer_reshape(target_shape = c(7, 7, 32))
    self$deconv1 <-
      layer_conv_2d_transpose(
        filters = 64,
        kernel_size = 3,
        strides = 2,
        padding = "similar",
        activation = "relu"
      )
    self$deconv2 <-
      layer_conv_2d_transpose(
        filters = 32,
        kernel_size = 3,
        strides = 2,
        padding = "similar",
        activation = "relu"
      )
    self$deconv3 <-
      layer_conv_2d_transpose(
        filters = 1,
        kernel_size = 3,
        strides = 1,
        padding = "similar"
      )
    
    perform (x, masks = NULL) {
      x <- x %>%
        self$dense() %>%
        self$reshape() %>%
        self$deconv1() %>%
        self$deconv2() %>%
        self$deconv3()
      
      tfd$Impartial(tfd$Bernoulli(logits = x),
                      reinterpreted_batch_ndims = 3L)
      
    }
  })
}

Let’s do this out too.

decoder <- decoder_model()
decoder_likelihood <- decoder(approx_posterior_sample)

tfp.distributions.Impartial(
"IndependentBernoulli/", batch_shape=(256,), event_shape=(28, 28, 1), dtype=int32
)

This distribution might be used to generate the “reconstructions,” in addition to decide the loglikelihood of the unique samples.

KL loss and optimizer

Each VAEs mentioned under will want an optimizer …

optimizer <- tf$practice$AdamOptimizer(1e-4)

… and each will delegate to compute_kl_loss to compute the KL a part of the loss.

This helper perform merely subtracts the log probability of the samples underneath the prior from their loglikelihood underneath the approximate posterior.

compute_kl_loss <- perform(
  latent_prior,
  approx_posterior,
  approx_posterior_sample) {
  
  kl_div <- approx_posterior$log_prob(approx_posterior_sample) -
    latent_prior$log_prob(approx_posterior_sample)
  avg_kl_div <- tf$reduce_mean(kl_div)
  avg_kl_div
}

Now that we’ve regarded on the frequent elements, we first focus on how you can practice a VAE with a static prior.

On this VAE, we use TFP to create the same old isotropic Gaussian prior.
We then instantly pattern from this distribution within the coaching loop.

latent_prior <- tfd$MultivariateNormalDiag(
  loc  = tf$zeros(record(latent_dim)),
  scale_identity_multiplier = 1
)

And right here is the whole coaching loop. We’ll level out the essential TFP-related steps under.

for (epoch in seq_len(num_epochs)) {
  iter <- make_iterator_one_shot(train_dataset)
  
  total_loss <- 0
  total_loss_nll <- 0
  total_loss_kl <- 0
  
  until_out_of_range({
    x <-  iterator_get_next(iter)
    
    with(tf$GradientTape(persistent = TRUE) %as% tape, {
      approx_posterior <- encoder(x)
      approx_posterior_sample <- approx_posterior$pattern()
      decoder_likelihood <- decoder(approx_posterior_sample)
      
      nll <- -decoder_likelihood$log_prob(x)
      avg_nll <- tf$reduce_mean(nll)
      
      kl_loss <- compute_kl_loss(
        latent_prior,
        approx_posterior,
        approx_posterior_sample
      )

      loss <- kl_loss + avg_nll
    })
    
    total_loss <- total_loss + loss
    total_loss_nll <- total_loss_nll + avg_nll
    total_loss_kl <- total_loss_kl + kl_loss
    
    encoder_gradients <- tape$gradient(loss, encoder$variables)
    decoder_gradients <- tape$gradient(loss, decoder$variables)
    
    optimizer$apply_gradients(purrr::transpose(record(
      encoder_gradients, encoder$variables
    )),
    global_step = tf$practice$get_or_create_global_step())
    optimizer$apply_gradients(purrr::transpose(record(
      decoder_gradients, decoder$variables
    )),
    global_step = tf$practice$get_or_create_global_step())
 
  })
  
  cat(
    glue(
      "Losses (epoch): {epoch}:",
      "  {(as.numeric(total_loss_nll)/batches_per_epoch) %>% spherical(4)} nll",
      "  {(as.numeric(total_loss_kl)/batches_per_epoch) %>% spherical(4)} kl",
      "  {(as.numeric(total_loss)/batches_per_epoch) %>% spherical(4)} complete"
    ),
    "n"
  )
}

Above, enjoying round with the encoder and the decoder, we’ve already seen how

approx_posterior <- encoder(x)

provides us a distribution we will pattern from. We use it to acquire samples from the approximate posterior:

approx_posterior_sample <- approx_posterior$pattern()

These samples, we take them and feed them to the decoder, who provides us on-off-likelihoods for picture pixels.

decoder_likelihood <- decoder(approx_posterior_sample)

Now the loss consists of the same old ELBO elements: reconstruction loss and KL divergence.
The reconstruction loss we instantly get hold of from TFP, utilizing the realized decoder distribution to evaluate the probability of the unique enter.

nll <- -decoder_likelihood$log_prob(x)
avg_nll <- tf$reduce_mean(nll)

The KL loss we get from compute_kl_loss, the helper perform we noticed above:

kl_loss <- compute_kl_loss(
        latent_prior,
        approx_posterior,
        approx_posterior_sample
      )

We add each and arrive on the total VAE loss:

loss <- kl_loss + avg_nll

Aside from these modifications as a consequence of utilizing TFP, the coaching course of is simply regular backprop, the way in which it seems utilizing keen execution.

Now let’s see how as an alternative of utilizing the usual isotropic Gaussian, we might be taught a combination of Gaussians.
The selection of variety of distributions right here is fairly arbitrary. Simply as with latent_dim, you may need to experiment and discover out what works finest in your dataset.

mixture_components <- 16

learnable_prior_model <- perform(identify = NULL, latent_dim, mixture_components) {
  
  keras_model_custom(identify = identify, perform(self) {
    
    self$loc <-
      tf$get_variable(
        identify = "loc",
        form = record(mixture_components, latent_dim),
        dtype = tf$float32
      )
    self$raw_scale_diag <- tf$get_variable(
      identify = "raw_scale_diag",
      form = c(mixture_components, latent_dim),
      dtype = tf$float32
    )
    self$mixture_logits <-
      tf$get_variable(
        identify = "mixture_logits",
        form = c(mixture_components),
        dtype = tf$float32
      )
      
    perform (x, masks = NULL) {
        tfd$MixtureSameFamily(
          components_distribution = tfd$MultivariateNormalDiag(
            loc = self$loc,
            scale_diag = tf$nn$softplus(self$raw_scale_diag)
          ),
          mixture_distribution = tfd$Categorical(logits = self$mixture_logits)
        )
      }
    })
  }

In TFP terminology, components_distribution is the underlying distribution sort, and mixture_distribution holds the possibilities that particular person elements are chosen.

Be aware how self$loc, self$raw_scale_diag and self$mixture_logits are TensorFlow Variables and thus, persistent and updatable by backprop.

Now we create the mannequin.

latent_prior_model <- learnable_prior_model(
  latent_dim = latent_dim,
  mixture_components = mixture_components
)

How can we get hold of a latent prior distribution we will pattern from? A bit unusually, this mannequin might be referred to as with out an enter:

latent_prior <- latent_prior_model(NULL)
latent_prior

tfp.distributions.MixtureSameFamily(
"MixtureSameFamily/", batch_shape=(), event_shape=(2,), dtype=float32
)

Right here now’s the whole coaching loop. Be aware how we now have a 3rd mannequin to backprop via.

for (epoch in seq_len(num_epochs)) {
  iter <- make_iterator_one_shot(train_dataset)
  
  total_loss <- 0
  total_loss_nll <- 0
  total_loss_kl <- 0
  
  until_out_of_range({
    x <-  iterator_get_next(iter)
    
    with(tf$GradientTape(persistent = TRUE) %as% tape, {
      approx_posterior <- encoder(x)
      
      approx_posterior_sample <- approx_posterior$pattern()
      decoder_likelihood <- decoder(approx_posterior_sample)
      
      nll <- -decoder_likelihood$log_prob(x)
      avg_nll <- tf$reduce_mean(nll)
      
      latent_prior <- latent_prior_model(NULL)
      
      kl_loss <- compute_kl_loss(
        latent_prior,
        approx_posterior,
        approx_posterior_sample
      )

      loss <- kl_loss + avg_nll
    })
    
    total_loss <- total_loss + loss
    total_loss_nll <- total_loss_nll + avg_nll
    total_loss_kl <- total_loss_kl + kl_loss
    
    encoder_gradients <- tape$gradient(loss, encoder$variables)
    decoder_gradients <- tape$gradient(loss, decoder$variables)
    prior_gradients <-
      tape$gradient(loss, latent_prior_model$variables)
    
    optimizer$apply_gradients(purrr::transpose(record(
      encoder_gradients, encoder$variables
    )),
    global_step = tf$practice$get_or_create_global_step())
    optimizer$apply_gradients(purrr::transpose(record(
      decoder_gradients, decoder$variables
    )),
    global_step = tf$practice$get_or_create_global_step())
    optimizer$apply_gradients(purrr::transpose(record(
      prior_gradients, latent_prior_model$variables
    )),
    global_step = tf$practice$get_or_create_global_step())
    
  })
  
  checkpoint$save(file_prefix = checkpoint_prefix)
  
  cat(
    glue(
      "Losses (epoch): {epoch}:",
      "  {(as.numeric(total_loss_nll)/batches_per_epoch) %>% spherical(4)} nll",
      "  {(as.numeric(total_loss_kl)/batches_per_epoch) %>% spherical(4)} kl",
      "  {(as.numeric(total_loss)/batches_per_epoch) %>% spherical(4)} complete"
    ),
    "n"
  )
}  

And that’s it! For us, each VAEs yielded related outcomes, and we didn’t expertise nice variations from experimenting with latent dimensionality and the variety of combination distributions. However once more, we wouldn’t need to generalize to different datasets, architectures, and so on.

Talking of outcomes, how do they appear? Right here we see letters generated after 40 epochs of coaching. On the left are random letters, on the correct, the same old VAE grid show of latent area.

Hopefully, we’ve succeeded in exhibiting that TensorFlow Chance, keen execution, and Keras make for a pretty mixture! In case you relate complete quantity of code required to the complexity of the duty, in addition to depth of the ideas concerned, this could seem as a fairly concise implementation.

Within the nearer future, we plan to observe up with extra concerned functions of TensorFlow Chance, largely from the world of illustration studying. Keep tuned!