Getting began with TensorFlow Likelihood from R

With the abundance of nice libraries, in R, for statistical computing, why would you be excited about TensorFlow Likelihood (TFP, for brief)? Properly – 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 purposes is huge – and much too various to cowl as an entire in an introductory weblog put up.

As an alternative, our intention right here is to supply a primary introduction to TFP, specializing in direct applicability to and interoperability with deep studying.
We’ll rapidly present easy methods to 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 realm of semi-supervised illustration studying.

To put in TFP along with TensorFlow, merely append tensorflow-probability to the default checklist 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 typical 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 wish to should 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 traces in a contemporary (R) session:

… and import TFP, identical 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 normal regular once more.

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

Issues generally accomplished with a distribution embrace 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 try 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 a lot 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
)

Notice that within the final chunk, we’re asking for the log chances of 4 unbiased attracts.

Batch shapes and occasion shapes

In TFP, we are able to 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’d appear like, this isn’t a multivariate regular. As indicated by batch_shape=(3,), it is a “batch” of unbiased univariate distributions. The truth that these are univariate is seen in event_shape=(): Every of them lives in one-dimensional occasion house.

If as a substitute 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 are able to mix each, creating batches of multivariate distributions:

nd_batch <- tfd$MultivariateNormalFullCovariance(
  loc = checklist(c(0., 0.), c(1., 1.), c(2., 2.)),
  covariance_matrix = checklist(
    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 might sound, conditions come up the place we wish to remodel distribution shapes between these varieties – in reality, we’ll see such a case very quickly.

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

Here’s a batch of three unbiased 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$Unbiased(bs, reinterpreted_batch_ndims = 1L)
b

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

Right here reinterpreted_batch_ndims tells TFP how most of the batch dimensions are getting used for the occasion house, beginning to rely from the suitable of the form checklist.

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 chances. Optionally, our new VAE will be capable of study the prior distribution.

Concretely, the next exposition will encompass three components.
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 talk about 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 accessible as a part of the Keras examples so that you don’t have to repeat out code snippets. The code additionally comprises further performance not mentioned and replicated right here, reminiscent of for saving mannequin weights.

So, let’s begin with the frequent half.

On the threat of repeating ourselves, right here once more are the preparatory steps (together with just a few further library masses).

Dataset

For a change from MNIST and Style-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 immediately as tensors. As an alternative, it returns a batch of multivariate regular distributions:

# you would possibly wish to change this relying on the dataset
latent_dim <- 2

encoder_model <- operate(title = NULL) {

  keras_model_custom(title = title, operate(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)
    
    operate (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 learn about you, however we nonetheless benefit from the ease of inspecting values with keen execution – lots.

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

Decoder

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

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

Let’s do this out too.

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

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

This distribution can 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 operate merely subtracts the log chance of the samples underneath the prior from their loglikelihood underneath the approximate posterior.

compute_kl_loss <- operate(
  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 appeared on the frequent components, we first talk about easy methods to practice a VAE with a static prior.

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

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

And right here is the entire 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$apply_gradients(purrr::transpose(checklist(
      encoder_gradients, encoder$variables
    )),
    global_step = tf$practice$get_or_create_global_step())
    optimizer$apply_gradients(purrr::transpose(checklist(
      decoder_gradients, decoder$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)} whole"
    ),
    "n"
  )
}

Above, taking part in round with the encoder and the decoder, we’ve already seen how

approx_posterior <- encoder(x)

offers us a distribution we are able to 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 offers us on-off-likelihoods for picture pixels.

decoder_likelihood <- decoder(approx_posterior_sample)

Now the loss consists of the standard ELBO elements: reconstruction loss and KL divergence.
The reconstruction loss we immediately get hold of from TFP, utilizing the realized decoder distribution to evaluate the chance 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 operate 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 attributable to utilizing TFP, the coaching course of is simply regular backprop, the best way it seems to be utilizing keen execution.

Now let’s see how as a substitute of utilizing the usual isotropic Gaussian, we might study a combination of Gaussians.
The selection of variety of distributions right here is fairly arbitrary. Simply as with latent_dim, you would possibly wish to experiment and discover out what works greatest in your dataset.

mixture_components <- 16

learnable_prior_model <- operate(title = NULL, latent_dim, mixture_components) {
  
  keras_model_custom(title = title, operate(self) {
    
    self$loc <-
      tf$get_variable(
        title = "loc",
        form = checklist(mixture_components, latent_dim),
        dtype = tf$float32
      )
    self$raw_scale_diag <- tf$get_variable(
      title = "raw_scale_diag",
      form = c(mixture_components, latent_dim),
      dtype = tf$float32
    )
    self$mixture_logits <-
      tf$get_variable(
        title = "mixture_logits",
        form = c(mixture_components),
        dtype = tf$float32
      )
      
    operate (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 kind, and mixture_distribution holds the possibilities that particular person elements are chosen.

Notice 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 are able to pattern from? A bit unusually, this mannequin can be known 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 could be the entire coaching loop. Notice how we now have a 3rd mannequin to backprop by.

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$apply_gradients(purrr::transpose(checklist(
      encoder_gradients, encoder$variables
    )),
    global_step = tf$practice$get_or_create_global_step())
    optimizer$apply_gradients(purrr::transpose(checklist(
      decoder_gradients, decoder$variables
    )),
    global_step = tf$practice$get_or_create_global_step())
    optimizer$apply_gradients(purrr::transpose(checklist(
      prior_gradients, latent_prior_model$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)} whole"
    ),
    "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 wish to generalize to different datasets, architectures, and so forth.

Talking of outcomes, how do they give the impression of being? Right here we see letters generated after 40 epochs of coaching. On the left are random letters, on the suitable, the standard VAE grid show of latent house.

Hopefully, we’ve succeeded in exhibiting that TensorFlow Likelihood, keen execution, and Keras make for a horny mixture! For those who relate whole 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 purposes of TensorFlow Likelihood, largely from the realm of illustration studying. Keep tuned!