tf.contrib.distributions.bijectors.MaskedAutoregressiveFlow

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Affine MaskedAutoregressiveFlow bijector for vector-valued events.

Inherits From: Bijector

tf.contrib.distributions.bijectors.MaskedAutoregressiveFlow(
    shift_and_log_scale_fn, is_constant_jacobian=False, validate_args=False,
    unroll_loop=False, name=None
)

The affine autoregressive flow [(Papamakarios et al., 2016)][3] provides a relatively simple framework for user-specified (deep) architectures to learn a distribution over vector-valued events. Regarding terminology,

"Autoregressive models decompose the joint density as a product of conditionals, and model each conditional in turn. Normalizing flows transform a base density (e.g. a standard Gaussian) into the target density by an invertible transformation with tractable Jacobian." [(Papamakarios et al., 2016)][3]

In other words, the "autoregressive property" is equivalent to the decomposition, p(x) = prod{ p(x[i] | x[0:i]) : i=0, ..., d }. The provided shift_and_log_scale_fn, masked_autoregressive_default_template, achieves this property by zeroing out weights in its masked_dense layers.

In the tfp framework, a "normalizing flow" is implemented as a tfp.bijectors.Bijector. The forward "autoregression" is implemented using a tf.while_loop and a deep neural network (DNN) with masked weights such that the autoregressive property is automatically met in the inverse.

A TransformedDistribution using MaskedAutoregressiveFlow(...) uses the (expensive) forward-mode calculation to draw samples and the (cheap) reverse-mode calculation to compute log-probabilities. Conversely, a TransformedDistribution using Invert(MaskedAutoregressiveFlow(...)) uses the (expensive) forward-mode calculation to compute log-probabilities and the (cheap) reverse-mode calculation to compute samples. See "Example Use" [below] for more details.

Given a shift_and_log_scale_fn, the forward and inverse transformations are (a sequence of) affine transformations. A "valid" shift_and_log_scale_fn must compute each shift (aka loc or "mu" in [Germain et al. (2015)][1]) and log(scale) (aka "alpha" in [Germain et al. (2015)][1]) such that each are broadcastable with the arguments to forward and inverse, i.e., such that the calculations in forward, inverse [below] are possible.

For convenience, masked_autoregressive_default_template is offered as a possible shift_and_log_scale_fn function. It implements the MADE architecture [(Germain et al., 2015)][1]. MADE is a feed-forward network that computes a shift and log(scale) using masked_dense layers in a deep neural network. Weights are masked to ensure the autoregressive property. It is possible that this architecture is suboptimal for your task. To build alternative networks, either change the arguments to masked_autoregressive_default_template, use the masked_dense function to roll-out your own, or use some other architecture, e.g., using tf.layers.

Assuming shift_and_log_scale_fn has valid shape and autoregressive semantics, the forward transformation is

def forward(x):
  y = zeros_like(x)
  event_size = x.shape[-1]
  for _ in range(event_size):
    shift, log_scale = shift_and_log_scale_fn(y)
    y = x * math_ops.exp(log_scale) + shift
  return y

and the inverse transformation is

def inverse(y):
  shift, log_scale = shift_and_log_scale_fn(y)
  return (y - shift) / math_ops.exp(log_scale)

Notice that the inverse does not need a for-loop. This is because in the forward pass each calculation of shift and log_scale is based on the y calculated so far (not x). In the inverse, the y is fully known, thus is equivalent to the scaling used in forward after event_size passes, i.e., the "last" y used to compute shift, log_scale. (Roughly speaking, this also proves the transform is bijective.)

Examples

import tensorflow_probability as tfp
tfd = tfp.distributions
tfb = tfp.bijectors

dims = 5

# A common choice for a normalizing flow is to use a Gaussian for the base
# distribution. (However, any continuous distribution would work.) E.g.,
maf = tfd.TransformedDistribution(
    distribution=tfd.Normal(loc=0., scale=1.),
    bijector=tfb.MaskedAutoregressiveFlow(
        shift_and_log_scale_fn=tfb.masked_autoregressive_default_template(
            hidden_layers=[512, 512])),
    event_shape=[dims])

x = maf.sample()  # Expensive; uses `tf.while_loop`, no Bijector caching.
maf.log_prob(x)   # Almost free; uses Bijector caching.
maf.log_prob(0.)  # Cheap; no `tf.while_loop` despite no Bijector caching.

# [Papamakarios et al. (2016)][3] also describe an Inverse Autoregressive
# Flow [(Kingma et al., 2016)][2]:
iaf = tfd.TransformedDistribution(
    distribution=tfd.Normal(loc=0., scale=1.),
    bijector=tfb.Invert(tfb.MaskedAutoregressiveFlow(
        shift_and_log_scale_fn=tfb.masked_autoregressive_default_template(
            hidden_layers=[512, 512]))),
    event_shape=[dims])

x = iaf.sample()  # Cheap; no `tf.while_loop` despite no Bijector caching.
iaf.log_prob(x)   # Almost free; uses Bijector caching.
iaf.log_prob(0.)  # Expensive; uses `tf.while_loop`, no Bijector caching.

# In many (if not most) cases the default `shift_and_log_scale_fn` will be a
# poor choice. Here's an example of using a "shift only" version and with a
# different number/depth of hidden layers.
shift_only = True
maf_no_scale_hidden2 = tfd.TransformedDistribution(
    distribution=tfd.Normal(loc=0., scale=1.),
    bijector=tfb.MaskedAutoregressiveFlow(
        tfb.masked_autoregressive_default_template(
            hidden_layers=[32],
            shift_only=shift_only),
        is_constant_jacobian=shift_only),
    event_shape=[dims])

References

[1]: Mathieu Germain, Karol Gregor, Iain Murray, and Hugo Larochelle. MADE: Masked Autoencoder for Distribution Estimation. In International Conference on Machine Learning, 2015. https://arxiv.org/abs/1502.03509

[2]: Diederik P. Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, and Max Welling. Improving Variational Inference with Inverse Autoregressive Flow. In Neural Information Processing Systems, 2016. https://arxiv.org/abs/1606.04934

[3]: George Papamakarios, Theo Pavlakou, and Iain Murray. Masked Autoregressive Flow for Density Estimation. In Neural Information Processing Systems, 2017. https://arxiv.org/abs/1705.07057

Methods

forward

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forward(
    x, name='forward'
)

Returns the forward Bijector evaluation, i.e., X = g(Y).

forward_event_shape

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forward_event_shape(
    input_shape
)

Shape of a single sample from a single batch as a TensorShape.

Same meaning as forward_event_shape_tensor. May be only partially defined.

forward_event_shape_tensor

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forward_event_shape_tensor(
    input_shape, name='forward_event_shape_tensor'
)

Shape of a single sample from a single batch as an int32 1D Tensor.

forward_log_det_jacobian

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forward_log_det_jacobian(
    x, event_ndims, name='forward_log_det_jacobian'
)

Returns both the forward_log_det_jacobian.

inverse

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inverse(
    y, name='inverse'
)

Returns the inverse Bijector evaluation, i.e., X = g^{-1}(Y).

inverse_event_shape

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inverse_event_shape(
    output_shape
)

Shape of a single sample from a single batch as a TensorShape.

Same meaning as inverse_event_shape_tensor. May be only partially defined.

inverse_event_shape_tensor

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inverse_event_shape_tensor(
    output_shape, name='inverse_event_shape_tensor'
)

Shape of a single sample from a single batch as an int32 1D Tensor.

inverse_log_det_jacobian

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inverse_log_det_jacobian(
    y, event_ndims, name='inverse_log_det_jacobian'
)

Returns the (log o det o Jacobian o inverse)(y).

Mathematically, returns: log(det(dX/dY))(Y). (Recall that: X=g^{-1}(Y).)

Note that forward_log_det_jacobian is the negative of this function, evaluated at g^{-1}(y).