tfp.bijectors.MaskedAutoregressiveFlow

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Affine MaskedAutoregressiveFlow bijector.

Inherits From: Bijector

tfp.bijectors.MaskedAutoregressiveFlow(
    shift_and_log_scale_fn=None, bijector_fn=None, is_constant_jacobian=False,
    validate_args=False, unroll_loop=False, event_ndims=1, 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 continuous 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[perm[i]] | x[perm[0:i]]) : i=0, ..., d } where perm is some permutation of {0, ..., d}. In the simple case where the permutation is identity this reduces to: p(x) = prod{ p(x[i] | x[0:i]) : i=0, ..., d }.

In TensorFlow Probability, 'normalizing flows' are implemented as tfp.bijectors.Bijectors. 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, tfp.bijectors.AutoregressiveNetwork 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 tfp.bijectors.AutoregressiveNetwork or use some other architecture, e.g., using tf.keras.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[-event_dims:].num_elements()
  for _ in range(event_size):
    shift, log_scale = shift_and_log_scale_fn(y)
    y = x * tf.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) / tf.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.)

The bijector_fn argument allows specifying a more general coupling relation, such as the LSTM-inspired activation from [4], or Neural Spline Flow [5]. It must logically operate on each element of the input individually, and still obey the 'autoregressive property' described above. The forward transformation is

def forward(x):
  y = zeros_like(x)
  event_size = x.shape[-event_dims:].num_elements()
  for _ in range(event_size):
    bijector = bijector_fn(y)
    y = bijector.forward(x)
  return y

and inverse transformation is

def inverse(y):
    bijector = bijector_fn(y)
    return bijector.inverse(y)

Examples

tfd = tfp.distributions
tfb = tfp.bijectors

dims = 2

# 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.AutoregressiveNetwork(
            params=2, hidden_units=[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.
# Cheap; no `tf.while_loop` despite no Bijector caching.
maf.log_prob(tf.zeros(dims))

# [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.AutoregressiveNetwork(
            params=2, hidden_units=[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.
# Expensive; uses `tf.while_loop`, no Bijector caching.
iaf.log_prob(tf.zeros(dims))

# 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.
made = tfb.AutoregressiveNetwork(params=1, hidden_units=[32])
maf_no_scale_hidden2 = tfd.TransformedDistribution(
    distribution=tfd.Normal(loc=0., scale=1.),
    bijector=tfb.MaskedAutoregressiveFlow(
        lambda y: (made(y)[..., 0], None),
        is_constant_jacobian=True),
    event_shape=[dims])
maf_no_scale_hidden2._made = made  # Ensure maf_no_scale_hidden2.trainable
# NOTE: The last line ensures that maf_no_scale_hidden2.trainable_variables
# will include all variables from `made`.

Variable Tracking

A tfb.MaskedAutoregressiveFlow instance saves a reference to the values passed as shift_and_log_scale_fn and bijector_fn to its constructor. Thus, for most values passed as shift_and_log_scale_fn or bijector_fn, variables referenced by those values will be found and tracked by the tfb.MaskedAutoregressiveFlow instance. Please see the tf.Module documentation for further details.

However, if the value passed to shift_and_log_scale_fn or bijector_fn is a Python function, then tfb.MaskedAutoregressiveFlow cannot automatically track variables used inside shift_and_log_scale_fn or bijector_fn. To get tfb.MaskedAutoregressiveFlow to track such variables, either:

1. Replace the Python function with a tf.Module, tf.keras.Layer, or other callable object through which tf.Module can find variables.

2. Or, add a reference to the variables to the tfb.MaskedAutoregressiveFlow instance by setting an attribute -- for example:

made1 = tfb.AutoregressiveNetwork(params=1, hidden_units=[10, 10])
made2 = tfb.AutoregressiveNetwork(params=1, hidden_units=[10, 10])
maf = tfb.MaskedAutoregressiveFlow(lambda y: (made1(y), made2(y) + 1.))
maf._made_variables = made1.variables + made2.variables

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

[4]: Diederik P Kingma, Tim Salimans, Max Welling. Improving Variational Inference with Inverse Autoregressive Flow. In Neural Information Processing Systems, 2016. https://arxiv.org/abs/1606.04934

[5]: Conor Durkan, Artur Bekasov, Iain Murray, George Papamakarios. Neural Spline Flows, 2019. http://arxiv.org/abs/1906.04032

a = tf.Module()
b = tf.Module()
c = tf.Module()
a.b = b
b.c = c
list(a.submodules) == [b, c]
True
list(b.submodules) == [c]
True
list(c.submodules) == []
True

Methods

forward

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forward(
    x, name='forward', **kwargs
)

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

forward_dtype

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forward_dtype(
    dtype, name='forward_dtype', **kwargs
)

Returns the dtype of the output of the forward transformation.

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', **kwargs
)

Returns both the forward_log_det_jacobian.

inverse

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inverse(
    y, name='inverse', **kwargs
)

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

inverse_dtype

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inverse_dtype(
    dtype, name='inverse_dtype', **kwargs
)

Returns the dtype of the output of the inverse transformation.

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

View source

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', **kwargs
)

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).

with_name_scope

@classmethod
with_name_scope(
    method
)

Decorator to automatically enter the module name scope.

class MyModule(tf.Module):
  @tf.Module.with_name_scope
  def __call__(self, x):
    if not hasattr(self, 'w'):
      self.w = tf.Variable(tf.random.normal([x.shape[1], 3]))
    return tf.matmul(x, self.w)

Using the above module would produce tf.Variables and tf.Tensors whose names included the module name:

mod = MyModule()
mod(tf.ones([1, 2]))
<tf.Tensor: shape=(1, 3), dtype=float32, numpy=..., dtype=float32)>
mod.w
<tf.Variable 'my_module/Variable:0' shape=(2, 3) dtype=float32,
numpy=..., dtype=float32)>

__call__

View source

__call__(
    value, name=None, **kwargs
)

Applies or composes the Bijector, depending on input type.

This is a convenience function which applies the Bijector instance in three different ways, depending on the input:

1. If the input is a tfd.Distribution instance, return tfd.TransformedDistribution(distribution=input, bijector=self).
2. If the input is a tfb.Bijector instance, return tfb.Chain([self, input]).
3. Otherwise, return self.forward(input)

Examples

sigmoid = tfb.Reciprocal()(
    tfb.AffineScalar(shift=1.)(
      tfb.Exp()(
        tfb.AffineScalar(scale=-1.))))
# ==> `tfb.Chain([
#         tfb.Reciprocal(),
#         tfb.AffineScalar(shift=1.),
#         tfb.Exp(),
#         tfb.AffineScalar(scale=-1.),
#      ])`  # ie, `tfb.Sigmoid()`

log_normal = tfb.Exp()(tfd.Normal(0, 1))
# ==> `tfd.TransformedDistribution(tfd.Normal(0, 1), tfb.Exp())`

tfb.Exp()([-1., 0., 1.])
# ==> tf.exp([-1., 0., 1.])