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# tfp.bijectors.AutoregressiveNetwork

Masked Autoencoder for Distribution Estimation [Germain et al. (2015)][1].

A AutoregressiveNetwork takes as input a Tensor of shape [..., event_size] and returns a Tensor of shape [..., event_size, params].

The output satisfies the autoregressive property. That is, the layer is configured with some permutation ord of {0, ..., event_size-1} (i.e., an ordering of the input dimensions), and the output output[batch_idx, i, ...] for input dimension i depends only on inputs x[batch_idx, j] where ord(j) < ord(i). The autoregressive property allows us to use output[batch_idx, i] to parameterize conditional distributions: p(x[batch_idx, i] | x[batch_idx, j] for ord(j) < ord(i)) which give us a tractable distribution over input x[batch_idx]: p(x[batch_idx]) = prod_i p(x[batch_idx, ord(i)] | x[batch_idx, ord(0:i)])

For example, when params is 2, the output of the layer can parameterize the location and log-scale of an autoregressive Gaussian distribution.

#### Example

The AutoregressiveNetwork can be used to do density estimation as is shown in the below example:

# Generate data -- as in Figure 1 in [Papamakarios et al. (2017)][2]).
n = 2000
x2 = np.random.randn(n).astype(dtype=np.float32) * 2.
x1 = np.random.randn(n).astype(dtype=np.float32) + (x2 * x2 / 4.)
data = np.stack([x1, x2], axis=-1)

distribution = tfd.TransformedDistribution(
distribution=tfd.Sample(tfd.Normal(loc=0., scale=1.), sample_shape=[2]),

# Construct and fit model.
x_ = tfkl.Input(shape=(2,), dtype=tf.float32)
log_prob_ = distribution.log_prob(x_)
model = tfk.Model(x_, log_prob_)

loss=lambda _, log_prob: -log_prob)

batch_size = 25
model.fit(x=data,
y=np.zeros((n, 0), dtype=np.float32),
batch_size=batch_size,
epochs=1,
steps_per_epoch=1,  # Usually `n // batch_size`.
shuffle=True,
verbose=True)

# Use the fitted distribution.
distribution.sample((3, 1))
distribution.log_prob(np.ones((3, 2), dtype=np.float32))

The conditional argument can be used to instead build a conditional density estimator. To do this the conditioning variable must be passed as a kwarg:

# Generate data as the mixture of two distributions.
n = 2000
c = np.r_[
np.zeros(n//2),
np.ones(n//2)
]
mean_0, mean_1 = 0, 5
x = np.r_[
np.random.randn(n//2).astype(dtype=np.float32) + mean_0,
np.random.randn(n//2).astype(dtype=np.float32) + mean_1
]

params=2,
hidden_units=[2, 2],
event_shape=(1,),
conditional=True,
kernel_initializer=tfk.initializers.VarianceScaling(0.1),
conditional_event_shape=(1,)
)

distribution = tfd.TransformedDistribution(
distribution=tfd.Sample(tfd.Normal(loc=0., scale=1.), sample_shape=[1]),

# Construct and fit model.
x_ = tfkl.Input(shape=(1,), dtype=tf.float32)
c_ = tfkl.Input(shape=(1,), dtype=tf.float32)
log_prob_ = distribution.log_prob(
x_, bijector_kwargs={'conditional_input': c_})
model = tfk.Model([x_, c_], log_prob_)

loss=lambda _, log_prob: -log_prob)

batch_size = 25
model.fit(x=[x, c],
y=np.zeros((n, 0), dtype=np.float32),
batch_size=batch_size,
epochs=3,
steps_per_epoch=n // batch_size,
shuffle=True,
verbose=True)

# Use the fitted distribution to sample condition on c = 1
n_samples = 1000
cond = 1
samples = distribution.sample(
(n_samples,),
bijector_kwargs={'conditional_input': cond * np.ones((n_samples, 1))})

#### Examples: Handling Rank-2+ Tensors

AutoregressiveNetwork can be used as a building block to achieve different autoregressive structures over rank-2+ tensors. For example, suppose we want to build an autoregressive distribution over images with dimension [weight, height, channels] with channels = 3:

1. We can parameterize a 'fully autoregressive' distribution, with cross-channel and within-pixel autoregressivity:

r0    g0   b0     r0    g0   b0       r0   g0    b0
^   ^      ^         ^   ^   ^         ^      ^   ^
|  /  ____/           \  |  /           \____  \  |
| /__/                 \ | /                 \__\ |
r1    g1   b1     r1 <- g1   b1       r1   g1 <- b1
^          |
\_________/

as:

# Generate random images for training data.
images = np.random.uniform(size=(100, 8, 8, 3)).astype(np.float32)
n, width, height, channels = images.shape

# Reshape images to achieve desired autoregressivity.
event_shape = [height * width * channels]
reshaped_images = tf.reshape(images, [n, event_shape])

hidden_units=[20, 20], activation='relu')
distribution = tfd.TransformedDistribution(
distribution=tfd.Sample(
tfd.Normal(loc=0., scale=1.), sample_shape=[dims]),

# Construct and fit model.
x_ = tfkl.Input(shape=event_shape, dtype=tf.float32)
log_prob_ = distribution.log_prob(x_)
model = tfk.Model(x_, log_prob_)

loss=lambda _, log_prob: -log_prob)

batch_size = 10
model.fit(x=data,
y=np.zeros((n, 0), dtype=np.float32),
batch_size=batch_size,
epochs=10,
steps_per_epoch=n // batch_size,
shuffle=True,
verbose=True)

# Use the fitted distribution.
distribution.sample((3, 1))
distribution.log_prob(np.ones((5, 8, 8, 3), dtype=np.float32))

2. We can parameterize a distribution with neither cross-channel nor within-pixel autoregressivity:

r0    g0   b0
^     ^    ^
|     |    |
|     |    |
r1    g1   b1

as:

# Generate fake images.
images = np.random.choice([0, 1], size=(100, 8, 8, 3))
n, width, height, channels = images.shape

# Reshape images to achieve desired autoregressivity.
reshaped_images = np.transpose(
np.reshape(images, [n, width * height, channels]),
axes=[0, 2, 1])

made = tfb.AutoregressiveNetwork(params=1, event_shape=[width * height],
hidden_units=[20, 20], activation='relu')

#
# NOTE: Parameterize an autoregressive distribution over an event_shape of
# [channels, width * height], with univariate Bernoulli conditional
# distributions.
distribution = tfd.Autoregressive(
lambda x: tfd.Independent(
dtype=tf.float32),
reinterpreted_batch_ndims=2),
sample0=tf.zeros([channels, width * height], dtype=tf.float32))

# Construct and fit model.
x_ = tfkl.Input(shape=(channels, width * height), dtype=tf.float32)
log_prob_ = distribution.log_prob(x_)
model = tfk.Model(x_, log_prob_)

loss=lambda _, log_prob: -log_prob)

batch_size = 10
model.fit(x=reshaped_images,
y=np.zeros((n, 0), dtype=np.float32),
batch_size=batch_size,
epochs=10,
steps_per_epoch=n // batch_size,
shuffle=True,
verbose=True)

distribution.sample(7)
distribution.log_prob(np.ones((4, 8, 8, 3), dtype=np.float32))

Note that one set of weights is shared for the mapping for each channel from image to distribution parameters -- i.e., the mapping layer(reshaped_images[..., channel, :]), where channel is 0, 1, or 2.

To use separate weights for each channel, we could construct an AutoregressiveNetwork and TransformedDistribution for each channel, and combine them with a tfd.Blockwise distribution.

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

params Python integer specifying the number of parameters to output per input.
event_shape Python list-like of positive integers (or a single int), specifying the shape of the input to this layer, which is also the event_shape of the distribution parameterized by this layer. Currently only rank-1 shapes are supported. That is, event_shape must be a single integer. If not specified, the event shape is inferred when this layer is first called or built.
conditional Python boolean describing whether to add conditional inputs.
conditional_event_shape Python list-like of positive integers (or a single int), specifying the shape of the conditional input to this layer (without the batch dimensions). This must be specified if conditional is True.
conditional_input_layers Python str describing how to add conditional parameters to the autoregressive network. When "all_layers" the conditional input will be combined with the network at every layer, whilst "first_layer" combines the conditional input only at the first layer which is then passed through the network autoregressively. Default: 'all_layers'.
hidden_units Python list-like of non-negative integers, specifying the number of units in each hidden layer.
input_order Order of degrees to the input units: 'random', 'left-to-right', 'right-to-left', or an array of an explicit order. For example, 'left-to-right' builds an autoregressive model: p(x) = p(x1) p(x2 | x1) ... p(xD | x<D). Default: 'left-to-right'.
hidden_degrees Method for assigning degrees to the hidden units: 'equal', 'random'. If 'equal', hidden units in each layer are allocated equally (up to a remainder term) to each degree. Default: 'equal'.
activation An activation function. See tf.keras.layers.Dense. Default: None.
use_bias Whether or not the dense layers constructed in this layer should have a bias term. See tf.keras.layers.Dense. Default: True.
kernel_initializer Initializer for the Dense kernel weight matrices. Default: 'glorot_uniform'.
bias_initializer Initializer for the Dense bias vectors. Default: 'zeros'.
kernel_regularizer Regularizer function applied to the Dense kernel weight matrices. Default: None.
bias_regularizer Regularizer function applied to the Dense bias weight vectors. Default: None.
kernel_constraint Constraint function applied to the Dense kernel weight matrices. Default: None.
bias_constraint Constraint function applied to the Dense bias weight vectors. Default: None.
validate_args Python bool, default False. When True, layer parameters are checked for validity despite possibly degrading runtime performance. When False invalid inputs may silently render incorrect outputs.
**kwargs Additional keyword arguments passed to this layer (but not to the tf.keras.layer.Dense layers constructed by this layer).

activity_regularizer Optional regularizer function for the output of this layer.
compute_dtype The dtype of the layer's computations.

This is equivalent to Layer.dtype_policy.compute_dtype. Unless mixed precision is used, this is the same as Layer.dtype, the dtype of the weights.

Layers automatically cast their inputs to the compute dtype, which causes computations and the output to be in the compute dtype as well. This is done by the base Layer class in Layer.call, so you do not have to insert these casts if implementing your own layer.

Layers often perform certain internal computations in higher precision when compute_dtype is float16 or bfloat16 for numeric stability. The output will still typically be float16 or bfloat16 in such cases.

dtype The dtype of the layer weights.

This is equivalent to Layer.dtype_policy.variable_dtype. Unless mixed precision is used, this is the same as Layer.compute_dtype, the dtype of the layer's computations.

dtype_policy The dtype policy associated with this layer.

This is an instance of a tf.keras.mixed_precision.Policy.

dynamic Whether the layer is dynamic (eager-only); set in the constructor.
event_shape

input Retrieves the input tensor(s) of a layer.

Only applicable if the layer has exactly one input, i.e. if it is connected to one incoming layer.

input_spec InputSpec instance(s) describing the input format for this layer.

When you create a layer subclass, you can set self.input_spec to enable the layer to run input compatibility checks when it is called. Consider a Conv2D layer: it can only be called on a single input tensor of rank 4. As such, you can set, in __init__():

self.input_spec = tf.keras.layers.InputSpec(ndim=4)

Now, if you try to call the layer on an input that isn't rank 4 (for instance, an input of shape (2,), it will raise a nicely-formatted error:

ValueError: Input 0 of layer conv2d is incompatible with the layer:
expected ndim=4, found ndim=1. Full shape received: [2]

Input checks that can be specified via input_spec include:

• Structure (e.g. a single input, a list of 2 inputs, etc)
• Shape
• Rank (ndim)
• Dtype

Variable regularization tensors are created when this property is accessed, so it is eager safe: accessing losses under a tf.GradientTape will propagate gradients back to the corresponding variables.

class MyLayer(tf.keras.layers.Layer):
def call(self, inputs):
return inputs
l = MyLayer()
l(np.ones((10, 1)))
l.losses
[1.0]
inputs = tf.keras.Input(shape=(10,))
x = tf.keras.layers.Dense(10)(inputs)
outputs = tf.keras.layers.Dense(1)(x)
model = tf.keras.Model(inputs, outputs)
# Activity regularization.
len(model.losses)
0
len(model.losses)
1
inputs = tf.keras.Input(shape=(10,))
d = tf.keras.layers.Dense(10, kernel_initializer='ones')
x = d(inputs)
outputs = tf.keras.layers.Dense(1)(x)
model = tf.keras.Model(inputs, outputs)
# Weight regularization.
model.losses
[<tf.Tensor: shape=(), dtype=float32, numpy=1.0>]

input = tf.keras.layers.Input(shape=(3,))
d = tf.keras.layers.Dense(2)
output = d(input)
[m.name for m in d.metrics]
['max', 'min']

name Name of the layer (string), set in the constructor.
name_scope Returns a tf.name_scope instance for this class.
non_trainable_weights List of all non-trainable weights tracked by this layer.

Non-trainable weights are not updated during training. They are expected to be updated manually in call().

output Retrieves the output tensor(s) of a layer.

Only applicable if the layer has exactly one output, i.e. if it is connected to one incoming layer.

params

submodules Sequence of all sub-modules.

Submodules are modules which are properties of this module, or found as properties of modules which are properties of this module (and so on).

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

trainable

trainable_weights List of all trainable weights tracked by this layer.

Trainable weights are updated via gradient descent during training.

variable_dtype Alias of Layer.dtype, the dtype of the weights.
weights Returns the list of all layer variables/weights.

## Methods

Add loss tensor(s), potentially dependent on layer inputs.

Some losses (for instance, activity regularization losses) may be dependent on the inputs passed when calling a layer. Hence, when reusing the same layer on different inputs a and b, some entries in layer.losses may be dependent on a and some on b. This method automatically keeps track of dependencies.

This method can be used inside a subclassed layer or model's call function, in which case losses should be a Tensor or list of Tensors.

#### Example:

class MyLayer(tf.keras.layers.Layer):
def call(self, inputs):
return inputs

This method can also be called directly on a Functional Model during construction. In this case, any loss Tensors passed to this Model must be symbolic and be able to be traced back to the model's Inputs. These losses become part of the model's topology and are tracked in get_config.

#### Example:

inputs = tf.keras.Input(shape=(10,))
x = tf.keras.layers.Dense(10)(inputs)
outputs = tf.keras.layers.Dense(1)(x)
model = tf.keras.Model(inputs, outputs)
# Activity regularization.

If this is not the case for your loss (if, for example, your loss references a Variable of one of the model's layers), you can wrap your loss in a zero-argument lambda. These losses are not tracked as part of the model's topology since they can't be serialized.

#### Example:

inputs = tf.keras.Input(shape=(10,))
d = tf.keras.layers.Dense(10)
x = d(inputs)
outputs = tf.keras.layers.Dense(1)(x)
model = tf.keras.Model(inputs, outputs)
# Weight regularization.

Args
losses Loss tensor, or list/tuple of tensors. Rather than tensors, losses may also be zero-argument callables which create a loss tensor.
**kwargs Used for backwards compatibility only.

Adds metric tensor to the layer.

This method can be used inside the call() method of a subclassed layer or model.

class MyMetricLayer(tf.keras.layers.Layer):
def __init__(self):
super(MyMetricLayer, self).__init__(name='my_metric_layer')
self.mean = tf.keras.metrics.Mean(name='metric_1')

def call(self, inputs):
return inputs

This method can also be called directly on a Functional Model during construction. In this case, any tensor passed to this Model must be symbolic and be able to be traced back to the model's Inputs. These metrics become part of the model's topology and are tracked when you save the model via save().

inputs = tf.keras.Input(shape=(10,))
x = tf.keras.layers.Dense(10)(inputs)
outputs = tf.keras.layers.Dense(1)(x)
model = tf.keras.Model(inputs, outputs)
inputs = tf.keras.Input(shape=(10,))
x = tf.keras.layers.Dense(10)(inputs)
outputs = tf.keras.layers.Dense(1)(x)
model = tf.keras.Model(inputs, outputs)

Args
value Metric tensor.
name String metric name.
**kwargs Additional keyword arguments for backward compatibility. Accepted values: aggregation - When the value tensor provided is not the result of calling a keras.Metric instance, it will be aggregated by default using a keras.Metric.Mean.

### build

View source

See tfkl.Layer.build.

Args
inputs Tensor or list of tensors.
mask Tensor or list of tensors.

Returns
None or a tensor (or list of tensors, one per output tensor of the layer).

### compute_output_shape

View source

See tfkl.Layer.compute_output_shape.

### count_params

Count the total number of scalars composing the weights.

Returns
An integer count.

Raises
ValueError if the layer isn't yet built (in which case its weights aren't yet defined).

### from_config

Creates a layer from its config.

This method is the reverse of get_config, capable of instantiating the same layer from the config dictionary. It does not handle layer connectivity (handled by Network), nor weights (handled by set_weights).

Args
config A Python dictionary, typically the output of get_config.

Returns
A layer instance.

### get_config

Returns the config of the layer.

A layer config is a Python dictionary (serializable) containing the configuration of a layer. The same layer can be reinstantiated later (without its trained weights) from this configuration.

The config of a layer does not include connectivity information, nor the layer class name. These are handled by Network (one layer of abstraction above).

Note that get_config() does not guarantee to return a fresh copy of dict every time it is called. The callers should make a copy of the returned dict if they want to modify it.

Returns
Python dictionary.

### get_weights

Returns the current weights of the layer, as NumPy arrays.

The weights of a layer represent the state of the layer. This function returns both trainable and non-trainable weight values associated with this layer as a list of NumPy arrays, which can in turn be used to load state into similarly parameterized layers.

For example, a Dense layer returns a list of two values: the kernel matrix and the bias vector. These can be used to set the weights of another Dense layer:

layer_a = tf.keras.layers.Dense(1,
kernel_initializer=tf.constant_initializer(1.))
a_out = layer_a(tf.convert_to_tensor([[1., 2., 3.]]))
layer_a.get_weights()
[array([[1.],
[1.],
[1.]], dtype=float32), array([0.], dtype=float32)]
layer_b = tf.keras.layers.Dense(1,
kernel_initializer=tf.constant_initializer(2.))
b_out = layer_b(tf.convert_to_tensor([[10., 20., 30.]]))
layer_b.get_weights()
[array([[2.],
[2.],
[2.]], dtype=float32), array([0.], dtype=float32)]
layer_b.set_weights(layer_a.get_weights())
layer_b.get_weights()
[array([[1.],
[1.],
[1.]], dtype=float32), array([0.], dtype=float32)]

Returns
Weights values as a list of NumPy arrays.

### set_weights

Sets the weights of the layer, from NumPy arrays.

The weights of a layer represent the state of the layer. This function sets the weight values from numpy arrays. The weight values should be passed in the order they are created by the layer. Note that the layer's weights must be instantiated before calling this function, by calling the layer.

For example, a Dense layer returns a list of two values: the kernel matrix and the bias vector. These can be used to set the weights of another Dense layer:

layer_a = tf.keras.layers.Dense(1,
kernel_initializer=tf.constant_initializer(1.))
a_out = layer_a(tf.convert_to_tensor([[1., 2., 3.]]))
layer_a.get_weights()
[array([[1.],
[1.],
[1.]], dtype=float32), array([0.], dtype=float32)]
layer_b = tf.keras.layers.Dense(1,
kernel_initializer=tf.constant_initializer(2.))
b_out = layer_b(tf.convert_to_tensor([[10., 20., 30.]]))
layer_b.get_weights()
[array([[2.],
[2.],
[2.]], dtype=float32), array([0.], dtype=float32)]
layer_b.set_weights(layer_a.get_weights())
layer_b.get_weights()
[array([[1.],
[1.],
[1.]], dtype=float32), array([0.], dtype=float32)]

Args
weights a list of NumPy arrays. The number of arrays and their shape must match number of the dimensions of the weights of the layer (i.e. it should match the output of get_weights).

Raises
ValueError If the provided weights list does not match the layer's specifications.

### with_name_scope

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)>

Args
method The method to wrap.

Returns
The original method wrapped such that it enters the module's name scope.

### __call__

Wraps call, applying pre- and post-processing steps.

Args
*args Positional arguments to be passed to self.call.
**kwargs Keyword arguments to be passed to self.call.

Returns
Output tensor(s).

#### Note:

• The following optional keyword arguments are reserved for specific uses:
• training: Boolean scalar tensor of Python boolean indicating whether the call is meant for training or inference.