tf.contrib.distributions.ExpRelaxedOneHotCategorical

Class ExpRelaxedOneHotCategorical

Inherits From: Distribution

Defined in tensorflow/contrib/distributions/python/ops/relaxed_onehot_categorical.py.

ExpRelaxedOneHotCategorical distribution with temperature and logits.

An ExpRelaxedOneHotCategorical distribution is a log-transformed RelaxedOneHotCategorical distribution. The RelaxedOneHotCategorical is a distribution over random probability vectors, vectors of positive real values that sum to one, which continuously approximates a OneHotCategorical. The degree of approximation is controlled by a temperature: as the temperature goes to 0 the RelaxedOneHotCategorical becomes discrete with a distribution described by the logits, as the temperature goes to infinity the RelaxedOneHotCategorical becomes the constant distribution that is identically the constant vector of (1/event_size, ..., 1/event_size).

Because computing log-probabilities of the RelaxedOneHotCategorical can suffer from underflow issues, this class is one solution for loss functions that depend on log-probabilities, such as the KL Divergence found in the variational autoencoder loss. The KL divergence between two distributions is invariant under invertible transformations, so evaluating KL divergences of ExpRelaxedOneHotCategorical samples, which are always followed by a tf.exp op, is equivalent to evaluating KL divergences of RelaxedOneHotCategorical samples. See the appendix of Maddison et al., 2016 for more mathematical details, where this distribution is called the ExpConcrete.

Examples

Creates a continuous distribution, whose exp approximates a 3-class one-hot categorical distribution. The 2nd class is the most likely to be the largest component in samples drawn from this distribution. If those samples are followed by a tf.exp op, then they are distributed as a relaxed onehot categorical.

temperature = 0.5
p = [0.1, 0.5, 0.4]
dist = ExpRelaxedOneHotCategorical(temperature, probs=p)
samples = dist.sample()
exp_samples = tf.exp(samples)
# exp_samples has the same distribution as samples from
# RelaxedOneHotCategorical(temperature, probs=p)

Creates a continuous distribution, whose exp approximates a 3-class one-hot categorical distribution. The 2nd class is the most likely to be the largest component in samples drawn from this distribution.

temperature = 0.5
logits = [-2, 2, 0]
dist = ExpRelaxedOneHotCategorical(temperature, logits=logits)
samples = dist.sample()
exp_samples = tf.exp(samples)
# exp_samples has the same distribution as samples from
# RelaxedOneHotCategorical(temperature, probs=p)

Creates a continuous distribution, whose exp approximates a 3-class one-hot categorical distribution. Because the temperature is very low, samples from this distribution are almost discrete, with one component almost 0 and the others very negative. The 2nd class is the most likely to be the largest component in samples drawn from this distribution.

temperature = 1e-5
logits = [-2, 2, 0]
dist = ExpRelaxedOneHotCategorical(temperature, logits=logits)
samples = dist.sample()
exp_samples = tf.exp(samples)
# exp_samples has the same distribution as samples from
# RelaxedOneHotCategorical(temperature, probs=p)

Creates a continuous distribution, whose exp approximates a 3-class one-hot categorical distribution. Because the temperature is very high, samples from this distribution are usually close to the (-log(3), -log(3), -log(3)) vector. The 2nd class is still the most likely to be the largest component in samples drawn from this distribution.

temperature = 10
logits = [-2, 2, 0]
dist = ExpRelaxedOneHotCategorical(temperature, logits=logits)
samples = dist.sample()
exp_samples = tf.exp(samples)
# exp_samples has the same distribution as samples from
# RelaxedOneHotCategorical(temperature, probs=p)

Chris J. Maddison, Andriy Mnih, and Yee Whye Teh. The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables. 2016.

Properties

allow_nan_stats

Python bool describing behavior when a stat is undefined.

Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined.

Returns:

  • allow_nan_stats: Python bool.

batch_shape

Shape of a single sample from a single event index as a TensorShape.

May be partially defined or unknown.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

Returns:

  • batch_shape: TensorShape, possibly unknown.

dtype

The DType of Tensors handled by this Distribution.

event_shape

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

May be partially defined or unknown.

Returns:

  • event_shape: TensorShape, possibly unknown.

event_size

Scalar int32 tensor: the number of classes.

logits

Vector of coordinatewise logits.

name

Name prepended to all ops created by this Distribution.

parameters

Dictionary of parameters used to instantiate this Distribution.

probs

Vector of probabilities summing to one.

reparameterization_type

Describes how samples from the distribution are reparameterized.

Currently this is one of the static instances distributions.FULLY_REPARAMETERIZED or distributions.NOT_REPARAMETERIZED.

Returns:

An instance of ReparameterizationType.

temperature

Batchwise temperature tensor of a RelaxedCategorical.

validate_args

Python bool indicating possibly expensive checks are enabled.

Methods

__init__

__init__(
    temperature,
    logits=None,
    probs=None,
    dtype=None,
    validate_args=False,
    allow_nan_stats=True,
    name='ExpRelaxedOneHotCategorical'
)

Initialize ExpRelaxedOneHotCategorical using class log-probabilities.

Args:

  • temperature: An 0-D Tensor, representing the temperature of a set of ExpRelaxedCategorical distributions. The temperature should be positive.
  • logits: An N-D Tensor, N >= 1, representing the log probabilities of a set of ExpRelaxedCategorical distributions. The first N - 1 dimensions index into a batch of independent distributions and the last dimension represents a vector of logits for each class. Only one of logits or probs should be passed in.
  • probs: An N-D Tensor, N >= 1, representing the probabilities of a set of ExpRelaxedCategorical distributions. The first N - 1 dimensions index into a batch of independent distributions and the last dimension represents a vector of probabilities for each class. Only one of logits or probs should be passed in.
  • dtype: The type of the event samples (default: inferred from logits/probs).
  • validate_args: Python bool, default False. When True distribution parameters are checked for validity despite possibly degrading runtime performance. When False invalid inputs may silently render incorrect outputs.
  • allow_nan_stats: Python bool, default True. When True, statistics (e.g., mean, mode, variance) use the value "NaN" to indicate the result is undefined. When False, an exception is raised if one or more of the statistic's batch members are undefined.
  • name: Python str name prefixed to Ops created by this class.

batch_shape_tensor

batch_shape_tensor(name='batch_shape_tensor')

Shape of a single sample from a single event index as a 1-D Tensor.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

Args:

  • name: name to give to the op

Returns:

  • batch_shape: Tensor.

cdf

cdf(
    value,
    name='cdf'
)

Cumulative distribution function.

Given random variable X, the cumulative distribution function cdf is:

cdf(x) := P[X <= x]

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

  • cdf: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

copy

copy(**override_parameters_kwargs)

Creates a deep copy of the distribution.

Args:

  • **override_parameters_kwargs: String/value dictionary of initialization arguments to override with new values.

Returns:

  • distribution: A new instance of type(self) initialized from the union of self.parameters and override_parameters_kwargs, i.e., dict(self.parameters, **override_parameters_kwargs).

covariance

covariance(name='covariance')

Covariance.

Covariance is (possibly) defined only for non-scalar-event distributions.

For example, for a length-k, vector-valued distribution, it is calculated as,

Cov[i, j] = Covariance(X_i, X_j) = E[(X_i - E[X_i]) (X_j - E[X_j])]

where Cov is a (batch of) k x k matrix, 0 <= (i, j) < k, and E denotes expectation.

Alternatively, for non-vector, multivariate distributions (e.g., matrix-valued, Wishart), Covariance shall return a (batch of) matrices under some vectorization of the events, i.e.,

Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]

where Cov is a (batch of) k' x k' matrices, 0 <= (i, j) < k' = reduce_prod(event_shape), and Vec is some function mapping indices of this distribution's event dimensions to indices of a length-k' vector.

Args:

  • name: Python str prepended to names of ops created by this function.

Returns:

  • covariance: Floating-point Tensor with shape [B1, ..., Bn, k', k'] where the first n dimensions are batch coordinates and k' = reduce_prod(self.event_shape).

cross_entropy

cross_entropy(
    other,
    name='cross_entropy'
)

Computes the (Shannon) cross entropy.

Denote this distribution (self) by P and the other distribution by Q. Assuming P, Q are absolutely continuous with respect to one another and permit densities p(x) dr(x) and q(x) dr(x), (Shanon) cross entropy is defined as:

H[P, Q] = E_p[-log q(X)] = -int_F p(x) log q(x) dr(x)

where F denotes the support of the random variable X ~ P.

Args:

Returns:

  • cross_entropy: self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of (Shanon) cross entropy.

entropy

entropy(name='entropy')

Shannon entropy in nats.

event_shape_tensor

event_shape_tensor(name='event_shape_tensor')

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

Args:

  • name: name to give to the op

Returns:

  • event_shape: Tensor.

is_scalar_batch

is_scalar_batch(name='is_scalar_batch')

Indicates that batch_shape == [].

Args:

  • name: Python str prepended to names of ops created by this function.

Returns:

  • is_scalar_batch: bool scalar Tensor.

is_scalar_event

is_scalar_event(name='is_scalar_event')

Indicates that event_shape == [].

Args:

  • name: Python str prepended to names of ops created by this function.

Returns:

  • is_scalar_event: bool scalar Tensor.

kl_divergence

kl_divergence(
    other,
    name='kl_divergence'
)

Computes the Kullback--Leibler divergence.

Denote this distribution (self) by p and the other distribution by q. Assuming p, q are absolutely continuous with respect to reference measure r, the KL divergence is defined as:

KL[p, q] = E_p[log(p(X)/q(X))]
         = -int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
         = H[p, q] - H[p]

where F denotes the support of the random variable X ~ p, H[., .] denotes (Shanon) cross entropy, and H[.] denotes (Shanon) entropy.

Args:

Returns:

  • kl_divergence: self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of the Kullback-Leibler divergence.

log_cdf

log_cdf(
    value,
    name='log_cdf'
)

Log cumulative distribution function.

Given random variable X, the cumulative distribution function cdf is:

log_cdf(x) := Log[ P[X <= x] ]

Often, a numerical approximation can be used for log_cdf(x) that yields a more accurate answer than simply taking the logarithm of the cdf when x << -1.

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

  • logcdf: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

log_prob

log_prob(
    value,
    name='log_prob'
)

Log probability density/mass function.

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

  • log_prob: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

log_survival_function

log_survival_function(
    value,
    name='log_survival_function'
)

Log survival function.

Given random variable X, the survival function is defined:

log_survival_function(x) = Log[ P[X > x] ]
                         = Log[ 1 - P[X <= x] ]
                         = Log[ 1 - cdf(x) ]

Typically, different numerical approximations can be used for the log survival function, which are more accurate than 1 - cdf(x) when x >> 1.

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

mean

mean(name='mean')

Mean.

mode

mode(name='mode')

Mode.

param_shapes

param_shapes(
    cls,
    sample_shape,
    name='DistributionParamShapes'
)

Shapes of parameters given the desired shape of a call to sample().

This is a class method that describes what key/value arguments are required to instantiate the given Distribution so that a particular shape is returned for that instance's call to sample().

Subclasses should override class method _param_shapes.

Args:

  • sample_shape: Tensor or python list/tuple. Desired shape of a call to sample().
  • name: name to prepend ops with.

Returns:

dict of parameter name to Tensor shapes.

param_static_shapes

param_static_shapes(
    cls,
    sample_shape
)

param_shapes with static (i.e. TensorShape) shapes.

This is a class method that describes what key/value arguments are required to instantiate the given Distribution so that a particular shape is returned for that instance's call to sample(). Assumes that the sample's shape is known statically.

Subclasses should override class method _param_shapes to return constant-valued tensors when constant values are fed.

Args:

  • sample_shape: TensorShape or python list/tuple. Desired shape of a call to sample().

Returns:

dict of parameter name to TensorShape.

Raises:

  • ValueError: if sample_shape is a TensorShape and is not fully defined.

prob

prob(
    value,
    name='prob'
)

Probability density/mass function.

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

  • prob: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

quantile

quantile(
    value,
    name='quantile'
)

Quantile function. Aka "inverse cdf" or "percent point function".

Given random variable X and p in [0, 1], the quantile is:

quantile(p) := x such that P[X <= x] == p

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

  • quantile: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

sample

sample(
    sample_shape=(),
    seed=None,
    name='sample'
)

Generate samples of the specified shape.

Note that a call to sample() without arguments will generate a single sample.

Args:

  • sample_shape: 0D or 1D int32 Tensor. Shape of the generated samples.
  • seed: Python integer seed for RNG
  • name: name to give to the op.

Returns:

  • samples: a Tensor with prepended dimensions sample_shape.

stddev

stddev(name='stddev')

Standard deviation.

Standard deviation is defined as,

stddev = E[(X - E[X])**2]**0.5

where X is the random variable associated with this distribution, E denotes expectation, and stddev.shape = batch_shape + event_shape.

Args:

  • name: Python str prepended to names of ops created by this function.

Returns:

  • stddev: Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().

survival_function

survival_function(
    value,
    name='survival_function'
)

Survival function.

Given random variable X, the survival function is defined:

survival_function(x) = P[X > x]
                     = 1 - P[X <= x]
                     = 1 - cdf(x).

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

variance

variance(name='variance')

Variance.

Variance is defined as,

Var = E[(X - E[X])**2]

where X is the random variable associated with this distribution, E denotes expectation, and Var.shape = batch_shape + event_shape.

Args:

  • name: Python str prepended to names of ops created by this function.

Returns:

  • variance: Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().