# Copyright 2017-2020 The GPflow Contributors. All Rights Reserved.
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# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
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# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
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from typing import Any, Optional
import numpy as np
import tensorflow as tf
import tensorflow_probability as tfp
from ..base import MeanAndVariance, Module, Parameter, TensorType
from ..config import default_float
from ..quadrature import hermgauss
from ..utilities import to_default_float, to_default_int
from .base import Likelihood, MonteCarloLikelihood
[docs]class Softmax(MonteCarloLikelihood):
"""
The soft-max multi-class likelihood. It can only provide a stochastic
Monte-Carlo estimate of the variational expectations term, but this
added variance tends to be small compared to that due to mini-batching
(when using the SVGP model).
"""
def __init__(self, num_classes: int, **kwargs: Any) -> None:
super().__init__(latent_dim=num_classes, observation_dim=None, **kwargs)
self.num_classes = self.latent_dim
def _log_prob(self, F: TensorType, Y: TensorType) -> tf.Tensor:
return -tf.nn.sparse_softmax_cross_entropy_with_logits(logits=F, labels=Y[:, 0])
def _conditional_mean(self, F: TensorType) -> tf.Tensor:
return tf.nn.softmax(F)
def _conditional_variance(self, F: TensorType) -> tf.Tensor:
p = self.conditional_mean(F)
return p - p ** 2
[docs]class RobustMax(Module):
"""
This class represent a multi-class inverse-link function. Given a vector
f=[f_1, f_2, ... f_k], the result of the mapping is
y = [y_1 ... y_k]
with
y_i = (1-epsilon) i == argmax(f)
epsilon/(k-1) otherwise
where k is the number of classes.
"""
def __init__(self, num_classes: int, epsilon: float = 1e-3, **kwargs: Any) -> None:
"""
`epsilon` represents the fraction of 'errors' in the labels of the
dataset. This may be a hard parameter to optimize, so by default
it is set un-trainable, at a small value.
"""
super().__init__(**kwargs)
transform = tfp.bijectors.Sigmoid()
prior = tfp.distributions.Beta(to_default_float(0.2), to_default_float(5.0))
self.epsilon = Parameter(epsilon, transform=transform, prior=prior, trainable=False)
self.num_classes = num_classes
self._squash = 1e-6
def __call__(self, F: TensorType) -> tf.Tensor:
i = tf.argmax(F, 1)
return tf.one_hot(
i, self.num_classes, tf.squeeze(1.0 - self.epsilon), tf.squeeze(self.eps_k1)
)
@property
def eps_k1(self) -> tf.Tensor:
return self.epsilon / (self.num_classes - 1.0)
def safe_sqrt(self, val: TensorType) -> tf.Tensor:
return tf.sqrt(tf.clip_by_value(val, 1e-10, np.inf))
def prob_is_largest(
self, Y: TensorType, mu: TensorType, var: TensorType, gh_x: TensorType, gh_w: TensorType
) -> tf.Tensor:
Y = to_default_int(Y)
# work out what the mean and variance is of the indicated latent function.
oh_on = tf.cast(
tf.one_hot(tf.reshape(Y, (-1,)), self.num_classes, 1.0, 0.0), dtype=mu.dtype
)
mu_selected = tf.reduce_sum(oh_on * mu, 1)
var_selected = tf.reduce_sum(oh_on * var, 1)
# generate Gauss Hermite grid
X = tf.reshape(mu_selected, (-1, 1)) + gh_x * tf.reshape(
self.safe_sqrt(2.0 * var_selected), (-1, 1)
)
# compute the CDF of the Gaussian between the latent functions and the grid (including the selected function)
dist = (tf.expand_dims(X, 1) - tf.expand_dims(mu, 2)) / tf.expand_dims(
self.safe_sqrt(var), 2
)
cdfs = 0.5 * (1.0 + tf.math.erf(dist / np.sqrt(2.0)))
cdfs = cdfs * (1 - 2 * self._squash) + self._squash
# blank out all the distances on the selected latent function
oh_off = tf.cast(
tf.one_hot(tf.reshape(Y, (-1,)), self.num_classes, 0.0, 1.0), dtype=mu.dtype
)
cdfs = cdfs * tf.expand_dims(oh_off, 2) + tf.expand_dims(oh_on, 2)
# take the product over the latent functions, and the sum over the GH grid.
return tf.reduce_prod(cdfs, axis=[1]) @ tf.reshape(gh_w / np.sqrt(np.pi), (-1, 1))
[docs]class MultiClass(Likelihood):
def __init__(
self, num_classes: int, invlink: Optional[RobustMax] = None, **kwargs: Any
) -> None:
"""
A likelihood for multi-way classification. Currently the only valid
choice of inverse-link function (invlink) is an instance of RobustMax.
For most problems, the stochastic `Softmax` likelihood may be more
appropriate (note that you then cannot use Scipy optimizer).
"""
super().__init__(latent_dim=num_classes, observation_dim=None, **kwargs)
self.num_classes = num_classes
self.num_gauss_hermite_points = 20
if invlink is None:
invlink = RobustMax(self.num_classes)
if not isinstance(invlink, RobustMax):
raise NotImplementedError
self.invlink = invlink
def _log_prob(self, F: TensorType, Y: TensorType) -> tf.Tensor:
hits = tf.equal(tf.expand_dims(tf.argmax(F, 1), 1), tf.cast(Y, tf.int64))
yes = tf.ones(tf.shape(Y), dtype=default_float()) - self.invlink.epsilon
no = tf.zeros(tf.shape(Y), dtype=default_float()) + self.invlink.eps_k1
p = tf.where(hits, yes, no)
return tf.reduce_sum(tf.math.log(p), axis=-1)
def _variational_expectations(
self, Fmu: TensorType, Fvar: TensorType, Y: TensorType
) -> tf.Tensor:
gh_x, gh_w = hermgauss(self.num_gauss_hermite_points)
p = self.invlink.prob_is_largest(Y, Fmu, Fvar, gh_x, gh_w)
ve = p * tf.math.log(1.0 - self.invlink.epsilon) + (1.0 - p) * tf.math.log(
self.invlink.eps_k1
)
return tf.reduce_sum(ve, axis=-1)
def _predict_mean_and_var(self, Fmu: TensorType, Fvar: TensorType) -> MeanAndVariance:
possible_outputs = [
tf.fill(tf.stack([tf.shape(Fmu)[0], 1]), np.array(i, dtype=np.int64))
for i in range(self.num_classes)
]
ps = [self._predict_non_logged_density(Fmu, Fvar, po) for po in possible_outputs]
ps = tf.transpose(tf.stack([tf.reshape(p, (-1,)) for p in ps]))
return ps, ps - tf.square(ps)
def _predict_log_density(self, Fmu: TensorType, Fvar: TensorType, Y: TensorType) -> tf.Tensor:
return tf.reduce_sum(tf.math.log(self._predict_non_logged_density(Fmu, Fvar, Y)), axis=-1)
def _predict_non_logged_density(
self, Fmu: TensorType, Fvar: TensorType, Y: TensorType
) -> tf.Tensor:
gh_x, gh_w = hermgauss(self.num_gauss_hermite_points)
p = self.invlink.prob_is_largest(Y, Fmu, Fvar, gh_x, gh_w)
den = p * (1.0 - self.invlink.epsilon) + (1.0 - p) * (self.invlink.eps_k1)
return den
def _conditional_mean(self, F: TensorType) -> tf.Tensor:
return self.invlink(F)
def _conditional_variance(self, F: TensorType) -> tf.Tensor:
p = self.conditional_mean(F)
return p - tf.square(p)