def true_positives_lower_bound(labels, logits, weights, surrogate_type):
"""Calculate a lower bound on the number of true positives.
This lower bound on the number of true positives given `logits` and `labels`
is the same one used in the global objectives loss functions.
Args:
labels: A `Tensor` of shape [batch_size] or [batch_size, num_labels].
logits: A `Tensor` of shape [batch_size, num_labels] or
[batch_size, num_labels, num_anchors]. If the third dimension is present,
the lower bound is computed on each slice [:, :, k] independently.
weights: Per-example loss coefficients, with shape broadcast-compatible with
that of `labels`.
surrogate_type: Either 'xent' or 'hinge', specifying which upper bound
should be used for indicator functions.
Returns:
A `Tensor` of shape [num_labels] or [num_labels, num_anchors].
"""
maybe_log2 = tf.log(2.0) if surrogate_type == 'xent' else 1.0
maybe_log2 = tf.cast(maybe_log2, logits.dtype.base_dtype)
if logits.get_shape().ndims == 3 and labels.get_shape().ndims < 3:
labels = tf.expand_dims(labels, 2)
loss_on_positives = util.weighted_surrogate_loss(
labels, logits, surrogate_type, negative_weights=0.0) / maybe_log2
return tf.reduce_sum(weights * (labels - loss_on_positives), 0)
def precision_recall_auc_loss(labels,
logits,
precision_range=(0.0, 1.0),
num_anchors=20,
weights=1.0,
dual_rate_factor=0.1,
label_priors=None,
surrogate_type='xent',
lambdas_initializer=tf.constant_initializer(1.0),
reuse=None,
variables_collections=None,
trainable=True,
scope=None):
"""Computes precision-recall AUC loss.
The loss is based on a sum of losses for recall at a range of
precision values (anchor points). This sum is a Riemann sum that
approximates the area under the precision-recall curve.
The per-example `weights` argument changes not only the coefficients of
individual training examples, but how the examples are counted toward the
constraint. If `label_priors` is given, it MUST take `weights` into account.
That is,
label_priors = P / (P + N)
where
P = sum_i (wt_i on positives)
N = sum_i (wt_i on negatives).
Args:
labels: A `Tensor` of shape [batch_size] or [batch_size, num_labels].
logits: A `Tensor` with the same shape as `labels`.
precision_range: A length-two tuple, the range of precision values over
which to compute AUC. The entries must be nonnegative, increasing, and
less than or equal to 1.0.
num_anchors: The number of grid points used to approximate the Riemann sum.
weights: Coefficients for the loss. Must be a scalar or `Tensor` of shape
[batch_size] or [batch_size, num_labels].
dual_rate_factor: A floating point value which controls the step size for
the Lagrange multipliers.
label_priors: None, or a floating point `Tensor` of shape [num_labels]
containing the prior probability of each label (i.e. the fraction of the
training data consisting of positive examples). If None, the label
priors are computed from `labels` with a moving average. See the notes
above regarding the interaction with `weights` and do not set this unless
you have a good reason to do so.
surrogate_type: Either 'xent' or 'hinge', specifying which upper bound
should be used for indicator functions.
lambdas_initializer: An initializer for the Lagrange multipliers.
reuse: Whether or not the layer and its variables should be reused. To be
able to reuse the layer scope must be given.
variables_collections: Optional list of collections for the variables.
trainable: If `True` also add variables to the graph collection
`GraphKeys.TRAINABLE_VARIABLES` (see `tf.Variable`).
scope: Optional scope for `variable_scope`.
Returns:
loss: A `Tensor` of the same shape as `logits` with the component-wise
loss.
other_outputs: A dictionary of useful internal quantities for debugging. For
more details, see http://arxiv.org/pdf/1608.04802.pdf.
lambdas: A Tensor of shape [1, num_labels, num_anchors] consisting of the
Lagrange multipliers.
biases: A Tensor of shape [1, num_labels, num_anchors] consisting of the
learned bias term for each.
label_priors: A Tensor of shape [1, num_labels, 1] consisting of the prior
probability of each label learned by the loss, if not provided.
true_positives_lower_bound: Lower bound on the number of true positives
given `labels` and `logits`. This is the same lower bound which is used
in the loss expression to be optimized.
false_positives_upper_bound: Upper bound on the number of false positives
given `labels` and `logits`. This is the same upper bound which is used
in the loss expression to be optimized.
Raises:
ValueError: If `surrogate_type` is not `xent` or `hinge`.
"""
with tf.variable_scope(scope,
'precision_recall_auc',
[labels, logits, label_priors],
reuse=reuse):
labels, logits, weights, original_shape = _prepare_labels_logits_weights(
labels, logits, weights)
num_labels = util.get_num_labels(logits)
# Convert other inputs to tensors and standardize dtypes.
dual_rate_factor = util.convert_and_cast(dual_rate_factor,
'dual_rate_factor',
logits.dtype)
# Create Tensor of anchor points and distance between anchors.
precision_values, delta = _range_to_anchors_and_delta(
precision_range, num_anchors, logits.dtype)
# Create lambdas with shape [1, num_labels, num_anchors].
lambdas, lambdas_variable = _create_dual_variable(
'lambdas',
shape=[1, num_labels, num_anchors],
dtype=logits.dtype,
initializer=lambdas_initializer,
collections=variables_collections,
trainable=trainable,
dual_rate_factor=dual_rate_factor)
# Create biases with shape [1, num_labels, num_anchors].
biases = tf.contrib.framework.model_variable(
name='biases',
shape=[1, num_labels, num_anchors],
dtype=logits.dtype,
initializer=tf.zeros_initializer(),
collections=variables_collections,
trainable=trainable)
# Maybe create label_priors.
label_priors = maybe_create_label_priors(label_priors, labels, weights,
variables_collections)
label_priors = tf.reshape(label_priors, [1, num_labels, 1])
# Expand logits, labels, and weights to shape [batch_size, num_labels, 1].
logits = tf.expand_dims(logits, 2)
labels = tf.expand_dims(labels, 2)
weights = tf.expand_dims(weights, 2)
# Calculate weighted loss and other outputs. The log(2.0) term corrects for
# logloss not being an upper bound on the indicator function.
loss = weights * util.weighted_surrogate_loss(
labels,
logits + biases,
surrogate_type=surrogate_type,
positive_weights=1.0 + lambdas * (1.0 - precision_values),
negative_weights=lambdas * precision_values)
maybe_log2 = tf.log(2.0) if surrogate_type == 'xent' else 1.0
maybe_log2 = tf.cast(maybe_log2, logits.dtype.base_dtype)
lambda_term = lambdas * (1.0 -
precision_values) * label_priors * maybe_log2
per_anchor_loss = loss - lambda_term
per_label_loss = delta * tf.reduce_sum(per_anchor_loss, 2)
# Normalize the AUC such that a perfect score function will have AUC 1.0.
# Because precision_range is discretized into num_anchors + 1 intervals
# but only num_anchors terms are included in the Riemann sum, the
# effective length of the integration interval is `delta` less than the
# length of precision_range.
scaled_loss = tf.div(per_label_loss,
precision_range[1] - precision_range[0] - delta,
name='AUC_Normalize')
scaled_loss = tf.reshape(scaled_loss, original_shape)
other_outputs = {
'lambdas':
lambdas_variable,
'biases':
biases,
'label_priors':
label_priors,
'true_positives_lower_bound':
true_positives_lower_bound(labels, logits, weights,
surrogate_type),
'false_positives_upper_bound':
false_positives_upper_bound(labels, logits, weights,
surrogate_type)
}
return scaled_loss, other_outputs
def true_positive_rate_at_false_positive_rate_loss(
labels,
logits,
target_rate,
weights=1.0,
dual_rate_factor=0.1,
label_priors=None,
surrogate_type='xent',
lambdas_initializer=tf.constant_initializer(1.0),
reuse=None,
variables_collections=None,
trainable=True,
scope=None):
"""Computes true positive rate at false positive rate loss.
The loss is based on a surrogate of the form
wt * loss(+) + lambdas * (wt * loss(-) - r * (1 - pi))
where:
- loss(-) is the loss on the negative examples
- loss(+) is the loss on the positive examples
- wt is a scalar or tensor of per-example weights
- r is the target rate
- pi is the label_priors.
The per-example weights change not only the coefficients of individual
training examples, but how the examples are counted toward the constraint.
If `label_priors` is given, it MUST take `weights` into account. That is,
label_priors = P / (P + N)
where
P = sum_i (wt_i on positives)
N = sum_i (wt_i on negatives).
Args:
labels: A `Tensor` of shape [batch_size] or [batch_size, num_labels].
logits: A `Tensor` with the same shape as `labels`.
target_rate: The false positive rate at which to compute the loss. Can be a
floating point value between 0 and 1 for a single false positive rate, or
a `Tensor` of shape [num_labels] holding each label's false positive rate.
weights: Coefficients for the loss. Must be a scalar or `Tensor` of shape
[batch_size] or [batch_size, num_labels].
dual_rate_factor: A floating point value which controls the step size for
the Lagrange multipliers.
label_priors: None, or a floating point `Tensor` of shape [num_labels]
containing the prior probability of each label (i.e. the fraction of the
training data consisting of positive examples). If None, the label
priors are computed from `labels` with a moving average. See the notes
above regarding the interaction with `weights` and do not set this unless
you have a good reason to do so.
surrogate_type: Either 'xent' or 'hinge', specifying which upper bound
should be used for indicator functions. 'xent' will use the cross-entropy
loss surrogate, and 'hinge' will use the hinge loss.
lambdas_initializer: An initializer op for the Lagrange multipliers.
reuse: Whether or not the layer and its variables should be reused. To be
able to reuse the layer scope must be given.
variables_collections: Optional list of collections for the variables.
trainable: If `True` also add variables to the graph collection
`GraphKeys.TRAINABLE_VARIABLES` (see `tf.Variable`).
scope: Optional scope for `variable_scope`.
Returns:
loss: A `Tensor` of the same shape as `logits` with the component-wise
loss.
other_outputs: A dictionary of useful internal quantities for debugging. For
more details, see http://arxiv.org/pdf/1608.04802.pdf.
lambdas: A Tensor of shape [num_labels] consisting of the Lagrange
multipliers.
label_priors: A Tensor of shape [num_labels] consisting of the prior
probability of each label learned by the loss, if not provided.
true_positives_lower_bound: Lower bound on the number of true positives
given `labels` and `logits`. This is the same lower bound which is used
in the loss expression to be optimized.
false_positives_upper_bound: Upper bound on the number of false positives
given `labels` and `logits`. This is the same upper bound which is used
in the loss expression to be optimized.
Raises:
ValueError: If `surrogate_type` is not `xent` or `hinge`.
"""
with tf.variable_scope(scope,
'tpr_at_fpr', [labels, logits, label_priors],
reuse=reuse):
labels, logits, weights, original_shape = _prepare_labels_logits_weights(
labels, logits, weights)
num_labels = util.get_num_labels(logits)
# Convert other inputs to tensors and standardize dtypes.
target_rate = util.convert_and_cast(target_rate, 'target_rate',
logits.dtype)
dual_rate_factor = util.convert_and_cast(dual_rate_factor,
'dual_rate_factor',
logits.dtype)
# Create lambdas.
lambdas, lambdas_variable = _create_dual_variable(
'lambdas',
shape=[num_labels],
dtype=logits.dtype,
initializer=lambdas_initializer,
collections=variables_collections,
trainable=trainable,
dual_rate_factor=dual_rate_factor)
# Maybe create label_priors.
label_priors = maybe_create_label_priors(label_priors, labels, weights,
variables_collections)
# Loss op and other outputs. The log(2.0) term corrects for
# logloss not being an upper bound on the indicator function.
weighted_loss = weights * util.weighted_surrogate_loss(
labels,
logits,
surrogate_type=surrogate_type,
positive_weights=1.0,
negative_weights=lambdas)
maybe_log2 = tf.log(2.0) if surrogate_type == 'xent' else 1.0
maybe_log2 = tf.cast(maybe_log2, logits.dtype.base_dtype)
lambda_term = lambdas * target_rate * (1.0 - label_priors) * maybe_log2
loss = tf.reshape(weighted_loss - lambda_term, original_shape)
other_outputs = {
'lambdas':
lambdas_variable,
'label_priors':
label_priors,
'true_positives_lower_bound':
true_positives_lower_bound(labels, logits, weights,
surrogate_type),
'false_positives_upper_bound':
false_positives_upper_bound(labels, logits, weights,
surrogate_type)
}
return loss, other_outputs
def roc_auc_loss(labels,
logits,
weights=1.0,
surrogate_type='xent',
scope=None):
"""Computes ROC AUC loss.
The area under the ROC curve is the probability p that a randomly chosen
positive example will be scored higher than a randomly chosen negative
example. This loss approximates 1-p by using a surrogate (either hinge loss or
cross entropy) for the indicator function. Specifically, the loss is:
sum_i sum_j w_i*w_j*loss(logit_i - logit_j)
where i ranges over the positive datapoints, j ranges over the negative
datapoints, logit_k denotes the logit (or score) of the k-th datapoint, and
loss is either the hinge or log loss given a positive label.
Args:
labels: A `Tensor` of shape [batch_size] or [batch_size, num_labels].
logits: A `Tensor` with the same shape and dtype as `labels`.
weights: Coefficients for the loss. Must be a scalar or `Tensor` of shape
[batch_size] or [batch_size, num_labels].
surrogate_type: Either 'xent' or 'hinge', specifying which upper bound
should be used for the indicator function.
scope: Optional scope for `name_scope`.
Returns:
loss: A `Tensor` of the same shape as `logits` with the component-wise loss.
other_outputs: An empty dictionary, for consistency.
Raises:
ValueError: If `surrogate_type` is not `xent` or `hinge`.
"""
with tf.name_scope(scope, 'roc_auc', [labels, logits, weights]):
# Convert inputs to tensors and standardize dtypes.
labels, logits, weights, original_shape = _prepare_labels_logits_weights(
labels, logits, weights)
# Create tensors of pairwise differences for logits and labels, and
# pairwise products of weights. These have shape
# [batch_size, batch_size, num_labels].
logits_difference = tf.expand_dims(logits, 0) - tf.expand_dims(
logits, 1)
labels_difference = tf.expand_dims(labels, 0) - tf.expand_dims(
labels, 1)
weights_product = tf.expand_dims(weights, 0) * tf.expand_dims(
weights, 1)
signed_logits_difference = labels_difference * logits_difference
raw_loss = util.weighted_surrogate_loss(
labels=tf.ones_like(signed_logits_difference),
logits=signed_logits_difference,
surrogate_type=surrogate_type)
weighted_loss = weights_product * raw_loss
# Zero out entries of the loss where labels_difference zero (so loss is only
# computed on pairs with different labels).
loss = tf.reduce_mean(tf.abs(labels_difference) * weighted_loss,
0) * 0.5
loss = tf.reshape(loss, original_shape)
return loss, {}