npdl/objectives.py
# -*- coding: utf-8 -*-
"""
Provides some minimal help with building loss expressions for training or
validating a neural network.
These functions build element- or item-wise loss expressions from network
predictions and targets.
Examples
--------
Assuming you have a simple neural network for 3-way classification:
>>> import npdl
>>> model = npdl.model.Model()
>>> model.add(npdl.layers.Dense(n_out=100, n_in=50))
>>> model.add(npdl.layers.Dense(n_out=3, activation=npdl.activations.Softmax()))
>>> model.compile(loss=npdl.objectives.SCCE(), optimizer=npdl.optimizers.SGD(lr=0.005))
"""
import copy
import numpy as np
class Objective(object):
"""An objective function (or loss function, or optimization score
function) is one of the two parameters required to compile a model.
"""
def forward(self, outputs, targets):
""" Forward function.
"""
raise NotImplementedError()
def backward(self, outputs, targets):
"""Backward function.
Parameters
----------
outputs, targets : numpy.array
The arrays to compute the derivatives of them.
Returns
-------
numpy.array
An array of derivative.
"""
raise NotImplementedError()
def __str__(self):
return self.__class__.__name__
class MeanSquaredError(Objective):
"""Computes the element-wise squared difference between ``targets`` and ``outputs``.
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an
estimator (of a procedure for estimating an unobserved quantity) measures the
average of the squares of the errors or deviations—that is, the difference between
the estimator and what is estimated. MSE is a risk function, corresponding to the
expected value of the squared error loss or quadratic loss. The difference occurs
because of randomness or because the estimator doesn't account for information that
could produce a more accurate estimate. [1]_
The MSE is a measure of the quality of an estimator—it is always non-negative,
and values closer to zero are better.
The MSE is the second moment (about the origin) of the error, and thus incorporates
both the variance of the estimator and its bias. For an unbiased estimator, the MSE
is the variance of the estimator. Like the variance, MSE has the same units of
measurement as the square of the quantity being estimated. In an analogy to standard
deviation, taking the square root of MSE yields the root-mean-square error or
root-mean-square deviation (RMSE or RMSD), which has the same units as the quantity
being estimated; for an unbiased estimator, the RMSE is the square root of the
variance, known as the standard deviation.
Notes
-----
This is the loss function of choice for many regression problems
or auto-encoders with linear output units.
References
----------
.. [1] Lehmann, E. L.; Casella, George (1998). Theory of Point Estimation (2nd ed.).
New York: Springer. ISBN 0-387-98502-6. MR 1639875.
"""
def forward(self, outputs, targets):
"""MeanSquaredError forward propagation.
.. math:: L = (p - t)^2
Parameters
----------
outputs, targets : numpy.array
The arrays to compute the squared difference between.
Returns
-------
numpy.array
An expression for the element-wise squared difference.
"""
return 0.5 * np.mean(np.sum(np.power(outputs - targets, 2), axis=1))
def backward(self, outputs, targets):
"""MeanSquaredError backward propagation.
.. math:: dE = p - t
Parameters
----------
outputs, targets : numpy.array
The arrays to compute the derivative between them.
Returns
-------
numpy.array
Derivative.
"""
return outputs - targets
MSE = MeanSquaredError
class HellingerDistance(Objective):
"""Computes the multi-class hinge loss between predictions and targets.
In probability and statistics, the Hellinger distance (closely related to,
although different from, the Bhattacharyya distance) is used to quantify
the similarity between two probability distributions. It is a type of
f-divergence. The Hellinger distance is defined in terms of the Hellinger
integral, which was introduced by Ernst Hellinger in 1909.[1]_ [2]_
Notes
-----
This is an alternative to the categorical cross-entropy loss for
multi-class classification problems
References
----------
.. [1] Nikulin, M.S. (2001), "Hellinger distance", in Hazewinkel, Michiel,
Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
.. [2] Jump up ^ Hellinger, Ernst (1909), "Neue Begründung der Theorie
quadratischer Formen von unendlichvielen Veränderlichen", Journal
für die reine und angewandte Mathematik (in German), 136: 210–271,
doi:10.1515/crll.1909.136.210, JFM 40.0393.01
"""
def forward(self, outputs, targets):
"""HellingerDistance forward propagation.
Parameters
----------
outputs : numpy 2D array
outputs in (0, 1), such as softmax output of a neural network,
with data points in rows and class probabilities in columns.
targets : numpy 2D array
Either a vector of int giving the correct class index per data point
or a 2D tensor of one-hot encoding of the correct class in the same
layout as predictions (non-binary targets in [0, 1] do not work!)
Returns
-------
numpy 1D array
An expression for the Hellinger Distance
"""
root_difference = np.sqrt(outputs) - np.sqrt(targets)
return np.mean(np.sum(np.power(root_difference, 2), axis=1) / np.sqrt(2))
def backward(self, outputs, targets):
"""HellingerDistance forward propagation.
"""
root_difference = np.sqrt(outputs) - np.sqrt(targets)
return root_difference / (np.sqrt(2) * np.sqrt(outputs))
HeD = HellingerDistance
class BinaryCrossEntropy(Objective):
"""Computes the binary cross-entropy between predictions and targets.
Returns
-------
numpy array
An expression for the element-wise binary cross-entropy.
Notes
-----
This is the loss function of choice for binary classification problems
and sigmoid output units.
"""
def __init__(self, epsilon=1e-11):
self.epsilon = epsilon
def forward(self, outputs, targets):
"""Forward pass.
.. math:: L = -t \\log(p) - (1 - t) \\log(1 - p)
Parameters
----------
outputs : numpy.array
Predictions in (0, 1), such as sigmoidal output of a neural network.
targets : numpy.array
Targets in [0, 1], such as ground truth labels.
"""
outputs = np.clip(outputs, self.epsilon, 1 - self.epsilon)
return np.mean(-np.sum(targets * np.log(outputs) + (1 - targets) * np.log(1 - outputs), axis=1))
def backward(self, outputs, targets):
"""Backward pass.
Parameters
----------
outputs : numpy.array
Predictions in (0, 1), such as sigmoidal output of a neural network.
targets : numpy.array
Targets in [0, 1], such as ground truth labels.
"""
outputs = np.clip(outputs, self.epsilon, 1 - self.epsilon)
divisor = np.maximum(outputs * (1 - outputs), self.epsilon)
return (outputs - targets) / divisor
BCE = BinaryCrossEntropy
class SoftmaxCategoricalCrossEntropy(Objective):
"""Computes the categorical cross-entropy between predictions and targets.
Notes
-----
This is the loss function of choice for multi-class classification
problems and softmax output units. For hard targets, i.e., targets
that assign all of the probability to a single class per data point,
providing a vector of int for the targets is usually slightly more
efficient than providing a matrix with a single 1.0 per row.
"""
def __init__(self, epsilon=1e-11):
self.epsilon = epsilon
def forward(self, outputs, targets):
"""SoftmaxCategoricalCrossEntropy forward propagation.
.. math:: L_i = - \\sum_j{t_{i,j} \\log(p_{i,j})}
Parameters
----------
outputs : numpy.array
Predictions in (0, 1), such as softmax output of a neural network,
with data points in rows and class probabilities in columns.
targets : numpy.array
Either targets in [0, 1] matching the layout of `outputs`, or
a vector of int giving the correct class index per data point.
Returns
-------
numpy 1D array
An expression for the item-wise categorical cross-entropy.
"""
outputs = np.clip(outputs, self.epsilon, 1 - self.epsilon)
return np.mean(-np.sum(targets * np.log(outputs), axis=1))
def backward(self, outputs, targets):
"""SoftmaxCategoricalCrossEntropy backward propagation.
.. math:: dE = p - t
Parameters
----------
outputs : numpy 2D array
Predictions in (0, 1), such as softmax output of a neural network,
with data points in rows and class probabilities in columns.
targets : numpy 2D array
Either targets in [0, 1] matching the layout of `outputs`, or
a vector of int giving the correct class index per data point.
Returns
-------
numpy 1D array
"""
outputs = np.clip(outputs, self.epsilon, 1 - self.epsilon)
return outputs - targets
SCCE = SoftmaxCategoricalCrossEntropy
def get(objective):
if objective.__class__.__name__ == 'str':
if objective in ['mse', 'MSE']:
return MSE()
if objective in ['mean_squared_error', 'MeanSquaredError']:
return MeanSquaredError()
if objective in ['hellinger_distance', 'HellingerDistance']:
return HellingerDistance()
if objective in ['hed', 'HeD']:
return HeD()
if objective in ['binary_cross_entropy', 'BinaryCrossEntropy']:
return BinaryCrossEntropy()
if objective in ['bce', 'BCE']:
return BCE()
if objective in ['softmax_categorical_cross_entropy', 'SoftmaxCategoricalCrossEntropy']:
return SoftmaxCategoricalCrossEntropy()
if objective in ['scce', 'SCCE']:
return SCCE()
raise ValueError('Unknown objective name: {}.'.format(objective))
elif isinstance(objective, Objective):
return copy.deepcopy(objective)
else:
raise ValueError("Unknown type: {}.".format(objective.__class__.__name__))