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/*******************************************************************************
* Copyright (c) 2015-2018 Skymind, Inc.
*
* This program and the accompanying materials are made available under the
* terms of the Apache License, Version 2.0 which is available at
* https://www.apache.org/licenses/LICENSE-2.0.
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
* WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the
* License for the specific language governing permissions and limitations
* under the License.
*
* SPDX-License-Identifier: Apache-2.0
******************************************************************************/
package org.nd4j.linalg.lossfunctions.impl;
import lombok.Data;
import lombok.EqualsAndHashCode;
import org.nd4j.linalg.activations.IActivation;
import org.nd4j.linalg.api.ndarray.INDArray;
import org.nd4j.linalg.api.ops.CustomOp;
import org.nd4j.linalg.api.ops.impl.transforms.custom.SoftMax;
import org.nd4j.linalg.factory.Nd4j;
import org.nd4j.linalg.indexing.INDArrayIndex;
import org.nd4j.linalg.indexing.NDArrayIndex;
import org.nd4j.linalg.lossfunctions.ILossFunction;
import org.nd4j.linalg.lossfunctions.LossUtil;
import org.nd4j.linalg.ops.transforms.Transforms;
import org.nd4j.common.primitives.Pair;
import org.nd4j.shade.jackson.annotation.JsonInclude;
import org.nd4j.shade.jackson.annotation.JsonProperty;
/**
* This is a cost function associated with a mixture-density network.
* For background, this is inspired by Bishop's work pioneering the mixture
* density network. The essence of the idea is that the cost function attempts
* to model the output as if it were a mixture of gaussian probability
* densities. The network attempts to converge on a cost function which
* involves the negative log likelihood of the output fitting a set of data
* and estimates the "alpha" contribution to of each of the distributions
* the 'mu' (mean) and 'sigma' (standard deviation) of each of the
* distributions.
*
* For a full description of the technique, refer to Bishop's work.
*
* Bishop CM. Mixture density networks,
* Neural Computing Research Group Report:
* NCRG/94/004, Aston University, Birmingham, 1994
* https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/bishop-ncrg-94-004.pdf
*
* There is no public constructor, please use the builder to create an
* approriate mixture loss function for the number of outputs and number
* of mixtures you would like to fit.
*
* Note that this means that the output
* layer must provide (labelWidth+2)*mixtures output values in order to describe
* the distributions of vectors of width labelWidth.
* Please ensure that the size of the output layer matches the number of
* mixtures provided.
*
* @author Jonathan Arney
*/
@EqualsAndHashCode
@JsonInclude(JsonInclude.Include.NON_NULL)
public class LossMixtureDensity implements ILossFunction {
private int mMixtures;
private int mLabelWidth;
private static final double SQRT_TWO_PI = Math.sqrt(2 * Math.PI);
public LossMixtureDensity() {
}
/**
* This method constructs a mixture density cost function
* which causes the network to learn a mixture of gaussian distributions
* for each network output. The network will learn the 'alpha' (weight
* for each distribution), the 'mu' or 'mean' of each distribution,
* and the 'sigma' (standard-deviation) of the mixture. Together,
* this distribution can be sampled according to the probability density
* learned by the network.
*
* @param mixtures Number of gaussian mixtures to model.
* @param labelWidth Size of the labels vector for each sample.
*/
private LossMixtureDensity(@JsonProperty("mixtures") int mixtures, @JsonProperty("labelWidth") int labelWidth) {
mMixtures = mixtures;
mLabelWidth = labelWidth;
}
/**
* This class is a data holder for the mixture density
* components for convenient manipulation.
* These are organized as rank-3 matrices with shape
* [nSamples, nLabelsPerSample, nMixturesPerLabel]
* and refer to the 'alpha' (weight of that gaussian), 'mu' (mean for that
* gaussian), and 'sigma' (standard-deviation for that gaussian).
*/
@Data
public static class MixtureDensityComponents {
private INDArray alpha;
private INDArray mu;
private INDArray sigma;
}
// This method extracts the "alpha", "mu", and "sigma" from the
// output of the neural network.
// This is done manually, but it should ultimately be done
// through Nd4j operations in order to increase performance.
public MixtureDensityComponents extractComponents(INDArray output) {
long outputSize = output.size(1);
if (outputSize != (mLabelWidth + 2) * mMixtures) {
throw new IllegalArgumentException(
"Network output size " + outputSize + " must be (labels+2)*mixtures where labels = "
+ mLabelWidth + " and mixtures = " + mMixtures);
}
MixtureDensityComponents mdc = new MixtureDensityComponents();
// Output is 2 dimensional (samples, labels)
//
// For each label vector of length 'labels', we will have
// an output vector of length '(labels + 2) * nMixtures.
// The first nMixtures outputs will correspond to the 'alpha' for each mixture.
// The second nMixtures outputs will correspond to the 'sigma' and the last nMixtures*labels
// will correspond to the 'mu' (mean) of the output.
// Reorganize these.
// alpha = samples, 0 to nMixtures
// mu = samples, nMixtures to 2*nMixtures
// sigma = samples, 2*nMixtures to (labels + 2)*nMixtures
// Alpha is then sub-divided through reshape by mixtures per label and samples.
mdc.alpha = output.get(NDArrayIndex.all(), NDArrayIndex.interval(0, mMixtures));
mdc.sigma = output.get(NDArrayIndex.all(), NDArrayIndex.interval(mMixtures, 2 * mMixtures));
mdc.mu = output.get(NDArrayIndex.all(), NDArrayIndex.interval(2 * mMixtures, (mLabelWidth + 2) * mMixtures))
.reshape(output.size(0), mMixtures, mLabelWidth);
// Alpha is a softmax because
// the alpha should all sum to 1 for a given gaussian mixture.
mdc.alpha = Nd4j.exec((CustomOp) new SoftMax(mdc.alpha, mdc.alpha, -1))[0];
// Mu comes directly from the network as an unmolested value.
// Note that this effectively means that the output layer of
// the network should have an activation function at least as large as
// the expected values. It is best for the output
// layer to be an IDENTITY activation function.
//mdc.mu = mdc.mu;
// Sigma comes from the network as an exponential in order to
// ensure that it is positive-definite.
mdc.sigma = Transforms.exp(mdc.sigma);
return mdc;
}
/**
* Computes the aggregate score as a sum of all of the individual scores of
* each of the labels against each of the outputs of the network. For
* the mixture density network, this is the negative log likelihood that
* the given labels fall within the probability distribution described by
* the mixture of gaussians of the network output.
* @param labels Labels to score against the network.
* @param preOutput Output of the network (before activation function has been called).
* @param activationFn Activation function for the network.
* @param mask Mask to be applied to labels (not used for MDN).
* @param average Whether or not to return an average instead of a total score (not used).
* @return Returns a single double which corresponds to the total score of all label values.
*/
@Override
public double computeScore(INDArray labels, INDArray preOutput, IActivation activationFn, INDArray mask,
boolean average) {
// The score overall consists of the
// sum of the negative log likelihoods for each
// of the individual labels.
INDArray scoreArr = computeScoreArray(labels, preOutput, activationFn, mask);
double score = scoreArr.sumNumber().doubleValue();
if (average) {
score /= scoreArr.size(0);
}
return score;
}
/**
* This method returns the score for each of the given outputs against the
* given set of labels. For a mixture density network, this is done by
* extracting the "alpha", "mu", and "sigma" components of each gaussian
* and computing the negative log likelihood that the labels fall within
* a linear combination of these gaussian distributions. The smaller
* the negative log likelihood, the higher the probability that the given
* labels actually would fall within the distribution. Therefore by
* minimizing the negative log likelihood, we get to a position of highest
* probability that the gaussian mixture explains the phenomenon.
*
* @param labels Labels give the sample output that the network should
* be trying to converge on.
* @param preOutput The output of the last layer (before applying the activation function).
* @param activationFn The activation function of the current layer.
* @param mask Mask to apply to score evaluation (not supported for this cost function).
* @return
*/
@Override
public INDArray computeScoreArray(INDArray labels, INDArray preOutput, IActivation activationFn, INDArray mask) {
labels = labels.castTo(preOutput.dataType()); //No-op if already correct dtype
INDArray output = activationFn.getActivation(preOutput.dup(), false);
MixtureDensityComponents mdc = extractComponents(output);
INDArray scoreArr = negativeLogLikelihood(labels, mdc.alpha, mdc.mu, mdc.sigma);
if (mask != null) {
LossUtil.applyMask(scoreArr, mask);
}
return scoreArr;
}
/**
* This method returns the gradient of the cost function with respect to the
* output from the previous layer. For this cost function, the gradient
* is derived from Bishop's paper "Mixture Density Networks" (1994) which
* gives an elegant closed-form expression for the derivatives with respect
* to each of the output components.
* @param labels Labels to train on.
* @param preOutput Output of neural network before applying the final activation function.
* @param activationFn Activation function of output layer.
* @param mask Mask to apply to gradients.
* @return Gradient of cost function with respect to preOutput parameters.
*/
@Override
public INDArray computeGradient(INDArray labels, INDArray preOutput, IActivation activationFn, INDArray mask) {
labels = labels.castTo(preOutput.dataType()); //No-op if already correct dtype
long nSamples = labels.size(0);
INDArray output = activationFn.getActivation(preOutput.dup(), false);
MixtureDensityComponents mdc = extractComponents(output);
INDArray gradient = Nd4j.zeros(nSamples, preOutput.columns());
INDArray labelsMinusMu = labelsMinusMu(labels, mdc.mu);
INDArray labelsMinusMuSquared = labelsMinusMu.mul(labelsMinusMu).sum(2);
// This computes pi_i, see Bishop equation (30).
// See http://www.plsyard.com/dealing-overflow-and-underflow-in-softmax-function/
// this post for why we calculate the pi_i in this way.
// With the exponential function here, we have to be very careful
// about overflow/underflow considerations even with
// fairly intermediate values. Subtracting the max
// here helps to ensure over/underflow does not happen here.
// This isn't exactly a softmax because there's an 'alpha' coefficient
// here, but the technique works, nonetheless.
INDArray variance = mdc.sigma.mul(mdc.sigma);
INDArray minustwovariance = variance.mul(2).negi();
INDArray normalPart = mdc.alpha.div(Transforms.pow(mdc.sigma.mul(SQRT_TWO_PI), mLabelWidth));
INDArray exponent = labelsMinusMuSquared.div(minustwovariance);
INDArray exponentMax = exponent.max(1);
exponent.subiColumnVector(exponentMax);
INDArray pi = Transforms.exp(exponent).muli(normalPart);
INDArray piDivisor = pi.sum(true,1);
pi.diviColumnVector(piDivisor);
// See Bishop equation (35)
//INDArray dLdZAlpha = Nd4j.zeros(nSamples, nLabelsPerSample, mMixturesPerLabel); //mdc.alpha.sub(pi);
INDArray dLdZAlpha = mdc.alpha.sub(pi);
// See Bishop equation (38)
INDArray dLdZSigma = (labelsMinusMuSquared.div(variance).subi(mLabelWidth)).muli(-1).muli(pi);
// See Bishop equation (39)
// This turned out to be way less efficient than
// the simple 'for' loop here.
//INDArray dLdZMu = pi
// .div(variance)
// .reshape(nSamples, mMixtures, 1)
// .repeat(2, mLabelWidth)
// .muli(labelsMinusMu)
// .negi()
// .reshape(nSamples, mMixtures * mLabelWidth);
INDArray dLdZMu = Nd4j.create(nSamples, mMixtures, mLabelWidth);
for (int k = 0; k < mLabelWidth; k++) {
dLdZMu.put(new INDArrayIndex[] {NDArrayIndex.all(), NDArrayIndex.all(), NDArrayIndex.point(k)},
labelsMinusMu.get(new INDArrayIndex[] {NDArrayIndex.all(), NDArrayIndex.all(),
NDArrayIndex.point(k)}).muli(pi).divi(variance).negi());
}
dLdZMu = dLdZMu.reshape(nSamples, mMixtures * mLabelWidth);
// Place components of gradient into gradient holder.
gradient.put(new INDArrayIndex[] {NDArrayIndex.all(), NDArrayIndex.interval(0, mMixtures)}, dLdZAlpha);
gradient.put(new INDArrayIndex[] {NDArrayIndex.all(), NDArrayIndex.interval(mMixtures, mMixtures * 2)},
dLdZSigma);
gradient.put(new INDArrayIndex[] {NDArrayIndex.all(),
NDArrayIndex.interval(mMixtures * 2, (mLabelWidth + 2) * mMixtures)}, dLdZMu);
INDArray gradients = activationFn.backprop(preOutput, gradient).getFirst();
if (mask != null) {
LossUtil.applyMask(gradients, mask);
}
return gradients;
}
@Override
public Pair computeGradientAndScore(INDArray labels, INDArray preOutput, IActivation activationFn,
INDArray mask, boolean average) {
double score = computeScore(labels, preOutput, activationFn, mask, average);
INDArray gradient = computeGradient(labels, preOutput, activationFn, mask);
Pair returnCode = new Pair<>(score, gradient);
return returnCode;
}
/**
* The opName of this function
*
* @return
*/
@Override
public String name() {
return "lossmixturedensity";
}
/**
* This method returns an array consisting of each of the training samples,
* for each label in each sample, the negative log likelihood of that
* value falling within the given gaussian mixtures.
* @param alpha
* @param mu
* @param sigma
* @param labels
* @return
*/
private INDArray negativeLogLikelihood(INDArray labels, INDArray alpha, INDArray mu, INDArray sigma) {
INDArray labelsMinusMu = labelsMinusMu(labels, mu);
INDArray diffsquared = labelsMinusMu.mul(labelsMinusMu).sum(2);
INDArray phitimesalphasum = phi(diffsquared, sigma).muli(alpha).sum(true,1);
// result = See Bishop(28,29)
INDArray result = Transforms.log(phitimesalphasum).negi();
return result;
}
private INDArray labelsMinusMu(INDArray labels, INDArray mu) {
// Now that we have the mixtures, let's compute the negative
// log likelihodd of the label against the
long nSamples = labels.size(0);
long labelsPerSample = labels.size(1);
// This worked, but was actually much
// slower than the for loop below.
// labels = samples, mixtures, labels
// mu = samples, mixtures
// INDArray labelMinusMu = labels
// .reshape('f', nSamples, labelsPerSample, 1)
// .repeat(2, mMixtures)
// .permute(0, 2, 1)
// .subi(mu);
// The above code does the same thing as the loop below,
// but it does it with index magix instead of a for loop.
// It turned out to be way less efficient than the simple 'for' here.
INDArray labelMinusMu = Nd4j.zeros(nSamples, mMixtures, labelsPerSample);
for (int k = 0; k < mMixtures; k++) {
labelMinusMu.put(new INDArrayIndex[] {NDArrayIndex.all(), NDArrayIndex.point(k), NDArrayIndex.all()},
labels);
}
labelMinusMu.subi(mu);
return labelMinusMu;
}
/**
* This method calculates 'phi' which is the probability
* density function (see Bishop 23)
* @param diffSquared This is the 'x-mu' term of the Gaussian distribution (distance between 'x' and the mean value of the distribution).
* @param sigma This is the standard deviation of the Gaussian distribution.
* @return This returns an array of shape [nsamples, nlabels, ndistributions] which contains the probability density (phi) for each of the
* samples * labels * distributions for the given x, sigma, mu.
*/
private INDArray phi(INDArray diffSquared, INDArray sigma) {
// 1/sqrt(2PIs^2) * e^((in-u)^2/2*s^2)
INDArray minustwovariance = sigma.mul(sigma).muli(2).negi();
// This is phi_i(x,mu,sigma)
INDArray likelihoods = Transforms.exp(diffSquared.divi(minustwovariance))
.divi(Transforms.pow(sigma.mul(SQRT_TWO_PI), (double) mLabelWidth));
return likelihoods;
}
/**
* Returns the number of gaussians this loss function
* will attempt to find.
* @return Number of gaussians to find.
*/
@JsonProperty("mixtures")
public int getNMixtures() {
return mMixtures;
}
/**
* Returns the width of each label vector.
* @return Width of label vectors expected.
*/
@JsonProperty("labelWidth")
public int getLabelWidth() {
return mLabelWidth;
}
@Override
public String toString() {
return "LossMixtureDensity(mixtures=" + mMixtures + ", labels=" + mLabelWidth + ")";
}
public static Builder builder() {
return new Builder();
}
public static class Builder {
private int mGaussians = 0;
private int mLabelWidth = 0;
private Builder() {}
/**
* Specifies the number of gaussian functions to attempt
* fitting against the data.
* @param aGaussians Number of gaussian functions to fit.
* @return DynamicCustomOpsBuilder.
*/
public Builder gaussians(int aGaussians) {
mGaussians = aGaussians;
return this;
}
/**
* Specifies the width of the labels vector which also corresponds
* to the width of the 'mean' vector for each of the gaussian functions.
* @param aLabelWidth Width of the labels vector.
* @return DynamicCustomOpsBuilder.
*/
public Builder labelWidth(int aLabelWidth) {
mLabelWidth = aLabelWidth;
return this;
}
/**
* Creates a new instance of the mixture density
* cost function.
* @return A new mixture density cost function built with
* the specified parameters.
*/
public LossMixtureDensity build() {
if (mGaussians <= 0) {
throw new IllegalArgumentException(
"Mixture density cost function must specify the number of mixtures to fit");
}
if (mLabelWidth <= 0) {
throw new IllegalArgumentException(
"Mixture density cost function must specify the size of the labels vectors");
}
return new LossMixtureDensity(mGaussians, mLabelWidth);
}
}
}
