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/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://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.
*/
package org.apache.spark.mllib.optimization
import org.apache.spark.mllib.linalg.{DenseVector, Vector, Vectors}
import org.apache.spark.mllib.linalg.BLAS.{axpy, dot, scal}
import org.apache.spark.mllib.util.MLUtils
/**
* Class used to compute the gradient for a loss function, given a single data point.
*/
abstract class Gradient extends Serializable {
/**
* Compute the gradient and loss given the features of a single data point.
*
* @param data features for one data point
* @param label label for this data point
* @param weights weights/coefficients corresponding to features
*
* @return (gradient: Vector, loss: Double)
*/
def compute(data: Vector, label: Double, weights: Vector): (Vector, Double) = {
val gradient = Vectors.zeros(weights.size)
val loss = compute(data, label, weights, gradient)
(gradient, loss)
}
/**
* Compute the gradient and loss given the features of a single data point,
* add the gradient to a provided vector to avoid creating new objects, and return loss.
*
* @param data features for one data point
* @param label label for this data point
* @param weights weights/coefficients corresponding to features
* @param cumGradient the computed gradient will be added to this vector
*
* @return loss
*/
def compute(data: Vector, label: Double, weights: Vector, cumGradient: Vector): Double
}
/**
* Compute gradient and loss for a multinomial logistic loss function, as used
* in multi-class classification (it is also used in binary logistic regression).
*
* In `The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd Edition`
* by Trevor Hastie, Robert Tibshirani, and Jerome Friedman, which can be downloaded from
* http://statweb.stanford.edu/~tibs/ElemStatLearn/ , Eq. (4.17) on page 119 gives the formula of
* multinomial logistic regression model. A simple calculation shows that
*
*
*
* for K classes multiclass classification problem.
*
* The model weights \(w = (w_1, w_2, ..., w_{K-1})^T\) becomes a matrix which has dimension of
* (K-1) * (N+1) if the intercepts are added. If the intercepts are not added, the dimension
* will be (K-1) * N.
*
* As a result, the loss of objective function for a single instance of data can be written as
*
*
* where $\alpha(i) = 1$ if \(i \ne 0\), and
* $\alpha(i) = 0$ if \(i == 0\),
* \(margins_i = x w_i\).
*
* For optimization, we have to calculate the first derivative of the loss function, and
* a simple calculation shows that
*
*
*
* where $\delta_{i, j} = 1$ if \(i == j\),
* $\delta_{i, j} = 0$ if \(i != j\), and
* multiplier =
* $\exp(margins_i) / (1 + \sum_k^{K-1} \exp(margins_i)) - (1-\alpha(y)\delta_{y, i+1})$
*
* If any of margins is larger than 709.78, the numerical computation of multiplier and loss
* function will be suffered from arithmetic overflow. This issue occurs when there are outliers
* in data which are far away from hyperplane, and this will cause the failing of training once
* infinity / infinity is introduced. Note that this is only a concern when max(margins)
* {@literal >} 0.
*
* Fortunately, when max(margins) = maxMargin {@literal >} 0, the loss function and the multiplier
* can be easily rewritten into the following equivalent numerically stable formula.
*
*
*
* where sum = $\exp(-maxMargin) + \sum_i^{K-1}\exp(margins_i - maxMargin) - 1$.
*
* Note that each term, $(margins_i - maxMargin)$ in $\exp$ is smaller than zero; as a result,
* overflow will not happen with this formula.
*
* For multiplier, similar trick can be applied as the following,
*
*
*
* where each term in $\exp$ is also smaller than zero, so overflow is not a concern.
*
* For the detailed mathematical derivation, see the reference at
* http://www.slideshare.net/dbtsai/2014-0620-mlor-36132297
*
* @param numClasses the number of possible outcomes for k classes classification problem in
* Multinomial Logistic Regression. By default, it is binary logistic regression
* so numClasses will be set to 2.
*/
class LogisticGradient(numClasses: Int) extends Gradient {
def this() = this(2)
override def compute(
data: Vector,
label: Double,
weights: Vector,
cumGradient: Vector): Double = {
val dataSize = data.size
// (weights.size / dataSize + 1) is number of classes
require(weights.size % dataSize == 0 && numClasses == weights.size / dataSize + 1)
numClasses match {
case 2 =>
/**
* For Binary Logistic Regression.
*
* Although the loss and gradient calculation for multinomial one is more generalized,
* and multinomial one can also be used in binary case, we still implement a specialized
* binary version for performance reason.
*/
val margin = -1.0 * dot(data, weights)
val multiplier = (1.0 / (1.0 + math.exp(margin))) - label
axpy(multiplier, data, cumGradient)
if (label > 0) {
// The following is equivalent to log(1 + exp(margin)) but more numerically stable.
MLUtils.log1pExp(margin)
} else {
MLUtils.log1pExp(margin) - margin
}
case _ =>
/**
* For Multinomial Logistic Regression.
*/
val weightsArray = weights match {
case dv: DenseVector => dv.values
case _ =>
throw new IllegalArgumentException(
s"weights only supports dense vector but got type ${weights.getClass}.")
}
val cumGradientArray = cumGradient match {
case dv: DenseVector => dv.values
case _ =>
throw new IllegalArgumentException(
s"cumGradient only supports dense vector but got type ${cumGradient.getClass}.")
}
// marginY is margins(label - 1) in the formula.
var marginY = 0.0
var maxMargin = Double.NegativeInfinity
var maxMarginIndex = 0
val margins = Array.tabulate(numClasses - 1) { i =>
var margin = 0.0
data.foreachNonZero { (index, value) =>
margin += value * weightsArray((i * dataSize) + index)
}
if (i == label.toInt - 1) marginY = margin
if (margin > maxMargin) {
maxMargin = margin
maxMarginIndex = i
}
margin
}
/**
* When maxMargin > 0, the original formula will cause overflow as we discuss
* in the previous comment.
* We address this by subtracting maxMargin from all the margins, so it's guaranteed
* that all of the new margins will be smaller than zero to prevent arithmetic overflow.
*/
val sum = {
var temp = 0.0
if (maxMargin > 0) {
for (i <- 0 until numClasses - 1) {
margins(i) -= maxMargin
if (i == maxMarginIndex) {
temp += math.exp(-maxMargin)
} else {
temp += math.exp(margins(i))
}
}
} else {
for (i <- 0 until numClasses - 1) {
temp += math.exp(margins(i))
}
}
temp
}
for (i <- 0 until numClasses - 1) {
val multiplier = math.exp(margins(i)) / (sum + 1.0) - {
if (label != 0.0 && label == i + 1) 1.0 else 0.0
}
data.foreachNonZero { (index, value) =>
cumGradientArray(i * dataSize + index) += multiplier * value
}
}
val loss = if (label > 0.0) math.log1p(sum) - marginY else math.log1p(sum)
if (maxMargin > 0) {
loss + maxMargin
} else {
loss
}
}
}
}
/**
* Compute gradient and loss for a Least-squared loss function, as used in linear regression.
* This is correct for the averaged least squares loss function (mean squared error)
* L = 1/2n ||A weights-y||^2
* See also the documentation for the precise formulation.
*/
class LeastSquaresGradient extends Gradient {
override def compute(data: Vector, label: Double, weights: Vector): (Vector, Double) = {
val diff = dot(data, weights) - label
val loss = diff * diff / 2.0
val gradient = data.copy
scal(diff, gradient)
(gradient, loss)
}
override def compute(
data: Vector,
label: Double,
weights: Vector,
cumGradient: Vector): Double = {
val diff = dot(data, weights) - label
axpy(diff, data, cumGradient)
diff * diff / 2.0
}
}
/**
* Compute gradient and loss for a Hinge loss function, as used in SVM binary classification.
* See also the documentation for the precise formulation.
*
* @note This assumes that the labels are {0,1}
*/
class HingeGradient extends Gradient {
override def compute(data: Vector, label: Double, weights: Vector): (Vector, Double) = {
val dotProduct = dot(data, weights)
// Our loss function with {0, 1} labels is max(0, 1 - (2y - 1) (f_w(x)))
// Therefore the gradient is -(2y - 1)*x
val labelScaled = 2 * label - 1.0
if (1.0 > labelScaled * dotProduct) {
val gradient = data.copy
scal(-labelScaled, gradient)
(gradient, 1.0 - labelScaled * dotProduct)
} else {
(Vectors.sparse(weights.size, Array.empty, Array.empty), 0.0)
}
}
override def compute(
data: Vector,
label: Double,
weights: Vector,
cumGradient: Vector): Double = {
val dotProduct = dot(data, weights)
// Our loss function with {0, 1} labels is max(0, 1 - (2y - 1) (f_w(x)))
// Therefore the gradient is -(2y - 1)*x
val labelScaled = 2 * label - 1.0
if (1.0 > labelScaled * dotProduct) {
axpy(-labelScaled, data, cumGradient)
1.0 - labelScaled * dotProduct
} else {
0.0
}
}
}
