Github user srowen commented on a diff in the pull request:
https://github.com/apache/spark/pull/17094#discussion_r118475804
--- Diff:
mllib/src/main/scala/org/apache/spark/ml/optim/aggregator/LeastSquaresAggregator.scala
---
@@ -0,0 +1,224 @@
+/*
+ * 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.ml.optim.aggregator
+
+import org.apache.spark.broadcast.Broadcast
+import org.apache.spark.ml.feature.Instance
+import org.apache.spark.ml.linalg.{BLAS, Vector, Vectors}
+
+/**
+ * LeastSquaresAggregator computes the gradient and loss for a
Least-squared loss function,
+ * as used in linear regression for samples in sparse or dense vector in
an online fashion.
+ *
+ * Two LeastSquaresAggregator can be merged together to have a summary of
loss and gradient of
+ * the corresponding joint dataset.
+ *
+ * For improving the convergence rate during the optimization process, and
also preventing against
+ * features with very large variances exerting an overly large influence
during model training,
+ * package like R's GLMNET performs the scaling to unit variance and
removing the mean to reduce
+ * the condition number, and then trains the model in scaled space but
returns the coefficients in
+ * the original scale. See page 9 in
http://cran.r-project.org/web/packages/glmnet/glmnet.pdf
+ *
+ * However, we don't want to apply the `StandardScaler` on the training
dataset, and then cache
+ * the standardized dataset since it will create a lot of overhead. As a
result, we perform the
+ * scaling implicitly when we compute the objective function. The
following is the mathematical
+ * derivation.
+ *
+ * Note that we don't deal with intercept by adding bias here, because the
intercept
+ * can be computed using closed form after the coefficients are converged.
+ * See this discussion for detail.
+ *
http://stats.stackexchange.com/questions/13617/how-is-the-intercept-computed-in-glmnet
+ *
+ * When training with intercept enabled,
+ * The objective function in the scaled space is given by
+ *
+ * <blockquote>
+ * $$
+ * L = 1/2n ||\sum_i w_i(x_i - \bar{x_i}) / \hat{x_i} - (y - \bar{y}) /
\hat{y}||^2,
+ * $$
+ * </blockquote>
+ *
+ * where $\bar{x_i}$ is the mean of $x_i$, $\hat{x_i}$ is the standard
deviation of $x_i$,
+ * $\bar{y}$ is the mean of label, and $\hat{y}$ is the standard deviation
of label.
+ *
+ * If we fitting the intercept disabled (that is forced through 0.0),
+ * we can use the same equation except we set $\bar{y}$ and $\bar{x_i}$ to
0 instead
+ * of the respective means.
+ *
+ * This can be rewritten as
+ *
+ * <blockquote>
+ * $$
+ * \begin{align}
+ * L &= 1/2n ||\sum_i (w_i/\hat{x_i})x_i - \sum_i
(w_i/\hat{x_i})\bar{x_i} - y / \hat{y}
+ * + \bar{y} / \hat{y}||^2 \\
+ * &= 1/2n ||\sum_i w_i^\prime x_i - y / \hat{y} + offset||^2 = 1/2n
diff^2
+ * \end{align}
+ * $$
+ * </blockquote>
+ *
+ * where $w_i^\prime$ is the effective coefficients defined by
$w_i/\hat{x_i}$, offset is
+ *
+ * <blockquote>
+ * $$
+ * - \sum_i (w_i/\hat{x_i})\bar{x_i} + \bar{y} / \hat{y}.
+ * $$
+ * </blockquote>
+ *
+ * and diff is
+ *
+ * <blockquote>
+ * $$
+ * \sum_i w_i^\prime x_i - y / \hat{y} + offset
+ * $$
+ * </blockquote>
+ *
+ * Note that the effective coefficients and offset don't depend on
training dataset,
+ * so they can be precomputed.
+ *
+ * Now, the first derivative of the objective function in scaled space is
+ *
+ * <blockquote>
+ * $$
+ * \frac{\partial L}{\partial w_i} = diff/N (x_i - \bar{x_i}) /
\hat{x_i}
+ * $$
+ * </blockquote>
+ *
+ * However, $(x_i - \bar{x_i})$ will densify the computation, so it's not
+ * an ideal formula when the training dataset is sparse format.
+ *
+ * This can be addressed by adding the dense $\bar{x_i} / \hat{x_i}$ terms
+ * in the end by keeping the sum of diff. The first derivative of total
+ * objective function from all the samples is
+ *
+ *
+ * <blockquote>
+ * $$
+ * \begin{align}
+ * \frac{\partial L}{\partial w_i} &=
+ * 1/N \sum_j diff_j (x_{ij} - \bar{x_i}) / \hat{x_i} \\
+ * &= 1/N ((\sum_j diff_j x_{ij} / \hat{x_i}) - diffSum \bar{x_i}
/ \hat{x_i}) \\
+ * &= 1/N ((\sum_j diff_j x_{ij} / \hat{x_i}) + correction_i)
+ * \end{align}
+ * $$
+ * </blockquote>
+ *
+ * where $correction_i = - diffSum \bar{x_i} / \hat{x_i}$
+ *
+ * A simple math can show that diffSum is actually zero, so we don't even
+ * need to add the correction terms in the end. From the definition of
diff,
+ *
+ * <blockquote>
+ * $$
+ * \begin{align}
+ * diffSum &= \sum_j (\sum_i w_i(x_{ij} - \bar{x_i})
+ * / \hat{x_i} - (y_j - \bar{y}) / \hat{y}) \\
+ * &= N * (\sum_i w_i(\bar{x_i} - \bar{x_i}) / \hat{x_i} -
(\bar{y} - \bar{y}) / \hat{y}) \\
+ * &= 0
+ * \end{align}
+ * $$
+ * </blockquote>
+ *
+ * As a result, the first derivative of the total objective function only
depends on
+ * the training dataset, which can be easily computed in distributed
fashion, and is
+ * sparse format friendly.
+ *
+ * <blockquote>
+ * $$
+ * \frac{\partial L}{\partial w_i} = 1/N ((\sum_j diff_j x_{ij} /
\hat{x_i})
+ * $$
+ * </blockquote>
+ *
+ * @note The constructor is curried, since the cost function will
repeatedly create new versions
+ * of this class for different coefficient vectors.
+ *
+ * @param labelStd The standard deviation value of the label.
+ * @param labelMean The mean value of the label.
+ * @param fitIntercept Whether to fit an intercept term.
+ * @param bcFeaturesStd The broadcast standard deviation values of the
features.
+ * @param bcFeaturesMean The broadcast mean values of the features.
+ * @param bcCoefficients The broadcast coefficients corresponding to the
features.
+ */
+private[ml] class LeastSquaresAggregator(
+ labelStd: Double,
+ labelMean: Double,
+ fitIntercept: Boolean,
+ bcFeaturesStd: Broadcast[Array[Double]],
+ bcFeaturesMean: Broadcast[Array[Double]])(bcCoefficients:
Broadcast[Vector])
+ extends DifferentiableLossAggregator[Instance, LeastSquaresAggregator] {
+ require(labelStd > 0.0, s"${this.getClass.getName} requires the label
standard" +
+ s"deviation to be positive.")
--- End diff --
Add a space before 'deviation' or at the end of the previous line
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