Github user jkbradley commented on a diff in the pull request:
https://github.com/apache/spark/pull/10207#discussion_r47006676
--- Diff: docs/ml-classification-regression.md ---
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+---
+layout: global
+title: Classification and regression - spark.ml
+displayTitle: Classification and regression in spark.ml
+---
+
+
+`\[
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\E}{\mathbb{E}}
+\newcommand{\x}{\mathbf{x}}
+\newcommand{\y}{\mathbf{y}}
+\newcommand{\wv}{\mathbf{w}}
+\newcommand{\av}{\mathbf{\alpha}}
+\newcommand{\bv}{\mathbf{b}}
+\newcommand{\N}{\mathbb{N}}
+\newcommand{\id}{\mathbf{I}}
+\newcommand{\ind}{\mathbf{1}}
+\newcommand{\0}{\mathbf{0}}
+\newcommand{\unit}{\mathbf{e}}
+\newcommand{\one}{\mathbf{1}}
+\newcommand{\zero}{\mathbf{0}}
+\]`
+
+**Table of Contents**
+
+* This will become a table of contents (this text will be scraped).
+{:toc}
+
+In MLlib, we implement popular linear methods such as logistic
+regression and linear least squares with $L_1$ or $L_2$ regularization.
+Refer to [the linear methods in mllib](mllib-linear-methods.html) for
+details. In `spark.ml`, we also include Pipelines API for [Elastic
+net](http://en.wikipedia.org/wiki/Elastic_net_regularization), a hybrid
+of $L_1$ and $L_2$ regularization proposed in [Zou et al, Regularization
+and variable selection via the elastic
+net](http://users.stat.umn.edu/~zouxx019/Papers/elasticnet.pdf).
+Mathematically, it is defined as a convex combination of the $L_1$ and
+the $L_2$ regularization terms:
+`\[
+\alpha \left( \lambda \|\wv\|_1 \right) + (1-\alpha) \left(
\frac{\lambda}{2}\|\wv\|_2^2 \right) , \alpha \in [0, 1], \lambda \geq 0
+\]`
+By setting $\alpha$ properly, elastic net contains both $L_1$ and $L_2$
+regularization as special cases. For example, if a [linear
+regression](https://en.wikipedia.org/wiki/Linear_regression) model is
+trained with the elastic net parameter $\alpha$ set to $1$, it is
+equivalent to a
+[Lasso](http://en.wikipedia.org/wiki/Least_squares#Lasso_method) model.
+On the other hand, if $\alpha$ is set to $0$, the trained model reduces
+to a [ridge
+regression](http://en.wikipedia.org/wiki/Tikhonov_regularization) model.
+We implement Pipelines API for both linear regression and logistic
+regression with elastic net regularization.
+
+# Regression
+
+## Linear regression
+
+The interface for working with linear regression models and model
+summaries is similar to the logistic regression case. The following
+example demonstrates training an elastic net regularized linear
+regression model and extracting model summary statistics.
+
+<div class="codetabs">
+
+<div data-lang="scala" markdown="1">
+{% include_example
scala/org/apache/spark/examples/ml/LinearRegressionWithElasticNetExample.scala
%}
+</div>
+
+<div data-lang="java" markdown="1">
+{% include_example
java/org/apache/spark/examples/ml/JavaLinearRegressionWithElasticNetExample.java
%}
+</div>
+
+<div data-lang="python" markdown="1">
+<!--- TODO: Add python model summaries once implemented -->
+{% include_example python/ml/linear_regression_with_elastic_net.py %}
+</div>
+
+</div>
+
+## Survival regression
+
+
+In `spark.ml`, we implement the [Accelerated failure time
(AFT)](https://en.wikipedia.org/wiki/Accelerated_failure_time_model)
+model which is a parametric survival regression model for censored data.
+It describes a model for the log of survival time, so it's often called
+log-linear model for survival analysis. Different from
+[Proportional
hazards](https://en.wikipedia.org/wiki/Proportional_hazards_model) model
+designed for the same purpose, the AFT model is more easily to parallelize
+because each instance contribute to the objective function independently.
+
+Given the values of the covariates $x^{'}$, for random lifetime $t_{i}$ of
+subjects i = 1, ..., n, with possible right-censoring,
+the likelihood function under the AFT model is given as:
+`\[
+L(\beta,\sigma)=\prod_{i=1}^n[\frac{1}{\sigma}f_{0}(\frac{\log{t_{i}}-x^{'}\beta}{\sigma})]^{\delta_{i}}S_{0}(\frac{\log{t_{i}}-x^{'}\beta}{\sigma})^{1-\delta_{i}}
+\]`
+Where $\delta_{i}$ is the indicator of the event has occurred i.e.
uncensored or not.
+Using $\epsilon_{i}=\frac{\log{t_{i}}-x^{'}\beta}{\sigma}$, the
log-likelihood function
+assumes the form:
+`\[
+\iota(\beta,\sigma)=\sum_{i=1}^{n}[-\delta_{i}\log\sigma+\delta_{i}\log{f_{0}}(\epsilon_{i})+(1-\delta_{i})\log{S_{0}(\epsilon_{i})}]
+\]`
+Where $S_{0}(\epsilon_{i})$ is the baseline survivor function,
+and $f_{0}(\epsilon_{i})$ is corresponding density function.
+
+The most commonly used AFT model is based on the Weibull distribution of
the survival time.
+The Weibull distribution for lifetime corresponding to extreme value
distribution for
+log of the lifetime, and the $S_{0}(\epsilon)$ function is:
+`\[
+S_{0}(\epsilon_{i})=\exp(-e^{\epsilon_{i}})
+\]`
+the $f_{0}(\epsilon_{i})$ function is:
+`\[
+f_{0}(\epsilon_{i})=e^{\epsilon_{i}}\exp(-e^{\epsilon_{i}})
+\]`
+The log-likelihood function for AFT model with Weibull distribution of
lifetime is:
+`\[
+\iota(\beta,\sigma)=
-\sum_{i=1}^n[\delta_{i}\log\sigma-\delta_{i}\epsilon_{i}+e^{\epsilon_{i}}]
+\]`
+Due to minimizing the negative log-likelihood equivalent to maximum a
posteriori probability,
+the loss function we use to optimize is $-\iota(\beta,\sigma)$.
+The gradient functions for $\beta$ and $\log\sigma$ respectively are:
+`\[
+\frac{\partial (-\iota)}{\partial
\beta}=\sum_{1=1}^{n}[\delta_{i}-e^{\epsilon_{i}}]\frac{x_{i}}{\sigma}
+\]`
+`\[
+\frac{\partial (-\iota)}{\partial
(\log\sigma)}=\sum_{i=1}^{n}[\delta_{i}+(\delta_{i}-e^{\epsilon_{i}})\epsilon_{i}]
+\]`
+
+The AFT model can be formulated as a convex optimization problem,
+i.e. the task of finding a minimizer of a convex function
$-\iota(\beta,\sigma)$
+that depends coefficients vector $\beta$ and the log of scale parameter
$\log\sigma$.
+The optimization algorithm underlying the implementation is L-BFGS.
+The implementation matches the result from R's survival function
+[survreg](https://stat.ethz.ch/R-manual/R-devel/library/survival/html/survreg.html)
+
+## Example:
+
+<div class="codetabs">
+
+<div data-lang="scala" markdown="1">
+{% include_example
scala/org/apache/spark/examples/ml/AFTSurvivalRegressionExample.scala %}
+</div>
+
+<div data-lang="java" markdown="1">
+{% include_example
java/org/apache/spark/examples/ml/JavaAFTSurvivalRegressionExample.java %}
+</div>
+
+<div data-lang="python" markdown="1">
+{% include_example python/ml/aft_survival_regression.py %}
+</div>
+
+</div>
+
+
+# Classification
+
+## Logistic regression
+
+The following example shows how to train a logistic regression model
+with elastic net regularization. `elasticNetParam` corresponds to
+$\alpha$ and `regParam` corresponds to $\lambda$.
+
+<div class="codetabs">
+
+<div data-lang="scala" markdown="1">
+{% include_example
scala/org/apache/spark/examples/ml/LogisticRegressionWithElasticNetExample.scala
%}
+</div>
+
+<div data-lang="java" markdown="1">
+{% include_example
java/org/apache/spark/examples/ml/JavaLogisticRegressionWithElasticNetExample.java
%}
+</div>
+
+<div data-lang="python" markdown="1">
+{% include_example python/ml/logistic_regression_with_elastic_net.py %}
+</div>
+
+</div>
+
+The `spark.ml` implementation of logistic regression also supports
+extracting a summary of the model over the training set. Note that the
+predictions and metrics which are stored as `Dataframe` in
+`BinaryLogisticRegressionSummary` are annotated `@transient` and hence
+only available on the driver.
+
+<div class="codetabs">
+
+<div data-lang="scala" markdown="1">
+
+[`LogisticRegressionTrainingSummary`](api/scala/index.html#org.apache.spark.ml.classification.LogisticRegressionTrainingSummary)
+provides a summary for a
+[`LogisticRegressionModel`](api/scala/index.html#org.apache.spark.ml.classification.LogisticRegressionModel).
+Currently, only binary classification is supported and the
+summary must be explicitly cast to
+[`BinaryLogisticRegressionTrainingSummary`](api/scala/index.html#org.apache.spark.ml.classification.BinaryLogisticRegressionTrainingSummary).
+This will likely change when multiclass classification is supported.
+
+Continuing the earlier example:
+
+{% include_example
scala/org/apache/spark/examples/ml/LogisticRegressionSummaryExample.scala %}
+</div>
+
+<div data-lang="java" markdown="1">
+[`LogisticRegressionTrainingSummary`](api/java/org/apache/spark/ml/classification/LogisticRegressionTrainingSummary.html)
+provides a summary for a
+[`LogisticRegressionModel`](api/java/org/apache/spark/ml/classification/LogisticRegressionModel.html).
+Currently, only binary classification is supported and the
+summary must be explicitly cast to
+[`BinaryLogisticRegressionTrainingSummary`](api/java/org/apache/spark/ml/classification/BinaryLogisticRegressionTrainingSummary.html).
+This will likely change when multiclass classification is supported.
+
+Continuing the earlier example:
+
+{% include_example
java/org/apache/spark/examples/ml/JavaLogisticRegressionSummaryExample.java %}
+</div>
+
+<!--- TODO: Add python model summaries once implemented -->
+<div data-lang="python" markdown="1">
+Logistic regression model summary is not yet supported in Python.
+</div>
+
+</div>
+
+
+## Multilayer perceptron classifier
+
+Multilayer perceptron classifier (MLPC) is a classifier based on the
[feedforward artificial neural
network](https://en.wikipedia.org/wiki/Feedforward_neural_network).
+MLPC consists of multiple layers of nodes.
+Each layer is fully connected to the next layer in the network. Nodes in
the input layer represent the input data. All other nodes maps inputs to the
outputs
+by performing linear combination of the inputs with the node's weights
`$\wv$` and bias `$\bv$` and applying an activation function.
+It can be written in matrix form for MLPC with `$K+1$` layers as follows:
+`\[
+\mathrm{y}(\x) = \mathrm{f_K}(...\mathrm{f_2}(\wv_2^T\mathrm{f_1}(\wv_1^T
\x+b_1)+b_2)...+b_K)
+\]`
+Nodes in intermediate layers use sigmoid (logistic) function:
+`\[
+\mathrm{f}(z_i) = \frac{1}{1 + e^{-z_i}}
+\]`
+Nodes in the output layer use softmax function:
+`\[
+\mathrm{f}(z_i) = \frac{e^{z_i}}{\sum_{k=1}^N e^{z_k}}
+\]`
+The number of nodes `$N$` in the output layer corresponds to the number of
classes.
+
+MLPC employes backpropagation for learning the model. We use logistic loss
function for optimization and L-BFGS as optimization routine.
+
+**Examples**
+
+<div class="codetabs">
+
+<div data-lang="scala" markdown="1">
+{% include_example
scala/org/apache/spark/examples/ml/MultilayerPerceptronClassifierExample.scala
%}
+</div>
+
+<div data-lang="java" markdown="1">
+{% include_example
java/org/apache/spark/examples/ml/JavaMultilayerPerceptronClassifierExample.java
%}
+</div>
+
+<div data-lang="python" markdown="1">
+{% include_example python/ml/multilayer_perceptron_classification.py %}
+</div>
+
+</div>
+
+
+## One-vs-Rest classifier (a.k.a. One-vs-All)
+
+[OneVsRest](http://en.wikipedia.org/wiki/Multiclass_classification#One-vs.-rest)
is an example of a machine learning reduction for performing multiclass
classification given a base classifier that can perform binary classification
efficiently. It is also known as "One-vs-All."
+
+`OneVsRest` is implemented as an `Estimator`. For the base classifier it
takes instances of `Classifier` and creates a binary classification problem for
each of the k classes. The classifier for class i is trained to predict whether
the label is i or not, distinguishing class i from all other classes.
+
+Predictions are done by evaluating each binary classifier and the index of
the most confident classifier is output as label.
+
+### Example
+
+The example below demonstrates how to load the
+[Iris
dataset](http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/multiclass/iris.scale),
parse it as a DataFrame and perform multiclass classification using
`OneVsRest`. The test error is calculated to measure the algorithm accuracy.
+
+<div class="codetabs">
+<div data-lang="scala" markdown="1">
+
+Refer to the [Scala API
docs](api/scala/index.html#org.apache.spark.ml.classifier.OneVsRest) for more
details.
+
+{% include_example
scala/org/apache/spark/examples/ml/OneVsRestExample.scala %}
+</div>
+
+<div data-lang="java" markdown="1">
+
+Refer to the [Java API
docs](api/java/org/apache/spark/ml/classification/OneVsRest.html) for more
details.
+
+{% include_example
java/org/apache/spark/examples/ml/JavaOneVsRestExample.java %}
+</div>
+</div>
+
+
+
+# Decision trees
--- End diff --
I'd prefer to split trees and ensembles into subsections of classification
& regression. General info about trees and ensembles could be put into a
separate section, with links to it from the classification & regression
subsections.
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