Merge pull request #566 from martinjaggi/copy-MLlib-d.

new MLlib documentation for optimization, regression and classification

new documentation with tex formulas, hopefully improving usability and reproducibility of the offered MLlib methods.
also did some minor changes in the code for consistency. scala tests pass.

this is the rebased branch, i deleted the old PR

jira:
https://spark-project.atlassian.net/browse/MLLIB-19

Author: Martin Jaggi <m.jaggi@gmail.com>

Closes #566 and squashes the following commits:

5f0f31e [Martin Jaggi] line wrap at 100 chars
4e094fb [Martin Jaggi] better description of GradientDescent
1d6965d [Martin Jaggi] remove broken url
ea569c3 [Martin Jaggi] telling what updater actually does
964732b [Martin Jaggi] lambda R() in documentation
a6c6228 [Martin Jaggi] better comments in SGD code for regression
b32224a [Martin Jaggi] new optimization documentation
d5dfef7 [Martin Jaggi] new classification and regression documentation
b07ead6 [Martin Jaggi] correct scaling for MSE loss
ba6158c [Martin Jaggi] use d for the number of features
bab2ed2 [Martin Jaggi] renaming LeastSquaresGradient

1 files changed, 255 insertions, 39 deletions

diff --git a/docs/mllib-classification-regression.md b/docs/mllib-classification-regression.md
index edb9338907..18a3e8e075 100644
--- a/docs/mllib-classification-regression.md
+++ b/docs/mllib-classification-regression.md
@@ -7,45 +7,256 @@ title: MLlib - Classification and Regression
 {:toc}
 
 
-# Binary Classification
-
-Binary classification is a supervised learning problem in which we want to
-classify entities into one of two distinct categories or labels, e.g.,
-predicting whether or not emails are spam.  This problem involves executing a
-learning *Algorithm* on a set of *labeled* examples, i.e., a set of entities
-represented via (numerical) features along with underlying category labels.
-The algorithm returns a trained *Model* that can predict the label for new
-entities for which the underlying label is unknown. 
- 
-MLlib currently supports two standard model families for binary classification,
-namely [Linear Support Vector Machines
-(SVMs)](http://en.wikipedia.org/wiki/Support_vector_machine) and [Logistic
-Regression](http://en.wikipedia.org/wiki/Logistic_regression), along with [L1
-and L2 regularized](http://en.wikipedia.org/wiki/Regularization_(mathematics))
-variants of each model family.  The training algorithms all leverage an
-underlying gradient descent primitive (described
-[below](#gradient-descent-primitive)), and take as input a regularization
-parameter (*regParam*) along with various parameters associated with gradient
-descent (*stepSize*, *numIterations*, *miniBatchFraction*). 
+`\[
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\E}{\mathbb{E}} 
+\newcommand{\x}{\mathbf{x}}
+\newcommand{\y}{\mathbf{y}}
+\newcommand{\wv}{\mathbf{w}}
+\newcommand{\av}{\mathbf{\alpha}}
+\newcommand{\bv}{\mathbf{b}}
+\newcommand{\N}{\mathbb{N}}
+\newcommand{\id}{\mathbf{I}} 
+\newcommand{\ind}{\mathbf{1}} 
+\newcommand{\0}{\mathbf{0}} 
+\newcommand{\unit}{\mathbf{e}} 
+\newcommand{\one}{\mathbf{1}} 
+\newcommand{\zero}{\mathbf{0}}
+\]`
+
+
+# Supervised Machine Learning
+Supervised machine learning is the setting where we are given a set of training data examples
+`$\{\x_i\}$`, each example `$\x_i$` coming with a corresponding label `$y_i$`.
+Given the training data `$\{(\x_i,y_i)\}$`, we want to learn a function to predict these labels.
+The two most well known classes of methods are
+[classification](http://en.wikipedia.org/wiki/Statistical_classification), and
+[regression](http://en.wikipedia.org/wiki/Regression_analysis).
+In classification, the label is a category (e.g. whether or not emails are spam), whereas in
+regression, the label is real value, and we want our prediction to be as close to the true value
+as possible.
+
+Supervised Learning involves executing a learning *Algorithm* on a set of *labeled* training
+examples. The algorithm returns a trained *Model* (such as for example a linear function) that
+can predict the label for new data examples for which the label is unknown.
+
+
+## Mathematical Formulation
+Many standard *machine learning* methods can be formulated as a convex optimization problem, i.e.
+the task of finding a minimizer of a convex function `$f$` that depends on a variable vector
+`$\wv$` (called `weights` in the code), which has `$d$` entries. 
+Formally, we can write this as the optimization problem `$\min_{\wv \in\R^d} \; f(\wv)$`, where
+the objective function is of the form
+`\begin{equation}
+    f(\wv) := 
+    \lambda\, R(\wv) +
+    \frac1n \sum_{i=1}^n L(\wv;\x_i,y_i) 
+    \label{eq:regPrimal}
+    \ .
+\end{equation}`
+Here the vectors `$\x_i\in\R^d$` are the training data examples, for `$1\le i\le n$`, and
+`$y_i\in\R$` are their corresponding labels, which we want to predict. 
+
+The objective function `$f$` has two parts:
+The *loss-function* measures the error of the model on the training data. The loss-function
+`$L(\wv;.)$` must be a convex function in `$\wv$`.
+The purpose of the [regularizer](http://en.wikipedia.org/wiki/Regularization_(mathematics)) is to
+encourage simple models, by punishing the complexity of the model `$\wv$`, in order to e.g. avoid
+over-fitting.
+Usually, the regularizer `$R(.)$` is chosen as either the standard (Euclidean) L2-norm, `$R(\wv)
+:= \frac{1}{2}\|\wv\|^2$`, or the L1-norm, `$R(\wv) := \|\wv\|_1$`, see
+[below](#using-different-regularizers) for more details.
+
+The fixed regularization parameter `$\lambda\ge0$` (`regParam` in the code) defines the trade-off
+between the two goals of small loss and small model complexity.
+
+
+## Binary Classification
+
+**Input:** Datapoints `$\x_i\in\R^{d}$`, labels `$y_i\in\{+1,-1\}$`, for `$1\le i\le n$`.
+
+**Distributed Datasets.**
+For all currently implemented optimization methods for classification, the data must be
+distributed between the worker machines *by examples*. Every machine holds a consecutive block of
+the `$n$` example/label pairs `$(\x_i,y_i)$`. 
+In other words, the input distributed dataset
+([RDD](scala-programming-guide.html#resilient-distributed-datasets-rdds)) must be the set of
+vectors `$\x_i\in\R^d$`.
+
+### Support Vector Machine
+The linear [Support Vector Machine (SVM)](http://en.wikipedia.org/wiki/Support_vector_machine)
+has become a standard choice for classification tasks.
+Here the loss function in formulation `$\eqref{eq:regPrimal}$` is given by the hinge-loss 
+`\[
+L(\wv;\x_i,y_i) := \max \{0, 1-y_i \wv^T \x_i \} \ .
+\]`
+
+By default, SVMs are trained with an L2 regularization, which gives rise to the large-margin
+interpretation if these classifiers. We also support alternative L1 regularization. In this case,
+the primal optimization problem becomes an [LP](http://en.wikipedia.org/wiki/Linear_programming).
+
+### Logistic Regression
+Despite its name, [Logistic Regression](http://en.wikipedia.org/wiki/Logistic_regression) is a
+binary classification method, again when the labels are given by binary values
+`$y_i\in\{+1,-1\}$`. The logistic loss function in formulation `$\eqref{eq:regPrimal}$` is
+defined as
+`\[
+L(\wv;\x_i,y_i) :=  \log(1+\exp( -y_i \wv^T \x_i)) \ .
+\]`
+
+
+## Linear Regression (Least Squares, Lasso and Ridge Regression)
+
+**Input:** Data matrix `$A\in\R^{n\times d}$`, right hand side vector `$\y\in\R^n$`.
+
+**Distributed Datasets.**
+For all currently implemented optimization methods for regression, the data matrix
+`$A\in\R^{n\times d}$` must be distributed between the worker machines *by rows* of `$A$`. In
+other words, the input distributed dataset
+([RDD](scala-programming-guide.html#resilient-distributed-datasets-rdds)) must be the set of the
+`$n$` rows `$A_{i:}$` of `$A$`.
+
+Least Squares Regression refers to the setting where we try to fit a vector `$\y\in\R^n$` by
+linear combination of our observed data `$A\in\R^{n\times d}$`, which is given as a matrix.
+
+It comes in 3 flavors:
+
+### Least Squares
+Plain old [least squares](http://en.wikipedia.org/wiki/Least_squares) linear regression is the
+problem of minimizing 
+  `\[ f_{\text{LS}}(\wv) := \frac1n \|A\wv-\y\|_2^2 \ . \]`
+
+### Lasso
+The popular [Lasso](http://en.wikipedia.org/wiki/Lasso_(statistics)#Lasso_method) (alternatively
+also known as  `$L_1$`-regularized least squares regression) is given by
+  `\[ f_{\text{Lasso}}(\wv) := \frac1n \|A\wv-\y\|_2^2  + \lambda \|\wv\|_1 \ . \]`
+
+### Ridge Regression
+[Ridge regression](http://en.wikipedia.org/wiki/Ridge_regression) uses the same loss function but
+with a L2 regularizer term:
+  `\[ f_{\text{Ridge}}(\wv) := \frac1n \|A\wv-\y\|_2^2  + \frac{\lambda}{2}\|\wv\|^2 \ . \]`
+
+**Loss Function.**
+For all 3, the loss function (i.e. the measure of model fit) is given by the squared deviations
+from the right hand side `$\y$`.
+`\[
+\frac1n \|A\wv-\y\|_2^2
+= \frac1n \sum_{i=1}^n (A_{i:} \wv - y_i )^2
+= \frac1n \sum_{i=1}^n L(\wv;\x_i,y_i)
+\]`
+This is also known as the [mean squared error](http://en.wikipedia.org/wiki/Mean_squared_error).
+In our generic problem formulation `$\eqref{eq:regPrimal}$`, this means the loss function is
+`$L(\wv;\x_i,y_i) := (A_{i:} \wv - y_i )^2$`, each depending only on a single row `$A_{i:}$` of
+the data matrix `$A$`.
+
+
+## Using Different Regularizers
+
+As we have mentioned above, the purpose of *regularizer* in `$\eqref{eq:regPrimal}$` is to
+encourage simple models, by punishing the complexity of the model `$\wv$`, in order to e.g. avoid
+over-fitting.
+All machine learning methods for classification and regression that we have mentioned above are
+of interest for different types of regularization, the 3 most common ones being
+
+* **L2-Regularization.**
+`$R(\wv) := \frac{1}{2}\|\wv\|^2$`.
+This regularizer is most commonly used for SVMs, logistic regression and ridge regression.
+
+* **L1-Regularization.**
+`$R(\wv) := \|\wv\|_1$`. The L1 norm `$\|\wv\|_1$` is the sum of the absolut values of the
+entries of a vector `$\wv$`. 
+This regularizer is most commonly used for sparse methods, and feature selection, such as the
+Lasso.
+
+* **Non-Regularized.**
+`$R(\wv):=0$`.
+Of course we can also train the models without any regularization, or equivalently by setting the
+regularization parameter `$\lambda:=0$`.
+
+The optimization problems of the form `$\eqref{eq:regPrimal}$` with convex regularizers such as
+the 3 mentioned here can be conveniently optimized with gradient descent type methods (such as
+SGD) which is implemented in `MLlib` currently, and explained in the next section.
+
+
+# Optimization Methods Working on the Primal Formulation
+
+**Stochastic subGradient Descent (SGD).**
+For optimization objectives `$f$` written as a sum, *stochastic subgradient descent (SGD)* can be
+an efficient choice of optimization method, as we describe in the <a
+href="mllib-optimization.html">optimization section</a> in more detail. 
+Because all methods considered here fit into the optimization formulation
+`$\eqref{eq:regPrimal}$`, this is especially natural, because the loss is written as an average
+of the individual losses coming from each datapoint.
+
+Picking one datapoint `$i\in[1..n]$` uniformly at random, we obtain a stochastic subgradient of
+`$\eqref{eq:regPrimal}$`, with respect to `$\wv$` as follows:
+`\[
+f'_{\wv,i} := L'_{\wv,i} + \lambda\, R'_\wv \ ,
+\]`
+where `$L'_{\wv,i} \in \R^d$` is a subgradient of the part of the loss function determined by the
+`$i$`-th datapoint, that is `$L'_{\wv,i} \in \frac{\partial}{\partial \wv}  L(\wv;\x_i,y_i)$`.
+Furthermore, `$R'_\wv$` is a subgradient of the regularizer `$R(\wv)$`, i.e. `$R'_\wv \in
+\frac{\partial}{\partial \wv} R(\wv)$`. The term `$R'_\wv$` does not depend on which random
+datapoint is picked.
+
+
+
+**Gradients.** 
+The following table summarizes the gradients (or subgradients) of all loss functions and
+regularizers that we currently support:
+
+<table class="table">
+  <thead>
+    <tr><th></th><th>Function</th><th>Stochastic (Sub)Gradient</th></tr>
+  </thead>
+  <tbody>
+    <tr>
+      <td>SVM Hinge Loss</td><td>$L(\wv;\x_i,y_i) := \max \{0, 1-y_i \wv^T \x_i \}$</td>
+      <td>$L'_{\wv,i} = \begin{cases}-y_i \x_i & \text{if $y_i \wv^T \x_i <1$}, \\ 0 &
+\text{otherwise}.\end{cases}$</td>
+    </tr>
+    <tr>
+      <td>Logistic Loss</td><td>$L(\wv;\x_i,y_i) :=  \log(1+\exp( -y_i \wv^T \x_i))$</td>
+      <td>$L'_{\wv,i} = -y_i \x_i  \left(1-\frac1{1+\exp(-y_i \wv^T \x_i)} \right)$</td>
+    </tr>
+    <tr>
+      <td>Least Squares Loss</td><td>$L(\wv;\x_i,y_i) := (A_{i:} \wv - y_i)^2$</td>
+      <td>$L'_{\wv,i} = 2 A_{i:}^T (A_{i:} \wv - y_i)$</td>
+    </tr>
+    <tr>
+      <td>Non-Regularized</td><td>$R(\wv) := 0$</td><td>$R'_\wv = \0$</td>
+    </tr>
+    <tr>
+      <td>L2 Regularizer</td><td>$R(\wv) := \frac{1}{2}\|\wv\|^2$</td><td>$R'_\wv = \wv$</td>
+    </tr>
+    <tr>
+      <td>L1 Regularizer</td><td>$R(\wv) := \|\wv\|_1$</td><td>$R'_\wv = \mathop{sign}(\wv)$</td>
+    </tr>
+  </tbody>
+</table>
+
+Here `$\mathop{sign}(\wv)$` is the vector consisting of the signs (`$\pm1$`) of all the entries
+of `$\wv$`.
+Also, note that `$A_{i:} \in \R^d$` is a row-vector, but the gradient is a column vector.
+
+
+
+## Implementation in MLlib
+
+For both classification and regression, `MLlib` implements a simple distributed version of
+stochastic subgradient descent (SGD), building on the underlying gradient descent primitive (as
+described in the
+<a href="mllib-optimization.html">optimization section</a>).
+All provided algorithms take as input a regularization parameter (`regParam`) along with various
+parameters associated with stochastic gradient
+descent (`stepSize`, `numIterations`, `miniBatchFraction`).
+For each of them, we support all 3 possible regularizations (none, L1 or L2).
 
 Available algorithms for binary classification:
 
 * [SVMWithSGD](api/mllib/index.html#org.apache.spark.mllib.classification.SVMWithSGD)
 * [LogisticRegressionWithSGD](api/mllib/index.html#org.apache.spark.mllib.classification.LogisticRegressionWithSGD)
 
-# Linear Regression
-
-Linear regression is another classical supervised learning setting.  In this
-problem, each entity is associated with a real-valued label (as opposed to a
-binary label as in binary classification), and we want to predict labels as
-closely as possible given numerical features representing entities.  MLlib
-supports linear regression as well as L1
-([lasso](http://en.wikipedia.org/wiki/Lasso_(statistics)#Lasso_method)) and L2
-([ridge](http://en.wikipedia.org/wiki/Ridge_regression)) regularized variants.
-The regression algorithms in MLlib also leverage the underlying gradient
-descent primitive (described [below](#gradient-descent-primitive)), and have
-the same parameters as the binary classification algorithms described above. 
-
 Available algorithms for linear regression: 
 
 * [LinearRegressionWithSGD](api/mllib/index.html#org.apache.spark.mllib.regression.LinearRegressionWithSGD)
@@ -59,6 +270,9 @@ gradient descent primitive in MLlib, see the
 * [GradientDescent](api/mllib/index.html#org.apache.spark.mllib.optimization.GradientDescent)
 
 
+
+
+
 # Usage in Scala
 
 Following code snippets can be executed in `spark-shell`.
@@ -115,9 +329,10 @@ val modelL1 = svmAlg.run(parsedData)
 {% endhighlight %}
 
 ## Linear Regression
-The following example demonstrate how to load training data, parse it as an RDD of LabeledPoint. The
-example then uses LinearRegressionWithSGD to build a simple linear model to predict label values. We
-compute the Mean Squared Error at the end to evaluate
+
+The following example demonstrate how to load training data, parse it as an RDD of LabeledPoint.
+The example then uses LinearRegressionWithSGD to build a simple linear model to predict label 
+values. We compute the Mean Squared Error at the end to evaluate
 [goodness of fit](http://en.wikipedia.org/wiki/Goodness_of_fit)
 
 {% highlight scala %}
@@ -157,6 +372,7 @@ Spark Java API uses a separate `JavaRDD` class. You can convert a Java RDD to a
 calling `.rdd()` on your `JavaRDD` object.
 
 # Usage in Python
+
 Following examples can be tested in the PySpark shell.
 
 ## Binary Classification
@@ -182,9 +398,9 @@ print("Training Error = " + str(trainErr))
 {% endhighlight %}
 
 ## Linear Regression
-The following example demonstrate how to load training data, parse it as an RDD of LabeledPoint. The
-example then uses LinearRegressionWithSGD to build a simple linear model to predict label values. We
-compute the Mean Squared Error at the end to evaluate
+The following example demonstrate how to load training data, parse it as an RDD of LabeledPoint.
+The example then uses LinearRegressionWithSGD to build a simple linear model to predict label 
+values. We compute the Mean Squared Error at the end to evaluate
 [goodness of fit](http://en.wikipedia.org/wiki/Goodness_of_fit)
 
 {% highlight python %}