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

3 files changed, 405 insertions, 58 deletions

diff --git a/docs/_layouts/global.html b/docs/_layouts/global.html
index b65686c0b1..7114e1f5dd 100755
--- a/docs/_layouts/global.html
+++ b/docs/_layouts/global.html
@@ -196,6 +196,11 @@
 
     </body>
     <!-- MathJax Section -->
+    <script type="text/x-mathjax-config">
+	  MathJax.Hub.Config({
+	    TeX: { equationNumbers: { autoNumber: "AMS" } }
+	  });
+	</script>
     <script type="text/javascript"
      src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
     <script>
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 %}
diff --git a/docs/mllib-optimization.md b/docs/mllib-optimization.md
index 428284ef29..396b98d52a 100644
--- a/docs/mllib-optimization.md
+++ b/docs/mllib-optimization.md
@@ -6,35 +6,161 @@ title: MLlib - Optimization
 * Table of contents
 {:toc}
 
+`\[
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\E}{\mathbb{E}} 
+\newcommand{\x}{\mathbf{x}}
+\newcommand{\y}{\mathbf{y}}
+\newcommand{\wv}{\mathbf{w}}
+\newcommand{\av}{\mathbf{\alpha}}
+\newcommand{\bv}{\mathbf{b}}
+\newcommand{\N}{\mathbb{N}}
+\newcommand{\id}{\mathbf{I}} 
+\newcommand{\ind}{\mathbf{1}} 
+\newcommand{\0}{\mathbf{0}} 
+\newcommand{\unit}{\mathbf{e}} 
+\newcommand{\one}{\mathbf{1}} 
+\newcommand{\zero}{\mathbf{0}}
+\]`
 
-# Gradient Descent Primitive
 
-[Gradient descent](http://en.wikipedia.org/wiki/Gradient_descent) (along with
-stochastic variants thereof) are first-order optimization methods that are
-well-suited for large-scale and distributed computation. Gradient descent
-methods aim to find a local minimum of a function by iteratively taking steps
-in the direction of the negative gradient of the function at the current point,
-i.e., the current parameter value. Gradient descent is included as a low-level
-primitive in MLlib, upon which various ML algorithms are developed, and has the
-following parameters:
 
-* *gradient* is a class that computes the stochastic gradient of the function
+# Mathematical Description
+
+## (Sub)Gradient Descent
+The simplest method to solve optimization problems of the form `$\min_{\wv \in\R^d} \; f(\wv)$`
+is [gradient descent](http://en.wikipedia.org/wiki/Gradient_descent).
+Such first-order optimization methods (including gradient descent and stochastic variants
+thereof) are well-suited for large-scale and distributed computation.
+
+Gradient descent methods aim to find a local minimum of a function by iteratively taking steps in
+the direction of steepest descent, which is the negative of the derivative (called the
+[gradient](http://en.wikipedia.org/wiki/Gradient)) of the function at the current point, i.e., at
+the current parameter value.
+If the objective function `$f$` is not differentiable at all arguments, but still convex, then a
+*subgradient* 
+is the natural generalization of the gradient, and assumes the role of the step direction.
+In any case, computing a gradient or subgradient of `$f$` is expensive --- it requires a full
+pass through the complete dataset, in order to compute the contributions from all loss terms.
+
+## Stochastic (Sub)Gradient Descent (SGD)
+Optimization problems whose objective function `$f$` is written as a sum are particularly
+suitable to be solved using *stochastic subgradient descent (SGD)*. 
+In our case, for the optimization formulations commonly used in <a
+href="mllib-classification-regression.html">supervised machine learning</a>,
+`\begin{equation}
+    f(\wv) := 
+    \lambda\, R(\wv) +
+    \frac1n \sum_{i=1}^n L(\wv;\x_i,y_i) 
+    \label{eq:regPrimal}
+    \ .
+\end{equation}`
+this is especially natural, because the loss is written as an average of the individual losses
+coming from each datapoint.
+
+A stochastic subgradient is a randomized choice of a vector, such that in expectation, we obtain
+a true subgradient of the original objective function.
+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.
+Clearly, in expectation over the random choice of `$i\in[1..n]$`, we have that `$f'_{\wv,i}$` is
+a subgradient of the original objective `$f$`, meaning that `$\E\left[f'_{\wv,i}\right] \in
+\frac{\partial}{\partial \wv} f(\wv)$`.
+
+Running SGD now simply becomes walking in the direction of the negative stochastic subgradient
+`$f'_{\wv,i}$`, that is
+`\begin{equation}\label{eq:SGDupdate}
+\wv^{(t+1)} := \wv^{(t)}  - \gamma \; f'_{\wv,i} \ .
+\end{equation}`
+**Step-size.**
+The parameter `$\gamma$` is the step-size, which in the default implementation is chosen
+decreasing with the square root of the iteration counter, i.e. `$\gamma := \frac{s}{\sqrt{t}}$`
+in the `$t$`-th iteration, with the input parameter `$s=$ stepSize`. Note that selecting the best
+step-size for SGD methods can often be delicate in practice and is a topic of active research.
+
+**Gradients.**
+A table of (sub)gradients of the machine learning methods implemented in MLlib, is available in
+the <a href="mllib-classification-regression.html">classification and regression</a> section.
+
+
+**Proximal Updates.**
+As an alternative to just use the subgradient `$R'(\wv)$` of the regularizer in the step
+direction, an improved update for some cases can be obtained by using the proximal operator
+instead.
+For the L1-regularizer, the proximal operator is given by soft thresholding, as implemented in
+[L1Updater](api/mllib/index.html#org.apache.spark.mllib.optimization.L1Updater).
+
+
+## Update Schemes for Distributed SGD
+The SGD implementation in
+[GradientDescent](api/mllib/index.html#org.apache.spark.mllib.optimization.GradientDescent) uses
+a simple (distributed) sampling of the data examples.
+We recall that the loss part of the optimization problem `$\eqref{eq:regPrimal}$` is
+`$\frac1n \sum_{i=1}^n L(\wv;\x_i,y_i)$`, and therefore `$\frac1n \sum_{i=1}^n L'_{\wv,i}$` would
+be the true (sub)gradient.
+Since this would require access to the full data set, the parameter `miniBatchFraction` specifies
+which fraction of the full data to use instead.
+The average of the gradients over this subset, i.e.
+`\[
+\frac1{|S|} \sum_{i\in S} L'_{\wv,i} \ ,
+\]`
+is a stochastic gradient. Here `$S$` is the sampled subset of size `$|S|=$ miniBatchFraction
+$\cdot n$`.
+
+In each iteration, the sampling over the distributed dataset
+([RDD](scala-programming-guide.html#resilient-distributed-datasets-rdds)), as well as the
+computation of the sum of the partial results from each worker machine is performed by the
+standard spark routines.
+
+If the fraction of points `miniBatchFraction` is set to 1 (default), then the resulting step in
+each iteration is exact (sub)gradient descent. In this case there is no randomness and no
+variance in the used step directions.
+On the other extreme, if `miniBatchFraction` is chosen very small, such that only a single point
+is sampled, i.e. `$|S|=$ miniBatchFraction $\cdot n = 1$`, then the algorithm is equivalent to
+standard SGD. In that case, the step direction depends from the uniformly random sampling of the
+point.
+
+
+
+# Implementation in MLlib
+
+Gradient descent methods including stochastic subgradient descent (SGD) as
+included as a low-level primitive in `MLlib`, upon which various ML algorithms 
+are developed, see the 
+<a href="mllib-classification-regression.html">classification and regression</a> 
+section for example.
+
+The SGD method
+[GradientDescent.runMiniBatchSGD](api/mllib/index.html#org.apache.spark.mllib.optimization.GradientDescent)
+has the following parameters:
+
+* `gradient` is a class that computes the stochastic gradient of the function
 being optimized, i.e., with respect to a single training example, at the
 current parameter value. MLlib includes gradient classes for common loss
 functions, e.g., hinge, logistic, least-squares.  The gradient class takes as
 input a training example, its label, and the current parameter value. 
-* *updater* is a class that updates weights in each iteration of gradient
-descent. MLlib includes updaters for cases without regularization, as well as
+* `updater` is a class that performs the actual gradient descent step, i.e. 
+updating the weights in each iteration, for a given gradient of the loss part.
+The updater is also responsible to perform the update from the regularization 
+part. MLlib includes updaters for cases without regularization, as well as
 L1 and L2 regularizers.
-* *stepSize* is a scalar value denoting the initial step size for gradient
+* `stepSize` is a scalar value denoting the initial step size for gradient
 descent. All updaters in MLlib use a step size at the t-th step equal to
-stepSize / sqrt(t). 
-* *numIterations* is the number of iterations to run.
-* *regParam* is the regularization parameter when using L1 or L2 regularization.
-* *miniBatchFraction* is the fraction of the data used to compute the gradient
-at each iteration.
+`stepSize $/ \sqrt{t}$`. 
+* `numIterations` is the number of iterations to run.
+* `regParam` is the regularization parameter when using L1 or L2 regularization.
+* `miniBatchFraction` is the fraction of the total data that is sampled in 
+each iteration, to compute the gradient direction.
 
 Available algorithms for gradient descent:
 
-* [GradientDescent](api/mllib/index.html#org.apache.spark.mllib.optimization.GradientDescent)
+* [GradientDescent.runMiniBatchSGD](api/mllib/index.html#org.apache.spark.mllib.optimization.GradientDescent)
+