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, 145 insertions, 19 deletions

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)
+