docs/mllib-linear-methods.md

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title: Linear Methods - RDD-based API
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* Table of contents
{:toc}


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## 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.
We call the method *linear* if $L(\wv; \x, y)$ can be expressed as a function of $\wv^T x$ and $y$.
Several of `spark.mllib`'s classification and regression algorithms fall into this category,
and are discussed here.

The objective function `$f$` has two parts:
the regularizer that controls the complexity of the model,
and the loss that measures the error of the model on the training data.
The loss function `$L(\wv;.)$` is typically a convex function in `$\wv$`.  The
fixed regularization parameter `$\lambda \ge 0$` (`regParam` in the code)
defines the trade-off between the two goals of minimizing the loss (i.e.,
training error) and minimizing model complexity (i.e., to avoid overfitting).

### Loss functions

The following table summarizes the loss functions and their gradients or sub-gradients for the
methods `spark.mllib` supports:

<table class="table">
  <thead>
    <tr><th></th><th>loss function $L(\wv; \x, y)$</th><th>gradient or sub-gradient</th></tr>
  </thead>
  <tbody>
    <tr>
      <td>hinge loss</td><td>$\max \{0, 1-y \wv^T \x \}, \quad y \in \{-1, +1\}$</td>
      <td>$\begin{cases}-y \cdot \x &amp; \text{if $y \wv^T \x &lt;1$}, \\ 0 &amp;
\text{otherwise}.\end{cases}$</td>
    </tr>
    <tr>
      <td>logistic loss</td><td>$\log(1+\exp( -y \wv^T \x)), \quad y \in \{-1, +1\}$</td>
      <td>$-y \left(1-\frac1{1+\exp(-y \wv^T \x)} \right) \cdot \x$</td>
    </tr>
    <tr>
      <td>squared loss</td><td>$\frac{1}{2} (\wv^T \x - y)^2, \quad y \in \R$</td>
      <td>$(\wv^T \x - y) \cdot \x$</td>
    </tr>
  </tbody>
</table>

Note that, in the mathematical formulation above, a binary label $y$ is denoted as either
$+1$ (positive) or $-1$ (negative), which is convenient for the formulation.
*However*, the negative label is represented by $0$ in `spark.mllib` instead of $-1$, to be consistent with
multiclass labeling.

### Regularizers

The purpose of the
[regularizer](http://en.wikipedia.org/wiki/Regularization_(mathematics)) is to
encourage simple models and avoid overfitting.  We support the following
regularizers in `spark.mllib`:

<table class="table">
  <thead>
    <tr><th></th><th>regularizer $R(\wv)$</th><th>gradient or sub-gradient</th></tr>
  </thead>
  <tbody>
    <tr>
      <td>zero (unregularized)</td><td>0</td><td>$\0$</td>
    </tr>
    <tr>
      <td>L2</td><td>$\frac{1}{2}\|\wv\|_2^2$</td><td>$\wv$</td>
    </tr>
    <tr>
      <td>L1</td><td>$\|\wv\|_1$</td><td>$\mathrm{sign}(\wv)$</td>
    </tr>
    <tr>
      <td>elastic net</td><td>$\alpha \|\wv\|_1 + (1-\alpha)\frac{1}{2}\|\wv\|_2^2$</td><td>$\alpha \mathrm{sign}(\wv) + (1-\alpha) \wv$</td>
    </tr>
  </tbody>
</table>

Here `$\mathrm{sign}(\wv)$` is the vector consisting of the signs (`$\pm1$`) of all the entries
of `$\wv$`.

L2-regularized problems are generally easier to solve than L1-regularized due to smoothness.
However, L1 regularization can help promote sparsity in weights leading to smaller and more interpretable models, the latter of which can be useful for feature selection.
[Elastic net](http://en.wikipedia.org/wiki/Elastic_net_regularization) is a combination of L1 and L2 regularization. It is not recommended to train models without any regularization,
especially when the number of training examples is small.

### Optimization

Under the hood, linear methods use convex optimization methods to optimize the objective functions.
`spark.mllib` uses two methods, SGD and L-BFGS, described in the [optimization section](mllib-optimization.html).
Currently, most algorithm APIs support Stochastic Gradient Descent (SGD), and a few support L-BFGS.
Refer to [this optimization section](mllib-optimization.html#Choosing-an-Optimization-Method) for guidelines on choosing between optimization methods.

## Classification

[Classification](http://en.wikipedia.org/wiki/Statistical_classification) aims to divide items into
categories.
The most common classification type is
[binary classification](http://en.wikipedia.org/wiki/Binary_classification), where there are two
categories, usually named positive and negative.
If there are more than two categories, it is called
[multiclass classification](http://en.wikipedia.org/wiki/Multiclass_classification).
`spark.mllib` supports two linear methods for classification: linear Support Vector Machines (SVMs)
and logistic regression.
Linear SVMs supports only binary classification, while logistic regression supports both binary and
multiclass classification problems.
For both methods, `spark.mllib` supports L1 and L2 regularized variants.
The training data set is represented by an RDD of [LabeledPoint](mllib-data-types.html#labeled-point) in MLlib,
where labels are class indices starting from zero: $0, 1, 2, \ldots$.

### Linear Support Vector Machines (SVMs)

The [linear SVM](http://en.wikipedia.org/wiki/Support_vector_machine#Linear_SVM)
is a standard method for large-scale classification tasks. It is a linear method as described above in equation `$\eqref{eq:regPrimal}$`, with the loss function in the formulation given by the hinge loss:

`\[
L(\wv;\x,y) := \max \{0, 1-y \wv^T \x \}.
\]`
By default, linear SVMs are trained with an L2 regularization.
We also support alternative L1 regularization. In this case,
the problem becomes a [linear program](http://en.wikipedia.org/wiki/Linear_programming).

The linear SVMs algorithm outputs an SVM model. Given a new data point,
denoted by $\x$, the model makes predictions based on the value of $\wv^T \x$.
By the default, if $\wv^T \x \geq 0$ then the outcome is positive, and negative
otherwise.

**Examples**

<div class="codetabs">

<div data-lang="scala" markdown="1">
The following code snippet illustrates how to load a sample dataset, execute a
training algorithm on this training data using a static method in the algorithm
object, and make predictions with the resulting model to compute the training
error.

Refer to the [`SVMWithSGD` Scala docs](api/scala/org/apache/spark/mllib/classification/SVMWithSGD.html) and [`SVMModel` Scala docs](api/scala/org/apache/spark/mllib/classification/SVMModel.html) for details on the API.

{% include_example scala/org/apache/spark/examples/mllib/SVMWithSGDExample.scala %}

The `SVMWithSGD.train()` method by default performs L2 regularization with the
regularization parameter set to 1.0. If we want to configure this algorithm, we
can customize `SVMWithSGD` further by creating a new object directly and
calling setter methods. All other `spark.mllib` algorithms support customization in
this way as well. For example, the following code produces an L1 regularized
variant of SVMs with regularization parameter set to 0.1, and runs the training
algorithm for 200 iterations.

{% highlight scala %}

import org.apache.spark.mllib.optimization.L1Updater

val svmAlg = new SVMWithSGD()
svmAlg.optimizer
  .setNumIterations(200)
  .setRegParam(0.1)
  .setUpdater(new L1Updater)
val modelL1 = svmAlg.run(training)
{% endhighlight %}

</div>

<div data-lang="java" markdown="1">
All of MLlib's methods use Java-friendly types, so you can import and call them there the same
way you do in Scala. The only caveat is that the methods take Scala RDD objects, while the
Spark Java API uses a separate `JavaRDD` class. You can convert a Java RDD to a Scala one by
calling `.rdd()` on your `JavaRDD` object. A self-contained application example
that is equivalent to the provided example in Scala is given below:

Refer to the [`SVMWithSGD` Java docs](api/java/org/apache/spark/mllib/classification/SVMWithSGD.html) and [`SVMModel` Java docs](api/java/org/apache/spark/mllib/classification/SVMModel.html) for details on the API.

{% include_example java/org/apache/spark/examples/mllib/JavaSVMWithSGDExample.java %}

The `SVMWithSGD.train()` method by default performs L2 regularization with the
regularization parameter set to 1.0. If we want to configure this algorithm, we
can customize `SVMWithSGD` further by creating a new object directly and
calling setter methods. All other `spark.mllib` algorithms support customization in
this way as well. For example, the following code produces an L1 regularized
variant of SVMs with regularization parameter set to 0.1, and runs the training
algorithm for 200 iterations.

{% highlight java %}
import org.apache.spark.mllib.optimization.L1Updater;

SVMWithSGD svmAlg = new SVMWithSGD();
svmAlg.optimizer()
  .setNumIterations(200)
  .setRegParam(0.1)
  .setUpdater(new L1Updater());
SVMModel modelL1 = svmAlg.run(training.rdd());
{% endhighlight %}

In order to run the above application, follow the instructions
provided in the [Self-Contained
Applications](quick-start.html#self-contained-applications) section of the Spark
quick-start guide. Be sure to also include *spark-mllib* to your build file as
a dependency.
</div>

<div data-lang="python" markdown="1">
The following example shows how to load a sample dataset, build SVM model,
and make predictions with the resulting model to compute the training error.

Refer to the [`SVMWithSGD` Python docs](api/python/pyspark.mllib.html#pyspark.mllib.classification.SVMWithSGD) and [`SVMModel` Python docs](api/python/pyspark.mllib.html#pyspark.mllib.classification.SVMModel) for more details on the API.

{% include_example python/mllib/svm_with_sgd_example.py %}
</div>
</div>

### Logistic regression

[Logistic regression](http://en.wikipedia.org/wiki/Logistic_regression) is widely used to predict a
binary response. It is a linear method as described above in equation `$\eqref{eq:regPrimal}$`,
with the loss function in the formulation given by the logistic loss:
`\[
L(\wv;\x,y) :=  \log(1+\exp( -y \wv^T \x)).
\]`

For binary classification problems, the algorithm outputs a binary logistic regression model.
Given a new data point, denoted by $\x$, the model makes predictions by
applying the logistic function
`\[
\mathrm{f}(z) = \frac{1}{1 + e^{-z}}
\]`
where $z = \wv^T \x$.
By default, if $\mathrm{f}(\wv^T x) > 0.5$, the outcome is positive, or
negative otherwise, though unlike linear SVMs, the raw output of the logistic regression
model, $\mathrm{f}(z)$, has a probabilistic interpretation (i.e., the probability
that $\x$ is positive).

Binary logistic regression can be generalized into
[multinomial logistic regression](http://en.wikipedia.org/wiki/Multinomial_logistic_regression) to
train and predict multiclass classification problems.
For example, for $K$ possible outcomes, one of the outcomes can be chosen as a "pivot", and the
other $K - 1$ outcomes can be separately regressed against the pivot outcome.
In `spark.mllib`, the first class $0$ is chosen as the "pivot" class.
See Section 4.4 of
[The Elements of Statistical Learning](http://statweb.stanford.edu/~tibs/ElemStatLearn/) for
references.
Here is a
[detailed mathematical derivation](http://www.slideshare.net/dbtsai/2014-0620-mlor-36132297).

For multiclass classification problems, the algorithm will output a multinomial logistic regression
model, which contains $K - 1$ binary logistic regression models regressed against the first class.
Given a new data points, $K - 1$ models will be run, and the class with largest probability will be
chosen as the predicted class.

We implemented two algorithms to solve logistic regression: mini-batch gradient descent and L-BFGS.
We recommend L-BFGS over mini-batch gradient descent for faster convergence.

**Examples**

<div class="codetabs">

<div data-lang="scala" markdown="1">
The following code illustrates how to load a sample multiclass dataset, split it into train and
test, and use
[LogisticRegressionWithLBFGS](api/scala/org/apache/spark/mllib/classification/LogisticRegressionWithLBFGS.html)
to fit a logistic regression model.
Then the model is evaluated against the test dataset and saved to disk.

Refer to the [`LogisticRegressionWithLBFGS` Scala docs](api/scala/org/apache/spark/mllib/classification/LogisticRegressionWithLBFGS.html) and [`LogisticRegressionModel` Scala docs](api/scala/org/apache/spark/mllib/classification/LogisticRegressionModel.html) for details on the API.

{% include_example scala/org/apache/spark/examples/mllib/LogisticRegressionWithLBFGSExample.scala %}

</div>

<div data-lang="java" markdown="1">
The following code illustrates how to load a sample multiclass dataset, split it into train and
test, and use
[LogisticRegressionWithLBFGS](api/java/org/apache/spark/mllib/classification/LogisticRegressionWithLBFGS.html)
to fit a logistic regression model.
Then the model is evaluated against the test dataset and saved to disk.

Refer to the [`LogisticRegressionWithLBFGS` Java docs](api/java/org/apache/spark/mllib/classification/LogisticRegressionWithLBFGS.html) and [`LogisticRegressionModel` Java docs](api/java/org/apache/spark/mllib/classification/LogisticRegressionModel.html) for details on the API.

{% include_example java/org/apache/spark/examples/mllib/JavaLogisticRegressionWithLBFGSExample.java %}
</div>

<div data-lang="python" markdown="1">
The following example shows how to load a sample dataset, build Logistic Regression model,
and make predictions with the resulting model to compute the training error.

Note that the Python API does not yet support multiclass classification and model save/load but
will in the future.

Refer to the [`LogisticRegressionWithLBFGS` Python docs](api/python/pyspark.mllib.html#pyspark.mllib.classification.LogisticRegressionWithLBFGS) and [`LogisticRegressionModel` Python docs](api/python/pyspark.mllib.html#pyspark.mllib.classification.LogisticRegressionModel) for more details on the API.

{% include_example python/mllib/logistic_regression_with_lbfgs_example.py %}
</div>
</div>

# Regression

### Linear least squares, Lasso, and ridge regression


Linear least squares is the most common formulation for regression problems.
It is a linear method as described above in equation `$\eqref{eq:regPrimal}$`, with the loss
function in the formulation given by the squared loss:
`\[
L(\wv;\x,y) :=  \frac{1}{2} (\wv^T \x - y)^2.
\]`

Various related regression methods are derived by using different types of regularization:
[*ordinary least squares*](http://en.wikipedia.org/wiki/Ordinary_least_squares) or
[*linear least squares*](http://en.wikipedia.org/wiki/Linear_least_squares_(mathematics)) uses
 no regularization; [*ridge regression*](http://en.wikipedia.org/wiki/Ridge_regression) uses L2
regularization; and [*Lasso*](http://en.wikipedia.org/wiki/Lasso_(statistics)) uses L1
regularization.  For all of these models, the average loss or training error, $\frac{1}{n} \sum_{i=1}^n (\wv^T x_i - y_i)^2$, is
known as the [mean squared error](http://en.wikipedia.org/wiki/Mean_squared_error).

### Streaming linear regression

When data arrive in a streaming fashion, it is useful to fit regression models online,
updating the parameters of the model as new data arrives. `spark.mllib` currently supports
streaming linear regression using ordinary least squares. The fitting is similar
to that performed offline, except fitting occurs on each batch of data, so that
the model continually updates to reflect the data from the stream.

**Examples**

The following example demonstrates how to load training and testing data from two different
input streams of text files, parse the streams as labeled points, fit a linear regression model
online to the first stream, and make predictions on the second stream.

<div class="codetabs">

<div data-lang="scala" markdown="1">

First, we import the necessary classes for parsing our input data and creating the model.

Then we make input streams for training and testing data. We assume a StreamingContext `ssc`
has already been created, see [Spark Streaming Programming Guide](streaming-programming-guide.html#initializing)
for more info. For this example, we use labeled points in training and testing streams,
but in practice you will likely want to use unlabeled vectors for test data.

We create our model by initializing the weights to zero and register the streams for training and
testing then start the job. Printing predictions alongside true labels lets us easily see the
result.

Finally, we can save text files with data to the training or testing folders.
Each line should be a data point formatted as `(y,[x1,x2,x3])` where `y` is the label
and `x1,x2,x3` are the features. Anytime a text file is placed in `args(0)`
the model will update. Anytime a text file is placed in `args(1)` you will see predictions.
As you feed more data to the training directory, the predictions
will get better!

Here is a complete example:
{% include_example scala/org/apache/spark/examples/mllib/StreamingLinearRegressionExample.scala %}

</div>

<div data-lang="python" markdown="1">

First, we import the necessary classes for parsing our input data and creating the model.

Then we make input streams for training and testing data. We assume a StreamingContext `ssc`
has already been created, see [Spark Streaming Programming Guide](streaming-programming-guide.html#initializing)
for more info. For this example, we use labeled points in training and testing streams,
but in practice you will likely want to use unlabeled vectors for test data.

We create our model by initializing the weights to 0.

Now we register the streams for training and testing and start the job.

We can now save text files with data to the training or testing folders.
Each line should be a data point formatted as `(y,[x1,x2,x3])` where `y` is the label
and `x1,x2,x3` are the features. Anytime a text file is placed in `sys.argv[1]`
the model will update. Anytime a text file is placed in `sys.argv[2]` you will see predictions.
As you feed more data to the training directory, the predictions
will get better!

Here a complete example:
{% include_example python/mllib/streaming_linear_regression_example.py %}

</div>

</div>


# Implementation (developer)

Behind the scene, `spark.mllib` implements a simple distributed version of stochastic gradient descent
(SGD), building on the underlying gradient descent primitive (as described in the <a
href="mllib-optimization.html">optimization</a> section).  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 three possible regularizations (none, L1 or L2).

For Logistic Regression, [L-BFGS](api/scala/org/apache/spark/mllib/optimization/LBFGS.html)
version is implemented under [LogisticRegressionWithLBFGS](api/scala/org/apache/spark/mllib/classification/LogisticRegressionWithLBFGS.html), and this
version supports both binary and multinomial Logistic Regression while SGD version only supports
binary Logistic Regression. However, L-BFGS version doesn't support L1 regularization but SGD one
supports L1 regularization. When L1 regularization is not required, L-BFGS version is strongly
recommended since it converges faster and more accurately compared to SGD by approximating the
inverse Hessian matrix using quasi-Newton method.

Algorithms are all implemented in Scala:

* [SVMWithSGD](api/scala/org/apache/spark/mllib/classification/SVMWithSGD.html)
* [LogisticRegressionWithLBFGS](api/scala/org/apache/spark/mllib/classification/LogisticRegressionWithLBFGS.html)
* [LogisticRegressionWithSGD](api/scala/org/apache/spark/mllib/classification/LogisticRegressionWithSGD.html)
* [LinearRegressionWithSGD](api/scala/org/apache/spark/mllib/regression/LinearRegressionWithSGD.html)
* [RidgeRegressionWithSGD](api/scala/org/apache/spark/mllib/regression/RidgeRegressionWithSGD.html)
* [LassoWithSGD](api/scala/org/apache/spark/mllib/regression/LassoWithSGD.html)