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## What changes were proposed in this pull request?

1. Make a pluggable solver interface for `WeightedLeastSquares`
2. Add a `QuasiNewton` solver to handle elastic net regularization for `WeightedLeastSquares`
3. Add method `BLAS.dspmv` used by QN solver
4. Add mechanism for WLS to handle singular covariance matrices by falling back to QN solver when Cholesky fails.

## How was this patch tested?
Unit tests - see below.

## Design choices

**Pluggable Normal Solver**

Before, the `WeightedLeastSquares` package always used the Cholesky decomposition solver to compute the solution to the normal equations. Now, we specify the solver as a constructor argument to the `WeightedLeastSquares`. We introduce a new trait:

````scala
private[ml] sealed trait NormalEquationSolver {

  def solve(
      bBar: Double,
      bbBar: Double,
      abBar: DenseVector,
      aaBar: DenseVector,
      aBar: DenseVector): NormalEquationSolution
}
````

We extend this trait for different variants of normal equation solvers. In the future, we can easily add others (like QR) using this interface.

**Always train in the standardized space**

The normal solver did not previously standardize the data, but this patch introduces a change such that we always solve the normal equations in the standardized space. We convert back to the original space in the same way that is done for distributed L-BFGS/OWL-QN. We add test cases for zero variance features/labels.

**Use L-BFGS locally to solve normal equations for singular matrix**

When linear regression with the normal solver is called for a singular matrix, we initially try to solve with Cholesky. We use the output of `lapack.dppsv` to determine if the matrix is singular. If it is, we fall back to using L-BFGS locally to solve the normal equations. We add test cases for this as well.

## Test cases
I found it helpful to enumerate some of the test cases and hopefully it makes review easier.

**WeightedLeastSquares**

1. Constant columns - Cholesky solver fails with no regularization, Auto solver falls back to QN, and QN trains successfully.
2. Collinear features - Cholesky solver fails with no regularization, Auto solver falls back to QN, and QN trains successfully.
3. Label is constant zero - no training is performed regardless of intercept. Coefficients are zero and intercept is zero.
4. Label is constant - if fitIntercept, then no training is performed and intercept equals label mean. If not fitIntercept, then we train and return an answer that matches R's lm package.
5. Test with L1 - go through various combinations of L1/L2, standardization, fitIntercept and verify that output matches glmnet.
6. Initial intercept - verify that setting the initial intercept to label mean is correct by training model with strong L1 regularization so that all coefficients are zero and intercept converges to label mean.
7. Test diagInvAtWA - since we are standardizing features now during training, we should test that the inverse is computed to match R.

**LinearRegression**
1. For all existing L1 test cases, test the "normal" solver too.
2. Check that using the normal solver now handles singular matrices.
3. Check that using the normal solver with L1 produces an objective history in the model summary, but does not produce the inverse of AtA.

**BLAS**
1. Test new method `dspmv`.

## Performance Testing
This patch will speed up linear regression with L1/elasticnet penalties when the feature size is < 4096. I have not conducted performance tests at scale, only observed by testing locally that there is a speed improvement.

We should decide if this PR needs to be blocked before performance testing is conducted.

Author: sethah <seth.hendrickson16@gmail.com>

Closes #15394 from sethah/SPARK-17748.
78d740a0
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Apache Spark

Spark is a fast and general cluster computing system for Big Data. It provides high-level APIs in Scala, Java, Python, and R, and an optimized engine that supports general computation graphs for data analysis. It also supports a rich set of higher-level tools including Spark SQL for SQL and DataFrames, MLlib for machine learning, GraphX for graph processing, and Spark Streaming for stream processing.

http://spark.apache.org/

Online Documentation

You can find the latest Spark documentation, including a programming guide, on the project web page and project wiki. This README file only contains basic setup instructions.

Building Spark

Spark is built using Apache Maven. To build Spark and its example programs, run:

build/mvn -DskipTests clean package

(You do not need to do this if you downloaded a pre-built package.)

You can build Spark using more than one thread by using the -T option with Maven, see "Parallel builds in Maven 3". More detailed documentation is available from the project site, at "Building Spark". For developing Spark using an IDE, see Eclipse and IntelliJ.

Interactive Scala Shell

The easiest way to start using Spark is through the Scala shell:

./bin/spark-shell

Try the following command, which should return 1000:

scala> sc.parallelize(1 to 1000).count()

Interactive Python Shell

Alternatively, if you prefer Python, you can use the Python shell:

./bin/pyspark

And run the following command, which should also return 1000:

>>> sc.parallelize(range(1000)).count()

Example Programs

Spark also comes with several sample programs in the examples directory. To run one of them, use ./bin/run-example <class> [params]. For example:

./bin/run-example SparkPi

will run the Pi example locally.

You can set the MASTER environment variable when running examples to submit examples to a cluster. This can be a mesos:// or spark:// URL, "yarn" to run on YARN, and "local" to run locally with one thread, or "local[N]" to run locally with N threads. You can also use an abbreviated class name if the class is in the examples package. For instance:

MASTER=spark://host:7077 ./bin/run-example SparkPi

Many of the example programs print usage help if no params are given.

Running Tests

Testing first requires building Spark. Once Spark is built, tests can be run using:

./dev/run-tests

Please see the guidance on how to run tests for a module, or individual tests.

A Note About Hadoop Versions

Spark uses the Hadoop core library to talk to HDFS and other Hadoop-supported storage systems. Because the protocols have changed in different versions of Hadoop, you must build Spark against the same version that your cluster runs.

Please refer to the build documentation at "Specifying the Hadoop Version" for detailed guidance on building for a particular distribution of Hadoop, including building for particular Hive and Hive Thriftserver distributions.

Configuration

Please refer to the Configuration Guide in the online documentation for an overview on how to configure Spark.

## Contributing

Please review the Contribution to Spark wiki for information on how to get started contributing to the project.