Github user yanboliang commented on a diff in the pull request:

    https://github.com/apache/spark/pull/15394#discussion_r83022418
  
    --- Diff: 
mllib/src/main/scala/org/apache/spark/ml/optim/NormalEquationSolver.scala ---
    @@ -0,0 +1,165 @@
    +/*
    + * Licensed to the Apache Software Foundation (ASF) under one or more
    + * contributor license agreements.  See the NOTICE file distributed with
    + * this work for additional information regarding copyright ownership.
    + * The ASF licenses this file to You under the Apache License, Version 2.0
    + * (the "License"); you may not use this file except in compliance with
    + * the License.  You may obtain a copy of the License at
    + *
    + *    http://www.apache.org/licenses/LICENSE-2.0
    + *
    + * Unless required by applicable law or agreed to in writing, software
    + * distributed under the License is distributed on an "AS IS" BASIS,
    + * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
    + * See the License for the specific language governing permissions and
    + * limitations under the License.
    + */
    +package org.apache.spark.ml.optim
    +
    +import breeze.linalg.{DenseVector => BDV}
    +import breeze.optimize.{CachedDiffFunction, DiffFunction, LBFGS => 
BreezeLBFGS, OWLQN => BreezeOWLQN}
    +import scala.collection.mutable
    +
    +import org.apache.spark.ml.linalg.{BLAS, DenseVector, Vectors}
    +import org.apache.spark.mllib.linalg.CholeskyDecomposition
    +
    +private[ml] class NormalEquationSolution(
    +    val fitIntercept: Boolean,
    +    private val _coefficients: Array[Double],
    +    val aaInv: Option[DenseVector],
    +    val objectiveHistory: Option[Array[Double]]) {
    +
    +  def coefficients: DenseVector = {
    +    if (fitIntercept) {
    +      new DenseVector(_coefficients.slice(0, _coefficients.length - 1))
    +    } else {
    +      new DenseVector(_coefficients)
    +    }
    +  }
    +
    +  def intercept: Double = if (fitIntercept) _coefficients.last else 0.0
    +}
    +
    +/**
    + * Interface for classes that solve the normal equations locally.
    + */
    +private[ml] sealed trait NormalEquationSolver {
    +
    +  /** Solve the normal equations from summary statistics. */
    +  def solve(
    +      bBar: Double,
    +      bbBar: Double,
    +      abBar: DenseVector,
    +      aaBar: DenseVector,
    +      aBar: DenseVector): NormalEquationSolution
    +}
    +
    +/**
    + * A class that solves the normal equations directly, using Cholesky 
decomposition.
    + */
    +private[ml] class CholeskySolver(val fitIntercept: Boolean) extends 
NormalEquationSolver {
    +
    +  def solve(
    +      bBar: Double,
    +      bbBar: Double,
    +      abBar: DenseVector,
    +      aaBar: DenseVector,
    +      aBar: DenseVector): NormalEquationSolution = {
    +    val k = abBar.size
    +    val x = CholeskyDecomposition.solve(aaBar.values, abBar.values)
    +    val aaInv = CholeskyDecomposition.inverse(aaBar.values, k)
    +
    +    new NormalEquationSolution(fitIntercept, x, Some(new 
DenseVector(aaInv)), None)
    +  }
    +}
    +
    +/**
    + * A class for solving the normal equations using Quasi-Newton 
optimization methods.
    + */
    +private[ml] class QuasiNewtonSolver(
    +    val fitIntercept: Boolean,
    +    maxIter: Int,
    +    tol: Double,
    +    l1RegFunc: Option[(Int) => Double]) extends NormalEquationSolver {
    +
    +  def solve(
    +      bBar: Double,
    +      bbBar: Double,
    +      abBar: DenseVector,
    +      aaBar: DenseVector,
    +      aBar: DenseVector): NormalEquationSolution = {
    +    val numFeatures = aBar.size
    +    val numFeaturesPlusIntercept = if (fitIntercept) numFeatures + 1 else 
numFeatures
    +    val initialCoefficientsWithIntercept = new 
Array[Double](numFeaturesPlusIntercept)
    +    if (fitIntercept) {
    +      initialCoefficientsWithIntercept(numFeaturesPlusIntercept - 1) = bBar
    +    }
    +
    +    val costFun =
    +      new NormalEquationCostFun(bBar, bbBar, abBar, aaBar, aBar, 
fitIntercept, numFeatures)
    +    val optimizer = l1RegFunc.map { func =>
    +      new BreezeOWLQN[Int, BDV[Double]](maxIter, 10, func, tol)
    +    }.getOrElse(new BreezeLBFGS[BDV[Double]](maxIter, 10, tol))
    +
    +    val states = optimizer.iterations(new CachedDiffFunction(costFun),
    +      new BDV[Double](initialCoefficientsWithIntercept))
    +
    +    val arrayBuilder = mutable.ArrayBuilder.make[Double]
    +    var state: optimizer.State = null
    +    while (states.hasNext) {
    +      state = states.next()
    +      arrayBuilder += state.adjustedValue
    +    }
    +    val x = state.x.toArray.clone()
    +    new NormalEquationSolution(fitIntercept, x, None, 
Some(arrayBuilder.result()))
    +  }
    +
    +  /**
    +   * NormalEquationCostFun implements Breeze's DiffFunction[T] for the 
normal equation.
    +   * It returns the loss and gradient with L2 regularization at a 
particular point (coefficients).
    +   * It's used in Breeze's convex optimization routines.
    +   */
    +  private class NormalEquationCostFun(
    +      bBar: Double,
    +      bbBar: Double,
    +      ab: DenseVector,
    +      aa: DenseVector,
    +      aBar: DenseVector,
    +      fitIntercept: Boolean,
    +      numFeatures: Int) extends DiffFunction[BDV[Double]] {
    +
    +    private val numFeaturesPlusIntercept = if (fitIntercept) numFeatures + 
1 else numFeatures
    +
    +    override def calculate(coefficients: BDV[Double]): (Double, 
BDV[Double]) = {
    +      val coef = Vectors.fromBreeze(coefficients).toDense
    +      if (fitIntercept) {
    +        var j = 0
    +        var dotProd = 0.0
    +        val coefValues = coef.values
    +        val aBarValues = aBar.values
    +        while (j < numFeatures) {
    +          dotProd += coefValues(j) * aBarValues(j)
    +          j += 1
    +        }
    +        coefValues(numFeatures) = bBar - dotProd
    +      }
    +      val xxb = new DenseVector(new 
Array[Double](numFeaturesPlusIntercept))
    +      BLAS.dspmv(numFeaturesPlusIntercept, 1.0, aa, coef, xxb)
    +      // loss = 1/2 (Y^T W Y - 2 beta^T X^T W Y + beta^T X^T W X beta)
    --- End diff --
    
    Shall we use the consistent name convention like ```Ax = b```?


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