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

    https://github.com/apache/spark/pull/15394#discussion_r83671808
  
    --- 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[Array[Double]],
    +    val objectiveHistory: Option[Array[Double]]) {
    +
    +  def coefficients: Array[Double] = {
    +    if (fitIntercept) {
    +      _coefficients.slice(0, _coefficients.length - 1)
    +    } else {
    +      _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 {
    --- End diff --
    
    In the `NormalEquationCostFun` we set the last coefficient (the intercept) 
to the analytically correct value in every iteration. I'm not sure how we can 
do this without having knowledge of whether or not we are fitting an intercept 
term. This modification does in fact have an effect on the convergence of the 
algorithm, so I think it's important to keep. 
    
    We can still move it out of `CholeskySolver` and `NormalEquationSolution` 
if you prefer. Not sure about `QuasiNewtonSolver`. Thoughts?


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