Repository: spark Updated Branches: refs/heads/master 9ab725eab -> 82253617f
[SPARK-18705][ML][DOC] Update user guide to reflect one pass solver for L1 and elastic-net ## What changes were proposed in this pull request? WeightedLeastSquares now supports L1 and elastic net penalties and has an additional solver option: QuasiNewton. The docs are updated to reflect this change. ## How was this patch tested? Docs only. Generated documentation to make sure Latex looks ok. Author: sethah <seth.hendrickso...@gmail.com> Closes #16139 from sethah/SPARK-18705. Project: http://git-wip-us.apache.org/repos/asf/spark/repo Commit: http://git-wip-us.apache.org/repos/asf/spark/commit/82253617 Tree: http://git-wip-us.apache.org/repos/asf/spark/tree/82253617 Diff: http://git-wip-us.apache.org/repos/asf/spark/diff/82253617 Branch: refs/heads/master Commit: 82253617f5b3cdbd418c48f94e748651ee80077e Parents: 9ab725e Author: sethah <seth.hendrickso...@gmail.com> Authored: Wed Dec 7 19:41:32 2016 -0800 Committer: Yanbo Liang <yblia...@gmail.com> Committed: Wed Dec 7 19:41:32 2016 -0800 ---------------------------------------------------------------------- docs/ml-advanced.md | 24 ++++++++++++++++-------- 1 file changed, 16 insertions(+), 8 deletions(-) ---------------------------------------------------------------------- http://git-wip-us.apache.org/repos/asf/spark/blob/82253617/docs/ml-advanced.md ---------------------------------------------------------------------- diff --git a/docs/ml-advanced.md b/docs/ml-advanced.md index 12a03d3..2747f2d 100644 --- a/docs/ml-advanced.md +++ b/docs/ml-advanced.md @@ -59,17 +59,25 @@ Given $n$ weighted observations $(w_i, a_i, b_i)$: The number of features for each observation is $m$. We use the following weighted least squares formulation: `\[ -minimize_{x}\frac{1}{2} \sum_{i=1}^n \frac{w_i(a_i^T x -b_i)^2}{\sum_{k=1}^n w_k} + \frac{1}{2}\frac{\lambda}{\delta}\sum_{j=1}^m(\sigma_{j} x_{j})^2 +\min_{\mathbf{x}}\frac{1}{2} \sum_{i=1}^n \frac{w_i(\mathbf{a}_i^T \mathbf{x} -b_i)^2}{\sum_{k=1}^n w_k} + \frac{\lambda}{\delta}\left[\frac{1}{2}(1 - \alpha)\sum_{j=1}^m(\sigma_j x_j)^2 + \alpha\sum_{j=1}^m |\sigma_j x_j|\right] \]` -where $\lambda$ is the regularization parameter, $\delta$ is the population standard deviation of the label +where $\lambda$ is the regularization parameter, $\alpha$ is the elastic-net mixing parameter, $\delta$ is the population standard deviation of the label and $\sigma_j$ is the population standard deviation of the j-th feature column. -This objective function has an analytic solution and it requires only one pass over the data to collect necessary statistics to solve. -Unlike the original dataset which can only be stored in a distributed system, -these statistics can be loaded into memory on a single machine if the number of features is relatively small, and then we can solve the objective function through Cholesky factorization on the driver. +This objective function requires only one pass over the data to collect the statistics necessary to solve it. For an +$n \times m$ data matrix, these statistics require only $O(m^2)$ storage and so can be stored on a single machine when $m$ (the number of features) is +relatively small. We can then solve the normal equations on a single machine using local methods like direct Cholesky factorization or iterative optimization programs. -WeightedLeastSquares only supports L2 regularization and provides options to enable or disable regularization and standardization. -In order to make the normal equation approach efficient, WeightedLeastSquares requires that the number of features be no more than 4096. For larger problems, use L-BFGS instead. +Spark MLlib currently supports two types of solvers for the normal equations: Cholesky factorization and Quasi-Newton methods (L-BFGS/OWL-QN). Cholesky factorization +depends on a positive definite covariance matrix (i.e. columns of the data matrix must be linearly independent) and will fail if this condition is violated. Quasi-Newton methods +are still capable of providing a reasonable solution even when the covariance matrix is not positive definite, so the normal equation solver can also fall back to +Quasi-Newton methods in this case. This fallback is currently always enabled for the `LinearRegression` and `GeneralizedLinearRegression` estimators. + +`WeightedLeastSquares` supports L1, L2, and elastic-net regularization and provides options to enable or disable regularization and standardization. In the case where no +L1 regularization is applied (i.e. $\alpha = 0$), there exists an analytical solution and either Cholesky or Quasi-Newton solver may be used. When $\alpha > 0$ no analytical +solution exists and we instead use the Quasi-Newton solver to find the coefficients iteratively. + +In order to make the normal equation approach efficient, `WeightedLeastSquares` requires that the number of features be no more than 4096. For larger problems, use L-BFGS instead. ## Iteratively reweighted least squares (IRLS) @@ -83,6 +91,6 @@ It solves certain optimization problems iteratively through the following proced * solve a weighted least squares (WLS) problem by WeightedLeastSquares. * repeat above steps until convergence. -Since it involves solving a weighted least squares (WLS) problem by WeightedLeastSquares in each iteration, +Since it involves solving a weighted least squares (WLS) problem by `WeightedLeastSquares` in each iteration, it also requires the number of features to be no more than 4096. Currently IRLS is used as the default solver of [GeneralizedLinearRegression](api/scala/index.html#org.apache.spark.ml.regression.GeneralizedLinearRegression). --------------------------------------------------------------------- To unsubscribe, e-mail: commits-unsubscr...@spark.apache.org For additional commands, e-mail: commits-h...@spark.apache.org
