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author | sethah <seth.hendrickson16@gmail.com> | 2016-12-07 19:41:32 -0800 |
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committer | Yanbo Liang <ybliang8@gmail.com> | 2016-12-07 19:41:32 -0800 |
commit | 82253617f5b3cdbd418c48f94e748651ee80077e (patch) | |
tree | 9f380bf3de244c5a64e1741ebdc9d6a2ad5e28a3 /docs/ml-advanced.md | |
parent | 9ab725eabbb4ad515a663b395bd2f91bb5853a23 (diff) | |
download | spark-82253617f5b3cdbd418c48f94e748651ee80077e.tar.gz spark-82253617f5b3cdbd418c48f94e748651ee80077e.tar.bz2 spark-82253617f5b3cdbd418c48f94e748651ee80077e.zip |
[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.hendrickson16@gmail.com>
Closes #16139 from sethah/SPARK-18705.
Diffstat (limited to 'docs/ml-advanced.md')
-rw-r--r-- | docs/ml-advanced.md | 24 |
1 files changed, 16 insertions, 8 deletions
diff --git a/docs/ml-advanced.md b/docs/ml-advanced.md index 12a03d3c91..2747f2df7c 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). |