address review

docs/ml-advanced.md

@@ -59,21 +59,24 @@ 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:
 `\[   
-\min_{\mathbf{x}}\frac{1}{2} \sum_{i=1}^n \frac{w_i(\mathbf{a}_i^T \mathbf{x} -b_i)^2}{\sum_{k=1}^n w_k} + \frac{1}{2}\frac{\lambda}{\delta}\sum_{j=1}^m(\sigma_{j} x_{j})^2
+\min_{\mathbf{x}}\frac{1}{2} \sum_{i=1}^n \frac{w_i(\mathbf{a}_i^T \mathbf{x} -b_i)^2}{\sum_{k=1}^n w_k} + \frac{\lambda}{\delta}\left[\frac{1}{2}(1 - \alpha)\sum_{j=1}^m(\sigma_j x_j)^2 + \alpha\sum_{j=1}^m |\sigma_j x_j|\right]
 \]`
-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. For an
+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.
 
-Spark ML currently supports two types of solvers for the normal equations: Cholesky factorization and Quasi-Newton methods (L-BFGS/OWL-QN). Cholesky factorization
+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 (e.g. 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` estimator.
+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. 
 
-`WeightedLeastSquares` supports L1, L2, and elastic-net 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.
 
 ## Iteratively reweighted least squares (IRLS)