## Details

## Description

Currently ml.recommendation.ALS is optimized for gram matrix generation which scales to modest ranks. The problems that we can solve are in the normal equation/quadratic form: 0.5x'Hx + c'x + g(z)

g(z) can be one of the constraints from Breeze proximal library:

https://github.com/scalanlp/breeze/blob/master/math/src/main/scala/breeze/optimize/proximal/Proximal.scala

In this PR we will re-use ml.recommendation.ALS design and come up with ml.recommendation.ALM (Alternating Minimization). Thanks to Xiangrui Meng recent changes, it's straightforward to do it now !

ALM will be capable of solving the following problems: min f ( x ) + g ( z )

1. Loss function f ( x ) can be LeastSquareLoss and LoglikelihoodLoss. Most likely we will re-use the Gradient interfaces already defined and implement LoglikelihoodLoss

2. Constraints g ( z ) supported are same as above except that we don't support affine + bounds yet Aeq x = beq , lb <= x <= ub yet. Most likely we don't need that for ML applications

3. For solver we will use breeze.optimize.proximal.NonlinearMinimizer which in turn uses projection based solver (SPG) or proximal solvers (ADMM) based on convergence speed.

4. The factors will be SparseVector so that we keep shuffle size in check. For example we will run with 10K ranks but we will force factors to be 100-sparse.

This is closely related to Sparse LDA https://issues.apache.org/jira/browse/SPARK-5564 with the difference that we are not using graph representation here.

As we do scaling experiments, we will understand which flow is more suited as ratings get denser (my understanding is that since we already scaled ALS to 2 billion ratings and we will keep sparsity in check, the same 2 billion flow will scale to 10K ranks as well)...

This JIRA is intended to extend the capabilities of ml recommendation to generalized loss function.