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https://issues.apache.org/jira/browse/MAHOUT-672?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=13021145#comment-13021145
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Ted Dunning commented on MAHOUT-672:
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{quote}
As for a linear regression implementation using CG compared to one using SGD,
it would be hard for me to reach any conclusions without comparing the two
approaches head to head on the same data. CG would probably gain some benefit
from being easily parallelizable, but the individual updates in SGD seem very
fast and lightweight, so any speed advantage to CG would probably only come up
for truly massive datasets. The SGD implementation in your patch also has a lot
of regularization support that a simple CG implementation of LMS would lack
(ridge regression i.e. L2 regularization comes for free, but L1 is considerably
harder). I'm also unaware of how one would do the automatic
validation/hyperparameter tuning using CG that your SGD implementation does.
{quote}
The other big difference, btw, is that all of our parallel approaches require
at least one pass through the data. The SGD stuff can stop early and often
only needs a small fraction of the input to converge. That gives sub-linear
convergence time in terms of input size (which sounds whacky, but is real).
Any approach that needs to read the entire data set obviously can't touch that
scaling factor.
Offsetting this is the idea that if we don't need all the data for a given
complexity of model, then we probably don't want to stop but would rather just
have a more complex model. This is where the non-parametric approaches come
in. The would give simple answers with small inputs and more nuanced answers
with large data.
> Implementation of Conjugate Gradient for solving large linear systems
> ---------------------------------------------------------------------
>
> Key: MAHOUT-672
> URL: https://issues.apache.org/jira/browse/MAHOUT-672
> Project: Mahout
> Issue Type: New Feature
> Components: Math
> Affects Versions: 0.5
> Reporter: Jonathan Traupman
> Priority: Minor
> Attachments: MAHOUT-672.patch
>
> Original Estimate: 48h
> Remaining Estimate: 48h
>
> This patch contains an implementation of conjugate gradient, an iterative
> algorithm for solving large linear systems. In particular, it is well suited
> for large sparse systems where a traditional QR or Cholesky decomposition is
> infeasible. Conjugate gradient only works for matrices that are square,
> symmetric, and positive definite (basically the same types where Cholesky
> decomposition is applicable). Systems like these commonly occur in statistics
> and machine learning problems (e.g. regression).
> Both a standard (in memory) solver and a distributed hadoop-based solver
> (basically the standard solver run using a DistributedRowMatrix a la
> DistributedLanczosSolver) are included.
> There is already a version of this algorithm in taste package, but it doesn't
> operate on standard mahout matrix/vector objects, nor does it implement a
> distributed version. I believe this implementation will be more generically
> useful to the community than the specialized one in taste.
> This implementation solves the following types of systems:
> Ax = b, where A is square, symmetric, and positive definite
> A'Ax = b where A is arbitrary but A'A is positive definite. Directly solving
> this system is more efficient than computing A'A explicitly then solving.
> (A + lambda * I)x = b and (A'A + lambda * I)x = b, for systems where A or A'A
> is singular and/or not full rank. This occurs commonly if A is large and
> sparse. Solving a system of this form is used, for example, in ridge
> regression.
> In addition to the normal conjugate gradient solver, this implementation also
> handles preconditioning, and has a sample Jacobi preconditioner included as
> an example. More work will be needed to build more advanced preconditioners
> if desired.
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