Hi Xiangrui,
I did some outofbox comparisons with ECOS and PDCO from SOL.
Both of them seems to be running at par but I will do more detailed
analysis.
I used pdco's testQP randomized problem generation. pdcotestQP(m, n) means
m constraints and n variables
For ECOS runtime reference here is the paper
http://web.stanford.edu/~boyd/papers/pdf/ecos_ecc.pdf
It runs at par with gurobi but slower than MOSEK. Note that MOSEK is also a
SOCP solver.
K>> pdcotestQP(50, 100)
ECOSQP: Converting QP to SOCP...
ECOSQP: Time for Cholesky: 0.00 seconds
Conversion completed. Calling ECOS...
ECOS  (c) A. Domahidi, Automatic Control Laboratory, ETH Zurich, 20122014.
OPTIMAL (within feastol=1.0e05, reltol=1.0e06, abstol=1.0e06).
Runtime: 0.010340 seconds.

pdco.m Version of 23 Nov 2013
Primaldual barrier method to minimize a convex function
subject to linear constraints Ax + r = b, bl <= x <= bu
Michael Saunders SOL and ICME, Stanford University
Contributors: Byunggyoo Kim (SOL), Chris Maes (ICME)
Santiago Akle (ICME), Matt Zahr (ICME)

m = 50 n = 100 nnz(A) = 483
Method = 21 (1=chol 2=QR 3=LSMR 4=MINRES 21=SQD(LU)
22=SQD(MA57))
Elapsed time is 0.050226 seconds.
2. With a larger problem with 50 equality and 1000 variables:
K>> pdcotestQP(50, 1000)
ECOSQP: Converting QP to SOCP...
ECOSQP: Time for Cholesky: 0.05 seconds
Conversion completed. Calling ECOS...
ECOS  (c) A. Domahidi, Automatic Control Laboratory, ETH Zurich, 20122014.
OPTIMAL (within feastol=1.0e05, reltol=1.0e06, abstol=1.0e06).
Runtime: 6.333036 seconds.

pdco.m Version of 23 Nov 2013
Primaldual barrier method to minimize a convex function
subject to linear constraints Ax + r = b, bl <= x <= bu
Michael Saunders SOL and ICME, Stanford University
Contributors: Byunggyoo Kim (SOL), Chris Maes (ICME)
Santiago Akle (ICME), Matt Zahr (ICME)

The objective is defined by a function handle:
@(x)deal(0.5*(x'*H*x)+c'*x,H*x+c,H)
The matrix A is an explicit sparse matrix
m = 50 n = 1000 nnz(A) = 4842
Method = 21 (1=chol 2=QR 3=LSMR 4=MINRES 21=SQD(LU)
22=SQD(MA57))
Elapsed time is 7.531934 seconds.
I will change the Method = 21 (LU) to 1 (chol) and that should help PDCO.
If both the IPMs are at par it's still a good idea to choose ECOS as the
generic IPM since it can solve conic programs which are a superset of
quadratic programs (robust portfolio optimization from quantitative finance
is an example of standard conic program).
For the runtime comparisons with ADMM based decomposition for simpler
constraints, I am doing further profiling and see if the jnilib is causing
any performance issues for ECOS calls...
Please look at the proximal algorithm references
http://web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf. For many problems
like L1 constraint / positivity / bounds / huber / hyperplane projection
etc, the proximal operator is simple to evaluate and for these cases ADMM
decomposition has been shown to run faster than standard constraint solvers
like IPM. I am not very surprised that Sparse NMF runs in 4X runtime of
least squares using ADMM decomposition.
Distributed consensus is another ADMM decomposition which we are working
with right now. We will have some results on that soon. There the idea is
to use ADMM so that accumulator need not collect dense gradient vectors
from each worker. This development will further help the treeReduce work.
Should I open up Spark JIRA's so that we can document Quadratic
Minimization related runtime experiments/benchmarks and share the code for
review ?
Most likely the core solvers will go to breeze and in Spark mllib
optimization, I will add a QpSolver object which will call the underlying
breeze solvers based on the problem complexity....the ecos jnilib can be
part of breeze natives as it is GPL licensed (same as netlibjava
jniloader). I will push the jnilib as a PR to ecos repository
https://github.com/ifaethz/ecos
Thanks.
Deb
On Wed, Jul 2, 2014 at 1:52 AM, Xiangrui Meng <mengxr@gmail.com> wrote:
> Hi Deb,
>
> KNITRO and MOSEK are both commercial. If you are looking for
> opensource ones, you can take a look at PDCO from SOL:
>
> http://web.stanford.edu/group/SOL/software/pdco/
>
> Each subproblem is really just a small QP. ADMM is designed for the
> cases when data is distributively stored or the objective function is
> complex but splittable. Neither applies to this case.
>
> Best,
> Xiangrui
>
> On Tue, Jul 1, 2014 at 11:05 PM, Debasish Das <debasish.das83@gmail.com>
> wrote:
> > Hi Xiangrui,
> >
> > Could you please point to the IPM solver that you have positive results
> > with ? I was planning to compare with CVX, KNITRO from Professor Nocedal
> > and MOSEK probably...I don't have CPLEX license so I won't be able to do
> > that comparison...
> >
> > My experiments so far tells me that ADMM based solver is faster than IPM
> > for simpler constraints but then perhaps I did not choose the correct
> > IPM....
> >
> > Proximal algorithm paper also shows very similar results compared to CVX:
> >
> > http://web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf
> >
> > Thanks.
> > Deb
> >
> > On Wed, Jun 11, 2014 at 3:21 PM, Xiangrui Meng <mengxr@gmail.com> wrote:
> >
> >> You idea is close to what implicit feedback does. You can check the
> >> paper, which is short and concise. In the ALS setting, all subproblems
> >> are independent in each iteration. This is part of the reason why ALS
> >> is scalable. If you have some global constraints that make the
> >> subproblems no longer decoupled, that would certainly affects
> >> scalability. Xiangrui
> >>
> >> On Wed, Jun 11, 2014 at 2:20 AM, Debasish Das <debasish.das83@gmail.com
> >
> >> wrote:
> >> > I got it...ALS formulation is solving the matrix completion
> problem....
> >> >
> >> > To convert the problem to matrix factorization or take user feedback
> >> > (missing entries means the user hate the site ?), we should put 0 to
> the
> >> > missing entries (or may be 1)...in that case we have to use
> computeYtY
> >> and
> >> > accumulate over users in each block to generate full gram matrix...and
> >> > after that while computing userXy(index) we have to be careful in
> putting
> >> > 0/1 for rest of the features...
> >> >
> >> > Is implicit feedback trying to do something like this ?
> >> >
> >> > Basically I am trying to see if it is possible to cache the gram
> matrix
> >> and
> >> > it's cholesky factorization, and then call the QpSolver multiple times
> >> with
> >> > updated gradient term...I am expecting better runtimes than dposv when
> >> > ranks are high...
> >> >
> >> > But seems like that's not possible without a broadcast step which
> might
> >> > kill all the runtime gain...
> >> >
> >> >
> >> > On Wed, Jun 11, 2014 at 12:21 AM, Xiangrui Meng <mengxr@gmail.com>
> >> wrote:
> >> >
> >> >> For explicit feedback, ALS uses only observed ratings for
> computation.
> >> >> So XtXs are not the same. Xiangrui
> >> >>
> >> >> On Tue, Jun 10, 2014 at 8:58 PM, Debasish Das <
> debasish.das83@gmail.com
> >> >
> >> >> wrote:
> >> >> > Sorry last one went out by mistake:
> >> >> >
> >> >> > Is not for users (0 to numUsers), fullXtX is same ? In the ALS
> >> >> formulation
> >> >> > this is W^TW or H^TH which should be same for all the users ?
Why
> we
> >> are
> >> >> > reading userXtX(index) and adding it to fullXtX in the loop over
> all
> >> >> > numUsers ?
> >> >> >
> >> >> > // Solve the leastsquares problem for each user and return the
new
> >> >> feature
> >> >> > vectors
> >> >> >
> >> >> > Array.range(0, numUsers).map { index =>
> >> >> >
> >> >> > // Compute the full XtX matrix from the lowertriangular
> part we
> >> >> got
> >> >> > above
> >> >> >
> >> >> > fillFullMatrix(userXtX(index), fullXtX)
> >> >> >
> >> >> > // Add regularization
> >> >> >
> >> >> > var i = 0
> >> >> >
> >> >> > while (i < rank) {
> >> >> >
> >> >> > fullXtX.data(i * rank + i) += lambda
> >> >> >
> >> >> > i += 1
> >> >> >
> >> >> > }
> >> >> >
> >> >> > // Solve the resulting matrix, which is symmetric and
> >> >> > positivedefinite
> >> >> >
> >> >> > algo match {
> >> >> >
> >> >> > case ALSAlgo.Implicit =>
> >> >> > Solve.solvePositive(fullXtX.addi(YtY.get.value),
> >> >> > userXy(index)).data
> >> >> >
> >> >> > case ALSAlgo.Explicit => Solve.solvePositive(fullXtX,
> userXy
> >> >> > (index)).data
> >> >> >
> >> >> > }
> >> >> >
> >> >> > }
> >> >> >
> >> >> >
> >> >> > On Tue, Jun 10, 2014 at 8:56 PM, Debasish Das <
> >> debasish.das83@gmail.com>
> >> >> > wrote:
> >> >> >
> >> >> >> Hi,
> >> >> >>
> >> >> >> I am bit confused wiht the code here:
> >> >> >>
> >> >> >> // Solve the leastsquares problem for each user and return
the
> new
> >> >> >> feature vectors
> >> >> >>
> >> >> >> Array.range(0, numUsers).map { index =>
> >> >> >>
> >> >> >> // Compute the full XtX matrix from the lowertriangular
> part
> >> we
> >> >> >> got above
> >> >> >>
> >> >> >> fillFullMatrix(userXtX(index), fullXtX)
> >> >> >>
> >> >> >> // Add regularization
> >> >> >>
> >> >> >> var i = 0
> >> >> >>
> >> >> >> while (i < rank) {
> >> >> >>
> >> >> >> fullXtX.data(i * rank + i) += lambda
> >> >> >>
> >> >> >> i += 1
> >> >> >>
> >> >> >> }
> >> >> >>
> >> >> >> // Solve the resulting matrix, which is symmetric and
> >> >> >> positivedefinite
> >> >> >>
> >> >> >> algo match {
> >> >> >>
> >> >> >> case ALSAlgo.Implicit =>
> >> >> Solve.solvePositive(fullXtX.addi(YtY.get.value),
> >> >> >> userXy(index)).data
> >> >> >>
> >> >> >> case ALSAlgo.Explicit => Solve.solvePositive(fullXtX,
> userXy
> >> >> >> (index)).data
> >> >> >>
> >> >> >> }
> >> >> >>
> >> >> >> }
> >> >> >>
> >> >> >>
> >> >> >> On Fri, Jun 6, 2014 at 10:42 AM, Debasish Das <
> >> debasish.das83@gmail.com
> >> >> >
> >> >> >> wrote:
> >> >> >>
> >> >> >>> Hi Xiangrui,
> >> >> >>>
> >> >> >>> It's not the linear constraint, It is quadratic inequality
with
> >> slack,
> >> >> >>> first order taylor approximation of off diagonal cross
terms and
> a
> >> >> cyclic
> >> >> >>> coordinate descent, which we think will yield
> orthogonality....It's
> >> >> still
> >> >> >>> under works...
> >> >> >>>
> >> >> >>> Also we want to put a L1 constraint as set of linear equations
> when
> >> >> >>> solving for ALS...
> >> >> >>>
> >> >> >>> I will create the JIRA...as I see it, this will evolve
to a
> generic
> >> >> >>> constraint solver for machine learning problems that has
a QP
> >> >> >>> structure....ALS is one example....another example is
kernel
> SVMs...
> >> >> >>>
> >> >> >>> I did not know that lgpl solver can be added to the
> classpath....if
> >> it
> >> >> >>> can be then definitely we should add these in ALS.scala...
> >> >> >>>
> >> >> >>> Thanks.
> >> >> >>> Deb
> >> >> >>>
> >> >> >>>
> >> >> >>>
> >> >> >>> On Thu, Jun 5, 2014 at 11:31 PM, Xiangrui Meng <mengxr@gmail.com
> >
> >> >> wrote:
> >> >> >>>
> >> >> >>>> I don't quite understand why putting linear constraints
can
> promote
> >> >> >>>> orthogonality. For the interfaces, if the subproblem
is
> determined
> >> by
> >> >> >>>> Y^T Y and Y^T b for each iteration, then the least
squares
> solver,
> >> the
> >> >> >>>> nonnegative least squares solver, or your convex
solver is
> simply
> >> a
> >> >> >>>> function
> >> >> >>>>
> >> >> >>>> (A, b) > x.
> >> >> >>>>
> >> >> >>>> You can define it as an interface, and make the solver
> pluggable by
> >> >> >>>> adding a setter to ALS. If you want to use your lgpl
solver,
> just
> >> >> >>>> include it in the classpath. Creating two separate
files still
> >> seems
> >> >> >>>> unnecessary to me. Could you create a JIRA and we
can move our
> >> >> >>>> discussion there? Thanks!
> >> >> >>>>
> >> >> >>>> Best,
> >> >> >>>> Xiangrui
> >> >> >>>>
> >> >> >>>> On Thu, Jun 5, 2014 at 7:20 PM, Debasish Das <
> >> >> debasish.das83@gmail.com>
> >> >> >>>> wrote:
> >> >> >>>> > Hi Xiangrui,
> >> >> >>>> >
> >> >> >>>> > For orthogonality properties in the factors we
need a
> constraint
> >> >> solver
> >> >> >>>> > other than the usuals (l1, upper and lower bounds,
l2 etc)
> >> >> >>>> >
> >> >> >>>> > The interface of constraint solver is standard
and I can add
> it
> >> in
> >> >> >>>> mllib
> >> >> >>>> > optimization....
> >> >> >>>> >
> >> >> >>>> > But I am not sure how will I call the gpl licensed
ipm solver
> >> from
> >> >> >>>> > mllib....assume the solver interface is as follows:
> >> >> >>>> >
> >> >> >>>> > Qpsolver (densematrix h, array [double] f, int
linearEquality,
> >> int
> >> >> >>>> > linearInequality, bool lb, bool ub)
> >> >> >>>> >
> >> >> >>>> > And then I have functions to update equalities,
inequalities,
> >> bounds
> >> >> >>>> etc
> >> >> >>>> > followed by the run which generates the solution....
> >> >> >>>> >
> >> >> >>>> > For l1 constraints I have to use epigraph formulation
which
> >> needs a
> >> >> >>>> > variable transformation before the solve....
> >> >> >>>> >
> >> >> >>>> > I was thinking that for the problems that does
not need
> >> constraints
> >> >> >>>> people
> >> >> >>>> > will use ALS.scala and ConstrainedALS.scala will
have the
> >> >> constrained
> >> >> >>>> > formulations....
> >> >> >>>> >
> >> >> >>>> > I can point you to the code once it is ready
and then you can
> >> guide
> >> >> me
> >> >> >>>> how
> >> >> >>>> > to refactor it to mllib als ?
> >> >> >>>> >
> >> >> >>>> > Thanks.
> >> >> >>>> > Deb
> >> >> >>>> > Hi Deb,
> >> >> >>>> >
> >> >> >>>> > Why do you want to make those methods public?
If you only
> need to
> >> >> >>>> > replace the solver for subproblems. You can try
to make the
> >> solver
> >> >> >>>> > pluggable. Now it supports least squares and
nonnegative
> least
> >> >> >>>> > squares. You can define an interface for the
subproblem
> solvers
> >> and
> >> >> >>>> > maintain the IPM solver at your own code base,
if the only
> >> >> information
> >> >> >>>> > you need is Y^T Y and Y^T b.
> >> >> >>>> >
> >> >> >>>> > Btw, just curious, what is the use case for quadratic
> >> constraints?
> >> >> >>>> >
> >> >> >>>> > Best,
> >> >> >>>> > Xiangrui
> >> >> >>>> >
> >> >> >>>> > On Thu, Jun 5, 2014 at 3:38 PM, Debasish Das
<
> >> >> debasish.das83@gmail.com
> >> >> >>>> >
> >> >> >>>> > wrote:
> >> >> >>>> >> Hi,
> >> >> >>>> >>
> >> >> >>>> >> We are adding a constrained ALS solver in
Spark to solve
> matrix
> >> >> >>>> >> factorization usecases which needs additional
constraints
> >> (bounds,
> >> >> >>>> >> equality, inequality, quadratic constraints)
> >> >> >>>> >>
> >> >> >>>> >> We are using a native version of a primal
dual SOCP solver
> due
> >> to
> >> >> its
> >> >> >>>> > small
> >> >> >>>> >> memory footprint and sparse ccs matrix computation
it
> uses...The
> >> >> >>>> solver
> >> >> >>>> >> depends on AMD and LDL packages from Timothy
Davis for sparse
> >> ccs
> >> >> >>>> matrix
> >> >> >>>> >> algebra (released under lgpl)...
> >> >> >>>> >>
> >> >> >>>> >> Due to GPL dependencies, it won't be possible
to release the
> >> code
> >> >> as
> >> >> >>>> > Apache
> >> >> >>>> >> license for now...If we get good results
on our usecases, we
> >> will
> >> >> >>>> plan to
> >> >> >>>> >> write a version in breeze/modify joptimizer
for sparse ccs
> >> >> >>>> operations...
> >> >> >>>> >>
> >> >> >>>> >> I derived ConstrainedALS from Spark mllib
ALS and I am
> comparing
> >> >> the
> >> >> >>>> >> performance with default ALS and nonnegative
ALS as
> baseline.
> >> Plan
> >> >> >>>> is to
> >> >> >>>> >> release the code as GPL license for community
review...I have
> >> kept
> >> >> the
> >> >> >>>> >> package structure as org.apache.spark.mllib.recommendation
> >> >> >>>> >>
> >> >> >>>> >> There are some private functions defined
in ALS, which I
> would
> >> >> like to
> >> >> >>>> >> reuse....Is it possible to take the private
out from the
> >> following
> >> >> >>>> >> functions:
> >> >> >>>> >>
> >> >> >>>> >> 1. makeLinkRDDs
> >> >> >>>> >> 2. makeInLinkBlock
> >> >> >>>> >> 3. makeOutLinkBlock
> >> >> >>>> >> 4. randomFactor
> >> >> >>>> >> 5. unblockFactors
> >> >> >>>> >>
> >> >> >>>> >> I don't want to copy any code.... I can ask
for a PR to make
> >> these
> >> >> >>>> >> changes...
> >> >> >>>> >>
> >> >> >>>> >> Thanks.
> >> >> >>>> >> Deb
> >> >> >>>>
> >> >> >>>
> >> >> >>>
> >> >> >>
> >> >>
> >>
>
