Hi Xiangrui, I did some out-of-box 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, 2012-2014. OPTIMAL (within feastol=1.0e-05, reltol=1.0e-06, abstol=1.0e-06). Runtime: 0.010340 seconds. -------------------------------------------------------- pdco.m Version of 23 Nov 2013 Primal-dual 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, 2012-2014. OPTIMAL (within feastol=1.0e-05, reltol=1.0e-06, abstol=1.0e-06). Runtime: 6.333036 seconds. -------------------------------------------------------- pdco.m Version of 23 Nov 2013 Primal-dual 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 netlib-java jniloader). I will push the jnilib as a PR to ecos repository https://github.com/ifa-ethz/ecos Thanks. Deb On Wed, Jul 2, 2014 at 1:52 AM, Xiangrui Meng <men...@gmail.com> wrote: > Hi Deb, > > KNITRO and MOSEK are both commercial. If you are looking for > open-source 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.da...@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 <men...@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.da...@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 <men...@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.da...@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 least-squares problem for each user and return the new > >> >> feature > >> >> > vectors > >> >> > > >> >> > Array.range(0, numUsers).map { index => > >> >> > > >> >> > // Compute the full XtX matrix from the lower-triangular > 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 > >> >> > positive-definite > >> >> > > >> >> > 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.da...@gmail.com> > >> >> > wrote: > >> >> > > >> >> >> Hi, > >> >> >> > >> >> >> I am bit confused wiht the code here: > >> >> >> > >> >> >> // Solve the least-squares problem for each user and return the > new > >> >> >> feature vectors > >> >> >> > >> >> >> Array.range(0, numUsers).map { index => > >> >> >> > >> >> >> // Compute the full XtX matrix from the lower-triangular > 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 > >> >> >> positive-definite > >> >> >> > >> >> >> 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.da...@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 <men...@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 > >> >> >>>> non-negative 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.da...@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 non-negative > 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.da...@gmail.com > >> >> >>>> > > >> >> >>>> > wrote: > >> >> >>>> >> Hi, > >> >> >>>> >> > >> >> >>>> >> We are adding a constrained ALS solver in Spark to solve > matrix > >> >> >>>> >> factorization use-cases 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 use-cases, 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 non-negative 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 > >> >> >>>> > >> >> >>> > >> >> >>> > >> >> >> > >> >> > >> >
