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 Post subject: High-dimensional QP problem yields unreliable resultsPosted: Wed Mar 07, 2018 9:46 am

Joined: Thu Jan 25, 2018 7:38 am
Posts: 11
Hi,

Numeric instability of problem? Double precision insufficient? Does AlgLib have a quadruple precision version?

An SVM problem with 770 training points yields a 770-dimensional QP problem.

Using DenseAU, with a variety of scaling factors - the problem converges - yet the solution to the QP problem doesn't provide a good solution to the SVM one. In addition, when reformulating the problem only with the training points for which the Lagarangian multipliers were non-zero (a great deal of them are, as expected, zero - the non-zero ones are points on the boundaries of two hyper-planes that separate the "1"s from the "-1" ones) - the solution comes out completely different...

On the other hand, when taking a subset of the training points (every other point from a continuum of points - 385 points) - the QP problem converges and the sub-problem (with the points for which the Lagrangian multipliers were non-zero) provides nearly the exact same solution, and the sub-SVM problem is solved exactly - and the sub-solution nearly resolves the full SVM problem.

Numeric instability of problem? Double precision insufficient? Does AlgLib have a quadruple precision version?
Any suggestions would be appreciated...

Thanks,

Zev

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 Post subject: Re: High-dimensional QP problem yields unreliable resultsPosted: Wed Mar 07, 2018 10:29 am

Joined: Fri May 07, 2010 7:06 am
Posts: 849
Can you send your code to sergey.bochkanov at alglib dot net ?

I will try to find out, whether it is problem of the solver - or something else.

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 Post subject: Re: High-dimensional QP problem yields unreliable resultsPosted: Tue Mar 13, 2018 11:33 am

Joined: Thu Jan 25, 2018 7:38 am
Posts: 11
Hi,

Providing the data is impossible with the 60K limit and no attachments...

The AlgLib part of my application does nothing different than the demo code - see below.
'ir_svm' just creates the (symmetric) N-D Hessian and (N+1)-D constraints matrices etc.

The constraints are: all Xi > 0 and sigma(Xi * Yi, i=1..N) = 0,
where N, the number of training points, was at most 770 - but might need to be larger,
Xi, i=1..N, are the LaGarange multipliers - and the QP unknowns,
where Yi, i=1..N, are either 1 or -1 according to Ti being a 'true' or 'false' training point.
Scales were set uniformly for all N - as 0.1, 1, 10, 100, 1000 - with no appreciable effect.
No initial guess for Xi is available - so initialized with ones.

The QP having converged successfully, W, defining the the M-dimensional (58) hyperplane separating the 'true' training points from the 'false' ones can be calculated - and a scalar b such that
(1)
<w . Ti> - b should be >= 1 for 'true' M-dimensional Training points and
<w . Ti> - b should be <= -1 for 'false' Training points;
<w . Ti> here is the inner product between the two M-dimensional vectors.

The distance between the two sets of training points is 2 / ||W||.

The result of the QP yields small but non-zero distance between the two sets -
but the conditions (1) don't hold correctly - the training points for which the Lagrange multiplies came out identically zero full fill - not (1) - rather, in most cases, a weaker version
(2)
<w . Ti> - b should be >= f1 for 'true' Training points and
<w . Ti> - b should be <= -f2 for 'false' Training points;
where 0< f1 & f2 < 1
- and in isolated cases not even (2) holds...

int _tmain(int argc, _TCHAR* argv[])
{
try
{
double *pa, *pb, *pc, *ps, *px;
int *pct;

int N = ir_svm(&pa, &pb, &pc, &pct, &ps, &px);

real_2d_array a;
a.setcontent(N, N, pa);

real_1d_array b;
b.setcontent(N, pb);

real_1d_array s;
s.setcontent(N, ps);

real_2d_array c;
c.setcontent(N + 1, N + 1, pc);

integer_1d_array ct;
ct.setcontent(N + 1, pct);

real_1d_array x;
x.setcontent(N, px);

minqpstate state;
minqpreport rep;

// create solver, set quadratic/linear terms
minqpcreate(N, state);
minqpsetlinearterm(state, b);
minqpsetlc(state, c, ct);

// NOTE: for convex problems you may try using minqpsetscaleautodiag()
// which automatically determines variable scales.
minqpsetscale(state, s);

//
// Solve problem with BLEIC-based QP solver.
//
// This solver is intended for problems with moderate (up to 50) number
// of general linear constraints and unlimited number of box constraints.
//
// Default stopping criteria are used.
//

//minqpsetalgobleic(state, 0.0, 0.0, 0.0, 0);
minqpsetalgodenseaul(state, 1.0e-15, 5.0e+4, 10);

minqpoptimize(state);
minqpresults(state, x, rep);
printf("inneriterationscount %u\n"
"ncholesky %u, nmv %u\n"
"outeriterationscount %u\n"
"terminationtype %d\n"
, rep.inneriterationscount, rep.ncholesky, rep.nmv, rep.outeriterationscount, rep.terminationtype);
}
catch(ap_error)
{
printf("Exception!\n");
getchar();
}
return 0;
}

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 Post subject: Re: High-dimensional QP problem yields unreliable resultsPosted: Wed Mar 14, 2018 10:32 am

Joined: Thu Jan 25, 2018 7:38 am
Posts: 11
Hi,

Here is an SVM problem with 8 training points - 4 'true' and 4 'false'
that gets formulated as the following QP problem:
Hessian (A)
0 1 2 3 4 5 6 7
0 9008.04 8692.77 6441.76 5085.08 -6532.93 -6773.31 -7054.06 -7549.81
1 8692.77 8406.34 6238.24 4929.02 -6087.81 -6389.49 -6718.46 -7243.83
2 6441.76 6238.24 4639.70 3671.38 -4409.45 -4668.51 -4938.08 -5348.06
3 5085.08 4929.02 3671.38 2918.12 -3363.90 -3608.07 -3826.17 -4142.27
4 -6532.93 -6087.81 -4409.45 -3363.90 8897.43 7848.32 7156.12 6844.05
5 -6773.31 -6389.49 -4668.51 -3608.07 7848.32 7433.34 7016.67 6873.04
6 -7054.06 -6718.46 -4938.08 -3826.17 7156.12 7016.67 6897.34 7022.35
7 -7549.81 -7243.83 -5348.06 -4142.27 6844.05 6873.04 7022.35 7418.02

Constraints (C) + (CT)
alpha0 alpha1 alpha2 alpha3 alpha4 alpha5 alpha6 alpha7 right side type
0 1 0 0 0 0 0 0 0 0 1
1 0 1 0 0 0 0 0 0 0 1
2 0 0 1 0 0 0 0 0 0 1
3 0 0 0 1 0 0 0 0 0 1
4 0 0 0 0 1 0 0 0 0 1
5 0 0 0 0 0 1 0 0 0 1
6 0 0 0 0 0 0 1 0 0 1
7 0 0 0 0 0 0 0 1 0 1
8 1 1 1 1 -1 -1 -1 -1 0 0

For such a small problem one might expect to get a more exact SVM solution than:
<Yi> <W . Ti> - b SVM Decision
0 1 1.123992 1
1 1 1.200366 1
2 1 1.123993 1
3 1 1.183616 1
4 -1 -1.579777 -1
5 -1 -1.197507 -1
6 -1 -0.978675 -1
7 -1 -0.876008 -1

where, as described in the previous post, W & b define the separating hyperplane.
It is expected that <W.Ti> - b should have exact 1 and -1 results for some of the training points (those defining the region boundaries - those for which the LaGrange multipliers came out non zero).
Not only does this not occur with reasonable accuracy - but for the 'false' points we have values greater than -1...

This result persists even with hundreds of iterations and various scalings - here globally 1.e4 .

Does this QP problem look so inherently unstable as to yield such disappointing results?

Increasing the number of training points and things get much worse - complete rubbish for 685 points (although the QP problem converges)...

Zev

PS. your patience and interest in this issue is greatly appreciated - thanks!

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 Post subject: Re: High-dimensional QP problem yields unreliable resultsPosted: Thu Mar 15, 2018 10:41 am

Joined: Fri May 07, 2010 7:06 am
Posts: 849
Hi!

Thank you for providing such short and easy to debug example! I've started investigating the problem, will report as soon as first results are available!

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 Post subject: Re: High-dimensional QP problem yields unreliable resultsPosted: Thu Mar 15, 2018 10:48 am

Joined: Fri May 07, 2010 7:06 am
Posts: 849
BTW, can you report also exact values of dataset points? There are only 8 ones, it should be easy to post.

If dimension count is too large, you can send data file to sergey.bochkanov at alglib dot net

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 Post subject: Re: High-dimensional QP problem yields unreliable resultsPosted: Thu Mar 15, 2018 11:38 am

Joined: Thu Jan 25, 2018 7:38 am
Posts: 11
Hi,

I found a mathematical bug in my implementation.

Now, I get exact 1/-1 <W.Ti> - b values for the support vectors for the 8-D problem...
But for a 600-D+ problem - not...

I will increase dimensionality and see for which N things go wrong.

Thanks for your help & patience.

Zev

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 Post subject: Re: High-dimensional QP problem yields unreliable resultsPosted: Thu Mar 15, 2018 12:23 pm

Joined: Fri May 07, 2010 7:06 am
Posts: 849
Hi!

I noticed that you specified non-negativity constraints as general linear ones, with minqpsetlc() call. Thus, you have N+1 general linear constraint. However, it is suboptimal from both performance and precision points of view. Box constraints are handled in a special way, much faster and precisely than general linear ones.

I recommend you to split your constraint into box constrained part, set by minqpsetbc() call, and general linear one - just one linear constraint set with minqpsetlc() function. You will get much better performance and precision.

Another suggestion is to set non-zero lambda coefficient for the SVM problem.

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 Post subject: Re: High-dimensional QP problem yields unreliable resultsPosted: Thu Mar 15, 2018 1:03 pm

Joined: Thu Jan 25, 2018 7:38 am
Posts: 11
Hi,

Thanks for the suggestions.
- How can I use box constraints for the N "Xi >= 0"? I'll have to guess what upper boundaries to specify...
- I'm not familiar with the "non-zero lambda coefficient" - I'll look into it...

BTW, with my current formulation, the <W.Ti> - b holds with great precision for the support vectors up to N=150; for N=186 I get the expected +1/-1 only with a precision of 2 decimal places - and thereafter things diverge...

Zev

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 Post subject: Re: High-dimensional QP problem yields unreliable resultsPosted: Fri Mar 16, 2018 2:37 pm

Joined: Fri May 07, 2010 7:06 am
Posts: 849
You can specify +INF as upper bound, ALGLIB will correctly handle signed infinities in the boundary values. Or just 1.0e20, if you want so.

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