fminunc can't solve problem

I'm trying to build a softmax classifier and I can't seem to compute the weight vectors using fminunc.

W_0 = zeros(784,10);
O = @(W,j) softmax(X*W(:, j));
minus_log = @(W,j) -log(O(W,j));
H = @(W,j) sum((minus_log(W,j) .* Y(:,j))');
J = @(W,j) 1/n * sum(H(W,j));
weights = zeros(784, 10);
for j=1:10
    wj = fminunc(@(W)J(W,j), zeros(784,1));
    weights(:,j) = wj;
end

X is n by 784 and Y is n by 10. fminunc ends with the 'Index exceeds matrix dimensions.' error.

[Walter Roberson]

n and X are undefined. n seems to be a scalar, but I do not know what size X is supposed to be.

[Stefan Nastase]

n is 60000, it is defined before the code I've posted.

[John D'Errico]

So it is not that fminunc cannot solve the problem, but that you wrote the code improperly to use fminunc.

Before you throw something into an optimizer, first, verify that your objective function works! Test the objective. When you pass in a vector of size 784x1, does it return a scalar value? Or does your code fail with an error?

[Stefan Nastase]

when I plug in a 784x1 vector, it does return a scalar

[Walter Roberson]

What size is X ?

[Stefan Nastase]

X is 60000x784

[Walter Roberson]

How long does the code run before reporting the problem? What is j when the problem is encounted?

I am running the code with X = rand(n, 784); Y = rand(n, 10); and it is processing through... fminunc with 784 variables should be expected to be slow.

[Stefan Nastase]

after about 1 minute I get the error, I tried getting rid of the for loop, and passing the whole W = [w1 | w2 | ... ] as parameter to fminunc, but it just loops forever

[Walter Roberson]

Would it be possible for you to attach the X and Y in a .mat file? Perhaps the data makes a difference.

Also, which MATLAB release are you using?

[Stefan Nastase]

@Walter Roberson I am using R2016a. I can't add the X data here since it is too large.

Answer 1

[Walter Roberson]

Look at your code:

    wj = fminunc(@(W)J(W,j), zeros(784,1));

The size of W that will be passed in will be 784 x 1.

    J = @(W,j) 1/n * sum(H(W,j));

J will take that and pass it to H

    H = @(W,j) sum((minus_log(W,j) .* Y(:,j))');

H will take it and pass it to minus_log

    minus_log = @(W,j) -log(O(W,j));

minus_log will take it and pass it to O

    O = @(W,j) softmax(X*W(:, j));

O will take it and try to index it at column j. But W only has one column.

[Walter Roberson]

n = 60000;
M = 784;
maxj = 10;
load Y

X = rand(n, M);

W_0 = zeros(M, maxj);
O = @(W) softmax(X*W);
minus_log = @(W) -log(O(W));
H = @(W,j) sum((minus_log(W) .* Y(:,j)), 2);
J = @(W,j) 1/n * sum(H(W,j));
weights = zeros(M, maxj);

options = optimoptions('fminunc','Algorithm','quasi-newton');

for j=1:maxj
    wj = fminunc(@(W)J(W,j), zeros(M,1), options);
    weights(:,j) = wj;
end

[Stefan Nastase]

It returns 0 always, I get a problem with the dimensions here :

H = @(W,j) sum((minus_log(W) .* Y(:,j)), 2);

minus_log returns a 60000x10 matrix, while Y(:,j) is a 60000x1, in MATLAB R2016a, this operation does not work using .* operator.

[Stefan Nastase]

Maybe I defined the functions wrong?

Here is the function I tried to write:

where T is the same as Y.

[Stefan Nastase]

This is a correct written cost function. I tried to compute the cost with a W matrix I found using GD. It is indeed pretty low.

O = @(W) softmax(X*W);
minus_log = @(W,j) -log(O(W)) *e(j);
H = @(W,j) sum((minus_log(W,j) .* Y(:,j)), 2);
J = @(W,j) 1/n * sum(H(W,j));

But I get the same problem using fminunc, it loops forever.

[Walter Roberson]

Please confirm your release is R2016a (as you indicated before) and not R2015a.

60000x10 matrix .* 60000x1 matrix is valid starting from R2015b.

[Walter Roberson]

e is not defined for your e(j) .

On my system, without the e(j), in R2016b, the code is not exactly fast, but it does run to completion; progress can be traced by the occasional message similar to

Local minimum found.

Optimization completed because the size of the gradient is less than
the default value of the optimality tolerance.

<stopping criteria details>

which happens once for each value of j.

I am still using random X. Perhaps you could put your X into Google Drive or Dropbox and post the link to it?

[Stefan Nastase]

Yes, my version is R2016a. Here is e.

e = @(k) [zeros(k-1, 1); 1; zeros(10-k, 1)];

https://drive.google.com/file/d/1-WB7Tn6Qx6FL0AX5Ph8QC2IlOMMuqzV6/view?usp=sharing

Here is the X matrix.

[Walter Roberson]

e = @(k) [zeros(k-1, 1); 1; zeros(10-k, 1)];

e is going to give back a 10 x 1 vector. You use * with e(j) which would be something suitable for selecting a single column from a something by 10 matrix. But W is 748 x 1 not 784 x 10.

[Stefan Nastase]

but O is 60000 x 10, because W is 748 x 10, i tried to plug the whole matrix as a parameter in fminunc

[Walter Roberson]

    wj = fminunc(@(W)J(W,j), zeros(M,1), options);

so W will be 784 x 1. It will be passed into J

    J = @(W,j) 1/n * sum(H(W,j));

and is still 784 x 1. It will be passed into H

    H = @(W,j) sum((minus_log(W,j) .* Y(:,j)), 2);

and will still be 784 x 1. It will be passed into minus_log

    minus_log = @(W,j) -log(O(W)) *e(j);

and will still be 784 x 1. It will be passed to O

    O = @(W) softmax(X*W);

X is 60000 x 784, and you * it with W that is 784 x 1, giving you a 60000 x 1 result. That 60000 x 1 will be passed to softmax() which will return a 60000 x 1 array (the same size).

That 60000 x 1 will be returned from O, and will be * by e(j)

    e = @(k) [zeros(k-1, 1); 1; zeros(10-k, 1)];

e(j) returns a 10 x 1 vector.

So minus_log will be a 60000 x 1 * a 10 x 1 . That is an error.

I would suggest that as the e(j) is obviously intended to select one column of a matrix that only has one column, that you could just leave out the e(j)

[Walter Roberson]

Note: without the e(j), each j iteration is taking about 28 minutes on my system, for a total of about 4 2/3 hours run time.

Ah, I just noticed that it is running into a limit on the number of function evaluations. I will need to adjust the options for that.

[Stefan Nastase]

Thanks a lot. Could you provide me the W matrix you obtained?

[Walter Roberson]

I increased the maximum iterations and maximum function evaluations to 1e6, and did one iteration. Literally just as I started typing this message, it finished the one iteration, after 14197 seconds -- just short of four hours (for one iteration)

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[Stefan Nastase]

Thank you a lot for your help!

[Walter Roberson]

I tried running with fminsearch() . I killed it off after about 7 hours... I could see from the output that descent was pretty slow.