# Gradient descent, anything better than golden section line search

1. May 31, 2010

### alecrimi

Hi
This is a long story, I make it short:
I am working in a project where I need to find a matrix defined by a third degree polynomial, the solution can be found iteratively using a gradient descent technique, I am using the golden section line search already implemented in matlab (with the code described below).
The algorithm looks powerful (it finds automatically the perfect step), but unfortunately the golden section line search does not avoid being stuck in local minima. How can I implement more efficiently this. The problem is that sometime it converges and sometimes no (sometimes is not science :-).

%Initial third degree polynomial
Cest = initial_guess;
normdev=inf ;
stepsize=inf;

%Stopping condition
stopping_condition = 10^(-5) *norm(X*X'/no_samples,'fro');

while abs(normdev*stepsize) > stopping_condition
%Third degree polynomial
dnew = Cest - 1/no_samples*(X*X' - 2/sigma^2 * (Cest*Cest'*Cest-Cest*B'*Cest));
%Find the best stepsize as a minimum using the goldensection line search
stepsize = fminbnd( @(stepsize) step(stepsize,Cest,dnew,X*X',B,sigma,no_samples),-.1,.1);

%Update
Cest = Cest + stepsize*dnew;
normdev = norm(dnew,'fro');
end

function error = step(stepsize,Cest,dnew,XX,B,sigma,no_samples)
Cest = Cest + stepsize*dnew;
error = norm(Cest - 1/no_samples*(XX - 2/sigma^2 * (Cest^3-Cest*B*Cest)),'fro');

I tried :
%Quasi-Newton
stepsize = fminunc( @(stepsize) step(stepsize,Cest,dnew,X*X',B,sigma,no_samples),dnew);
But matlab get stuck (no heap memory) probably due the fact that this function is suppose to be used when we don't have a trust region.

Any suggestions ?

2. Jun 2, 2010

### alecrimi

So I moved to a stochastic approach (gradient descent approaches find only local minima, stochastic approaches can find global minima) like simulated annealing, now the problem is that with my objective function the default convergence never happens.
So I have to force annealing to stop. BUT WHEN ?
I tried after some iteration but the annealing has the defect that at some point it randomize his trusted region and maybe when I stop it, it is really far from the minima (I am using too many iterations). And I have the same problem with time limit. So when I should stop the minima the search ?

Code using the simulated annealing:
Cest = initial_point;

lb = -20*ones(size(Cest));
ub = 20*ones(size(Cest));

options = saoptimset('MaxIter',300) ;

Cest = simulannealbnd(@(funcobj) myfuncobj(Cest ,X*X',B,sigma,no_samples),Cest,lb,ub, options);

function error = myfuncobj( Cest, XX, B, sigma, no_samples)
error = norm(Cest - 1/no_samples*(XX - 2/sigma^2 * (Cest^3-Cest*B*Cest)),'fro');