Find Minimum Quadratic and Cubic.
This works similarly to fminunc, but is more efficient when we are dealing
with large number of parameters.
[theta, fX, i] = fmincg(@(t)(Cost(theta, X, y, lambda)), guess,
options)
Where:

Cost is a function which returns cost and a vector of partial derivatives.

theta are the parameters being found.

X is a matrix of m rows of training data, where each row is a vector of n
input values. n can be in the hundreds on a modern PC.

y is a vector of m correct values

lambda is a multiplier for regularizing the parameters.

guess is a starting set of parameters n wide

options is a optimset. e.g. optimset('MaxIter', 50);
Returns the best theta found and optionally function values "fX" indicating
the progress made and "i" the number of iterations. For a use example see
the Logistic Classifier
function [X, fX, i] = fmincg(f, X, options, P1, P2, P3, P4, P5)
% Minimize a continuous differentialble multivariate function. Starting point
% is given by "X" (D by 1), and the function named in the string "f", must
% return a function value and a vector of partial derivatives. The Polack
% Ribiere flavour of conjugate gradients is used to compute search directions,
% and a line search using quadratic and cubic polynomial approximations and the
% WolfePowell stopping criteria is used together with the slope ratio method
% for guessing initial step sizes. Additionally a bunch of checks are made to
% make sure that exploration is taking place and that extrapolation will not
% be unboundedly large. The "length" gives the length of the run: if it is
% positive, it gives the maximum number of line searches, if negative its
% absolute gives the maximum allowed number of function evaluations. You can
% (optionally) give "length" a second component, which will indicate the
% reduction in function value to be expected in the first linesearch (defaults
% to 1.0). The function returns when either its length is up, or if no further
% progress can be made (ie, we are at a minimum, or so close that due to
% numerical problems, we cannot get any closer). If the function terminates
% within a few iterations, it could be an indication that the function value
% and derivatives are not consistent (ie, there may be a bug in the
% implementation of your "f" function). The function returns the found
% solution "X", a vector of function values "fX" indicating the progress made
% and "i" the number of iterations (line searches or function evaluations,
% depending on the sign of "length") used.
%
% Usage: [X, fX, i] = fmincg(f, X, options, P1, P2, P3, P4, P5)
%
%
See also:
checkgrad
%
% Copyright (C) 2001 and 2002 by Carl Edward Rasmussen. Date 20020213
%
%
% (C) Copyright 1999, 2000 & 2001, Carl Edward Rasmussen
%
% Permission is granted for anyone to copy, use, or modify these
% programs and accompanying documents for purposes of research or
% education, provided this copyright notice is retained, and note is
% made of any changes that have been made.
%
% These programs and documents are distributed without any warranty,
% express or implied. As the programs were written for research
% purposes only, they have not been tested to the degree that would be
% advisable in any important application. All use of these programs is
% entirely at the user's own risk.
%
% [mlclass] Changes Made:
% 1) Function name and argument specifications
% 2) Output display
%
% Read options
if exist('options', 'var') && ~isempty(options) && isfield(options, 'MaxIter')
length = options.MaxIter;
else
length = 100;
end
RHO = 0.01; % a bunch of constants for line searches
SIG = 0.5; % RHO and SIG are the constants in the WolfePowell conditions
INT = 0.1; % don't reevaluate within 0.1 of the limit of the current bracket
EXT = 3.0; % extrapolate maximum 3 times the current bracket
MAX = 20; % max 20 function evaluations per line search
RATIO = 100; % maximum allowed slope ratio
argstr = ['feval(f, X']; % compose string used to call function
for i = 1:(nargin  3)
argstr = [argstr, ',P', int2str(i)];
end
argstr = [argstr, ')'];
if max(size(length)) == 2, red=length(2); length=length(1); else red=1; end
S=['Iteration '];
i = 0; % zero the run length counter
ls_failed = 0; % no previous line search has failed
fX = [];
[f1 df1] = eval(argstr); % get function value and gradient
i = i + (length<0); % count epochs?!
s = df1; % search direction is steepest
d1 = s'*s; % this is the slope
z1 = red/(1d1); % initial step is red/(s+1)
while i < abs(length) % while not finished
i = i + (length>0); % count iterations?!
X0 = X; f0 = f1; df0 = df1; % make a copy of current values
X = X + z1*s; % begin line search
[f2 df2] = eval(argstr);
i = i + (length<0); % count epochs?!
d2 = df2'*s;
f3 = f1; d3 = d1; z3 = z1; % initialize point 3 equal to point 1
if length>0, M = MAX; else M = min(MAX, lengthi); end
success = 0; limit = 1; % initialize quanteties
while 1
while ((f2 > f1+z1*RHO*d1)  (d2 > SIG*d1)) && (M > 0)
limit = z1; % tighten the bracket
if f2 > f1
z2 = z3  (0.5*d3*z3*z3)/(d3*z3+f2f3); % quadratic fit
else
A = 6*(f2f3)/z3+3*(d2+d3); % cubic fit
B = 3*(f3f2)z3*(d3+2*d2);
z2 = (sqrt(B*BA*d2*z3*z3)B)/A; % numerical error possible  ok!
end
if isnan(z2)  isinf(z2)
z2 = z3/2; % if we had a numerical problem then bisect
end
z2 = max(min(z2, INT*z3),(1INT)*z3); % don't accept too close to limits
z1 = z1 + z2; % update the step
X = X + z2*s;
[f2 df2] = eval(argstr);
M = M  1; i = i + (length<0); % count epochs?!
d2 = df2'*s;
z3 = z3z2; % z3 is now relative to the location of z2
end
if f2 > f1+z1*RHO*d1  d2 > SIG*d1
break; % this is a failure
elseif d2 > SIG*d1
success = 1; break; % success
elseif M == 0
break; % failure
end
A = 6*(f2f3)/z3+3*(d2+d3); % make cubic extrapolation
B = 3*(f3f2)z3*(d3+2*d2);
z2 = d2*z3*z3/(B+sqrt(B*BA*d2*z3*z3)); % num. error possible  ok!
if ~isreal(z2)  isnan(z2)  isinf(z2)  z2 < 0 % num prob or wrong sign?
if limit < 0.5 % if we have no upper limit
z2 = z1 * (EXT1); % the extrapolate the maximum amount
else
z2 = (limitz1)/2; % otherwise bisect
end
elseif (limit > 0.5) && (z2+z1 > limit) % extraplation beyond max?
z2 = (limitz1)/2; % bisect
elseif (limit < 0.5) && (z2+z1 > z1*EXT) % extrapolation beyond limit
z2 = z1*(EXT1.0); % set to extrapolation limit
elseif z2 < z3*INT
z2 = z3*INT;
elseif (limit > 0.5) && (z2 < (limitz1)*(1.0INT)) % too close to limit?
z2 = (limitz1)*(1.0INT);
end
f3 = f2; d3 = d2; z3 = z2; % set point 3 equal to point 2
z1 = z1 + z2; X = X + z2*s; % update current estimates
[f2 df2] = eval(argstr);
M = M  1; i = i + (length<0); % count epochs?!
d2 = df2'*s;
end % end of line search
if success % if line search succeeded
f1 = f2; fX = [fX' f1]';
fprintf('%s %4i  Cost: %4.6e\r', S, i, f1);
s = (df2'*df2df1'*df2)/(df1'*df1)*s  df2; % PolackRibiere direction
tmp = df1; df1 = df2; df2 = tmp; % swap derivatives
d2 = df1'*s;
if d2 > 0 % new slope must be negative
s = df1; % otherwise use steepest direction
d2 = s'*s;
end
z1 = z1 * min(RATIO, d1/(d2realmin)); % slope ratio but max RATIO
d1 = d2;
ls_failed = 0; % this line search did not fail
else
X = X0; f1 = f0; df1 = df0; % restore point from before failed line search
if ls_failed  i > abs(length) % line search failed twice in a row
break; % or we ran out of time, so we give up
end
tmp = df1; df1 = df2; df2 = tmp; % swap derivatives
s = df1; % try steepest
d1 = s'*s;
z1 = 1/(1d1);
ls_failed = 1; % this line search failed
end
if exist('OCTAVE_VERSION')
fflush(stdout);
end
end
fprintf('\n');
file: /Techref/method/ai/fmincg.htm, 9KB, , updated: 2015/8/2 18:05, local time: 2021/4/17 01:13,

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