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:
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');
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