man Algorithm::LBFGS (3): Perl extension for L-BFGS

SYNOPSIS

use Algorithm::LBFGS;
# create an L-BFGS optimizer
my $o = Algorithm::LBFGS->new;
# f(x) = (x1 - 1)^2 + (x2 + 2)^2
# grad f(x) = (2 * (x1 - 1), 2 * (x2 + 2));
my $eval_cb = sub {
my $x = shift;
my $f = ($x->[0] - 1) * ($x->[0] - 1) + ($x->[1] + 2) * ($x->[1] + 2);
my $g = [ 2 * ($x->[0] - 1), 2 * ($x->[1] + 2) ];
return ($f, $g);
};
my $x0 = [0.0, 0.0]; # initial point
my $x = $o->fmin($eval_cb, $x0); # $x is supposed to be [ 1, -2 ];

DESCRIPTION

L-BFGS (Limited-memory Broyden-Fletcher-Goldfarb-Shanno) is a quasi-Newton method for unconstrained optimization. This method is especially efficient on problems involving a large number of variables.

Generally, it solves a problem described as following:

  min f(x), x = (x1, x2, ..., xn)

Jorge Nocedal wrote a Fortran 77 version of this algorithm.

<http://www.ece.northwestern.edu/~nocedal/lbfgs.html>

And, Naoaki Okazaki rewrote it in pure C (liblbfgs).

<http://www.chokkan.org/software/liblbfgs/index.html>

This module is a Perl port of Naoaki Okazaki's C version.

new

"new" creates a L-BFGS optimizer with given parameters.

  my $o1 = new Algorithm::LBFGS(m => 5);
  my $o2 = new Algorithm::LBFGS(m => 3, eps => 1e-6);
  my $o3 = new Algorithm::LBFGS;

If no parameter is specified explicitly, their default values are used.

The parameter can be changed after the creation of the optimizer by ``set_param''. Also, they can be queryed by ``get_param''.

Please refer to the ``List of Parameters'' for details about parameters.

get_param

Query the value of a parameter.

   my $o = Algorithm::LBFGS->new;
   print $o->get_param('epsilon'); # 1e-5

set_param

Change the values of one or several parameters.

   my $o = Algorithm::LBFGS->new;
   $o->set_param(epsilon => 1e-6, m => 7);

fmin

The prototype of ``fmin'' is like

  x = fmin(evaluation_cb, x0, progress_cb, user_data)

As the name says, it finds a vector x which minimize the function f(x).

``evaluation_cb'' is a ref to the evaluation callback subroutine, ``x0'' is the initial point of the optimization algorithm, ``progress_cb'' (optional) is a ref to the progress callback subroutine, and ``user_data'' (optional) is a piece of extra data that client program want to pass to both ``evaluation_cb'' and ``progress_cb''.

Client program can use ``get_status'' to find if any problem occured during the optimization after their calling ``fmin''. When the status is ``LBFGS_OK'', the returning value "x" (array ref) contains the optimized variables, otherwise, there may be some problems occured and the value in the returning "x" is undefined.

evaluation_cb

The ref to the evaluation callback subroutine.

The evaluation callback subroutine is supposed to calculate the function value and gradient vector at a specified point "x". It is called automatically by ``fmin'' when an evaluation is needed.

The client program need to make sure their evaluation callback subroutine has a prototype like

  (f, g) = evaluation_cb(x, step, user_data)

"x" (array ref) is the current values of variables, "step" is the current step of the line search routine, ``user_data'' is the extra user data specified when calling ``fmin''.

The evaluation callback subroutine is supposed to return both the function value "f" and the gradient vector "g" (array ref) at current "x".

The initial point of the optimization algorithm. The final result may depend on your choice of "x0".

NOTE: The content of "x0" will be modified after calling ``fmin''. When the algorithm terminates successfully, the content of "x0" will be replaced by the optimized variables, otherwise, the content of "x0" is undefined.

progress_cb

The ref to the progress callback subroutine.

The progress callback subroutine is called by ``fmin'' at the end of each iteration, with information of current iteration. It is very useful for a client program to monitor the optimization progress.

The client program need to make sure their progress callback subroutine has a prototype like

  s = progress_cb(x, g, fx, xnorm, gnorm, step, k, ls, user_data)

"x" (array ref) is the current values of variables. "g" (array ref) is the current gradient vector. "fx" is the current function value. "xnorm" and "gnorm" is the L2 norm of "x" and "g". "step" is the line-search step used for this iteration. "k" is the iteration count. "ls" is the number of evaluations in this iteration. ``user_data'' is the extra user data specified when calling ``fmin''.

The progress callback subroutine is supposed to return an indicating value "s" for ``fmin'' to decide whether the optimization should continue or stop. "fmin" continues to the next iteration when "s=0", otherwise, it terminates with status code ``LBFGSERR_CANCELED''.

The client program can also pass string values to ``progress_cb'', which means it want to use a predefined progress callback subroutine. There are two predefined progress callback subroutines, 'verbose' and 'logging'. 'verbose' just prints out all information of each iteration, while 'logging' logs the same information in an array ref provided by ``user_data''.

  ...
  # print out the iterations
  fmin($eval_cb, $x0, 'verbose'); 
  # log iterations information in the array ref $log
  my $log = [];
  fmin($eval_cb, $x0, 'logging', $log);
  
  use Data::Dumper;
  print Dumper $log;

user_data

The extra user data. It will be sent to both ``evaluation_cb'' and ``progress_cb''.

get_status

Get the status of previous call of ``fmin''.

  ...
  $o->fmin(...);
  # check the status
  if ($o->get_status eq 'LBFGS_OK') {
     ...
  }
  # print the status out
  print $o->get_status;

The status code is a string, which could be one of those in the ``List of Status Codes''.

status_ok

This is a shortcut of saying ``get_status'' eq ``LBFGS_OK''.

  ...
  if ($o->fmin(...), $o->status_ok) {
      ...
  }

List of Parameters

The number of corrections to approximate the inverse hessian matrix.

The L-BFGS algorithm stores the computation results of previous ``m'' iterations to approximate the inverse hessian matrix of the current iteration. This parameter controls the size of the limited memories (corrections). The default value is 6. Values less than 3 are not recommended. Large values will result in excessive computing time.

epsilon

Epsilon for convergence test.

This parameter determines the accuracy with which the solution is to be found. A minimization terminates when

  ||grad f(x)|| < epsilon * max(1, ||x||)

where ||.|| denotes the Euclidean (L2) norm. The default value is 1e-5.

max_iterations

The maximum number of iterations.

The L-BFGS algorithm terminates an optimization process with ``LBFGSERR_MAXIMUMITERATION'' status code when the iteration count exceedes this parameter. Setting this parameter to zero continues an optimization process until a convergence or error. The default value is 0.

max_linesearch

The maximum number of trials for the line search.

This parameter controls the number of function and gradients evaluations per iteration for the line search routine. The default value is 20.

min_step

The minimum step of the line search routine.

The default value is 1e-20. This value need not be modified unless the exponents are too large for the machine being used, or unless the problem is extremely badly scaled (in which case the exponents should be increased).

max_step

The maximum step of the line search.

The default value is 1e+20. This value need not be modified unless the exponents are too large for the machine being used, or unless the problem is extremely badly scaled (in which case the exponents should be increased).

ftol

A parameter to control the accuracy of the line search routine.

The default value is 1e-4. This parameter should be greater than zero and smaller than 0.5.

gtol

A parameter to control the accuracy of the line search routine.

The default value is 0.9. If the function and gradient evaluations are inexpensive with respect to the cost of the iteration (which is sometimes the case when solving very large problems) it may be advantageous to set this parameter to a small value. A typical small value is 0.1. This parameter shuold be greater than the ftol parameter (1e-4) and smaller than 1.0.

xtol

The machine precision for floating-point values.

This parameter must be a positive value set by a client program to estimate the machine precision. The line search routine will terminate with the status code (``LBFGSERR_ROUNDING_ERROR'') if the relative width of the interval of uncertainty is less than this parameter.

orthantwise_c

Coeefficient for the L1 norm of variables.

This parameter should be set to zero for standard minimization problems. Setting this parameter to a positive value minimizes the objective function f(x) combined with the L1 norm |x| of the variables, f(x) + c|x|. This parameter is the coeefficient for the |x|, i.e., c. As the L1 norm |x| is not differentiable at zero, the module modify function and gradient evaluations from a client program suitably; a client program thus have only to return the function value f(x) and gradients grad f(x) as usual. The default value is zero.

List of Status Codes

LBFGS_OK

No error occured.

LBFGSERR_UNKNOWNERROR

Unknown error.

LBFGSERR_LOGICERROR

Logic error.

LBFGSERR_OUTOFMEMORY

Insufficient memory.

LBFGSERR_CANCELED

The minimization process has been canceled.

LBFGSERR_INVALID_N

Invalid number of variables specified.

LBFGSERR_INVALID_N_SSE

Invalid number of variables (for SSE) specified.

LBFGSERR_INVALID_MINSTEP

Invalid parameter ``max_step'' specified.

LBFGSERR_INVALID_MAXSTEP

Invalid parameter ``max_step'' specified.

LBFGSERR_INVALID_FTOL

Invalid parameter ``ftol'' specified.

LBFGSERR_INVALID_GTOL

Invalid parameter ``gtol'' specified.

LBFGSERR_INVALID_XTOL

Invalid parameter ``xtol'' specified.

LBFGSERR_INVALID_MAXLINESEARCH

Invalid parameter ``max_linesearch'' specified.

LBFGSERR_INVALID_ORTHANTWISE

Invalid parameter ``orthantwise_c'' specified.

LBFGSERR_OUTOFINTERVAL

The line-search step went out of the interval of uncertainty.

LBFGSERR_INCORRECT_TMINMAX

A logic error occurred; alternatively, the interval of uncertainty became too small.

LBFGSERR_ROUNDING_ERROR

A rounding error occurred; alternatively, no line-search step satisfies the sufficient decrease and curvature conditions.

LBFGSERR_MINIMUMSTEP

The line-search step became smaller than ``min_step''.

LBFGSERR_MAXIMUMSTEP

The line-search step became larger than ``max_step''.

LBFGSERR_MAXIMUMLINESEARCH

The line-search routine reaches the maximum number of evaluations.

LBFGSERR_MAXIMUMITERATION

The algorithm routine reaches the maximum number of iterations.

LBFGSERR_WIDTHTOOSMALL

Relative width of the interval of uncertainty is at most ``xtol''.

LBFGSERR_INVALIDPARAMETERS

A logic error (negative line-search step) occurred.

LBFGSERR_INCREASEGRADIENT

The current search direction increases the objective function value.

AUTHOR

Laye Suen, <[email protected]>

COPYRIGHT AND LICENSE

This library is distributed under the term of the MIT license.

<http://opensource.org/licenses/mit-license.php>

REFERENCE

J. Nocedal. Updating Quasi-Newton Matrices with Limited Storage (1980) , Mathematics of Computation 35, pp. 773-782.
D.C. Liu and J. Nocedal. On the Limited Memory Method for Large Scale Optimization (1989), Mathematical Programming B, 45, 3, pp. 503-528.
Jorge Nocedal's Fortran 77 implementation, <http://www.ece.northwestern.edu/~nocedal/lbfgs.html>
Naoaki Okazaki's C implementation (liblbfgs), <http://www.chokkan.org/software/liblbfgs/index.html>