NAME
Algorithm::LBFGS - 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.
And, Naoaki Okazaki rewrote it in pure C (liblbfgs).
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" specifies the evaluation callback subroutine, "x0" is
the initial point, "progress_cb" (optional) specifies 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
Specifies 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 value of "evaluation_cb" can be either a sub ref or an integer.
When "evaluation_cb" is a sub ref, the client program need to make sure
it points to a Perl subrountine that has a prototype like
(f, g) = pl_eval_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 subroutine is supposed to return both the function value "f" and the
gradient vector "g" (array ref) at current "x".
When "evaluation_cb" is an integer, it is interpreted as the address of
an external C evaluation callback, which should has a prototype like
double c_eval_cb(
void* user_data,
const double* x,
double* g,
int n,
double step
)
The meanings of arguments are mostly as same as they are in the Perl
callback. However, as C arrays do not indicate their lengths, "n" stores
the length of both "x" and "g".
The C callback is supposed to return the function value by its returning
value and gradient vector by "g".
x0
The initial point of the optimization algorithm. The final result may
depend on your choice of "x0".
NOTE: The content of "x0" could 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
Specifies 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
the those who want to monitor the optimization progress.
The value of "progress_cb" can be either a sub ref, an integer or a
string.
When "progress_cb" is a sub ref, the client program need to make sure it
points to a Perl subroutine which has a prototype like
s = pl_prgr_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 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".
When "progress_cb" is an integer, it is interpreted as the address of an
external C progress callback which has a prototype like
int c_prgr_cb(
void* user_data,
const double* x,
const double* g,
const double fx,
const double xnorm,
const double gnorm,
const double step,
int n,
int k,
int ls
)
The meanings of arguments are mostly as same as they are in the Perl
callback, while "n" again stores the lengths of "x" and "g".
The C callback is supposed to return the same indicating value "s", too.
When "progress_cb" is a string, it chooses a predefined progress
callback subroutine. There are two predefined progress callback
subroutines, 'verbose' and 'logging'. 'verbose' just prints out some
essential informations of each iteration, while 'logging' logs them 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
m
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.
An example of external C callbacks
C callbacks if for those who care much about the performance. There are
mainly 2 ways to define a C callback, one is by XS code, the other is by
the module Inline::C.
Here we give an example implemented by Inline::C.
use Inline 'C';
my $o = Algorithm::LBFGS->new;
my $x1 = $o->fmin(f1_eval_ptr(), [6]);
# now $x1 should equal to [0]
__END__
__C__
/* f1(x) = x^2 */
double f1_eval(
void* userdata,
const double* x,
double* g,
const int n,
const double step)
{
g[0] = 2 * x[0];
return x[0] * x[0];
}
void* f1_eval_ptr() { return &f1_eval; }
SEE ALSO
PDL, PDL::Opt::NonLinear
AUTHOR
Laye Suen,
COPYRIGHT AND LICENSE
Copyright (C) 1990, Jorge Nocedal
Copyright (C) 2007, Naoaki Okazaki
Copyright (C) 2008, Laye Suen
This library is distributed under the term of the MIT license.
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,
Naoaki Okazaki's C implementation (liblbfgs),