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/*
* C library of Limited memory BFGS (L-BFGS).
*
* Copyright (c) 1990, Jorge Nocedal
* Copyright (c) 2007,2008, Naoaki Okazaki
* All rights reserved.
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
* THE SOFTWARE.
*/
/* $Id$ */
#ifndef __LBFGS_H__
#define __LBFGS_H__
#ifdef __cplusplus
extern "C" {
#endif/*__cplusplus*/
/*
* The default precision of floating point values is 64bit (double).
*/
#ifndef LBFGS_FLOAT
#define LBFGS_FLOAT 64
#endif/*LBFGS_FLOAT*/
/*
* Activate optimization routines for IEEE754 floating point values.
*/
#ifndef LBFGS_IEEE_FLOAT
#define LBFGS_IEEE_FLOAT 1
#endif/*LBFGS_IEEE_FLOAT*/
#if LBFGS_FLOAT == 32
typedef float lbfgsfloatval_t;
#elif LBFGS_FLOAT == 64
typedef double lbfgsfloatval_t;
#else
#error "liblbfgs supports single (float; LBFGS_FLOAT = 32) or double (double; LBFGS_FLOAT=64) precision only."
#endif
/**
* \addtogroup liblbfgs_api libLBFGS API
* @{
*
* The libLBFGS API.
*/
/**
* Return values of lbfgs().
*/
enum {
/** False value. */
LBFGSFALSE = 0,
/** True value. */
LBFGSTRUE,
/** Unknown error. */
LBFGSERR_UNKNOWNERROR = -1024,
/** Logic error. */
LBFGSERR_LOGICERROR,
/** Insufficient memory. */
LBFGSERR_OUTOFMEMORY,
/** The minimization process has been canceled. */
LBFGSERR_CANCELED,
/** Invalid number of variables specified. */
LBFGSERR_INVALID_N,
/** Invalid number of variables (for SSE) specified. */
LBFGSERR_INVALID_N_SSE,
/** Invalid parameter lbfgs_parameter_t::linesearch specified. */
LBFGSERR_INVALID_LINESEARCH,
/** Invalid parameter lbfgs_parameter_t::max_step specified. */
LBFGSERR_INVALID_MINSTEP,
/** Invalid parameter lbfgs_parameter_t::max_step specified. */
LBFGSERR_INVALID_MAXSTEP,
/** Invalid parameter lbfgs_parameter_t::ftol specified. */
LBFGSERR_INVALID_FTOL,
/** Invalid parameter lbfgs_parameter_t::gtol specified. */
LBFGSERR_INVALID_GTOL,
/** Invalid parameter lbfgs_parameter_t::xtol specified. */
LBFGSERR_INVALID_XTOL,
/** Invalid parameter lbfgs_parameter_t::max_linesearch specified. */
LBFGSERR_INVALID_MAXLINESEARCH,
/** Invalid parameter lbfgs_parameter_t::orthantwise_c specified. */
LBFGSERR_INVALID_ORTHANTWISE,
/** The line-search step went out of the interval of uncertainty. */
LBFGSERR_OUTOFINTERVAL,
/** A logic error occurred; alternatively, the interval of uncertainty
became too small. */
LBFGSERR_INCORRECT_TMINMAX,
/** A rounding error occurred; alternatively, no line-search step
satisfies the sufficient decrease and curvature conditions. */
LBFGSERR_ROUNDING_ERROR,
/** The line-search step became smaller than lbfgs_parameter_t::min_step. */
LBFGSERR_MINIMUMSTEP,
/** The line-search step became larger than lbfgs_parameter_t::max_step. */
LBFGSERR_MAXIMUMSTEP,
/** The line-search routine reaches the maximum number of evaluations. */
LBFGSERR_MAXIMUMLINESEARCH,
/** The algorithm routine reaches the maximum number of iterations. */
LBFGSERR_MAXIMUMITERATION,
/** Relative width of the interval of uncertainty is at most
lbfgs_parameter_t::xtol. */
LBFGSERR_WIDTHTOOSMALL,
/** A logic error (negative line-search step) occurred. */
LBFGSERR_INVALIDPARAMETERS,
/** The current search direction increases the objective function value. */
LBFGSERR_INCREASEGRADIENT,
};
/**
* Line search algorithms.
*/
enum {
/** The default algorithm (MoreThuente method). */
LBFGS_LINESEARCH_DEFAULT = 0,
/** MoreThuente method proposd by More and Thuente. */
LBFGS_LINESEARCH_MORETHUENTE = 0,
/** Backtracking method. */
LBFGS_LINESEARCH_BACKTRACKING,
};
/**
* L-BFGS optimization parameters.
* Call lbfgs_parameter_init() function to initialize parameters to the
* default values.
*/
typedef struct {
/**
* The number of corrections to approximate the inverse hessian matrix.
* The L-BFGS routine stores the computation results of previous \ref 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 \c 6. Values less than \c 3 are
* not recommended. Large values will result in excessive computing time.
*/
int m;
/**
* Epsilon for convergence test.
* This parameter determines the accuracy with which the solution is to
* be found. A minimization terminates when
* ||g|| < \ref epsilon * max(1, ||x||),
* where ||.|| denotes the Euclidean (L2) norm. The default value is
* \c 1e-5.
*/
lbfgsfloatval_t epsilon;
/**
* The maximum number of iterations.
* The lbfgs() function 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 \c 0.
*/
int max_iterations;
/**
* The line search algorithm.
* This parameter specifies a line search algorithm to be used by the
* L-BFGS routine.
*/
int 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 \c 20.
*/
int max_linesearch;
/**
* The minimum step of the line search routine.
* The default value is \c 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).
*/
lbfgsfloatval_t min_step;
/**
* The maximum step of the line search.
* The default value is \c 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).
*/
lbfgsfloatval_t max_step;
/**
* A parameter to control the accuracy of the line search routine.
* The default value is \c 1e-4. This parameter should be greater
* than zero and smaller than \c 0.5.
*/
lbfgsfloatval_t ftol;
/**
* A parameter to control the accuracy of the line search routine.
* The default value is \c 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 \c 0.1. This parameter shuold be
* greater than the \ref ftol parameter (\c 1e-4) and smaller than
* \c 1.0.
*/
lbfgsfloatval_t gtol;
/**
* 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.
*/
lbfgsfloatval_t xtol;
/**
* 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 library
* 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 G(x) as usual. The default value is zero.
*/
lbfgsfloatval_t orthantwise_c;
} lbfgs_parameter_t;
/**
* Callback interface to provide objective function and gradient evaluations.
*
* The lbfgs() function call this function to obtain the values of objective
* function and its gradients when needed. A client program must implement
* this function to evaluate the values of the objective function and its
* gradients, given current values of variables.
*
* @param instance The user data sent for lbfgs() function by the client.
* @param x The current values of variables.
* @param g The gradient vector. The callback function must compute
* the gradient values for the current variables.
* @param n The number of variables.
* @param step The current step of the line search routine.
* @retval lbfgsfloatval_t The value of the objective function for the current
* variables.
*/
typedef lbfgsfloatval_t (*lbfgs_evaluate_t)(
void *instance,
const lbfgsfloatval_t *x,
lbfgsfloatval_t *g,
const int n,
const lbfgsfloatval_t step
);
/**
* Callback interface to receive the progress of the optimization process.
*
* The lbfgs() function call this function for each iteration. Implementing
* this function, a client program can store or display the current progress
* of the optimization process.
*
* @param instance The user data sent for lbfgs() function by the client.
* @param x The current values of variables.
* @param g The current gradient values of variables.
* @param fx The current value of the objective function.
* @param xnorm The Euclidean norm of the variables.
* @param gnorm The Euclidean norm of the gradients.
* @param step The line-search step used for this iteration.
* @param n The number of variables.
* @param k The iteration count.
* @param ls The number of evaluations called for this iteration.
* @retval int Zero to continue the optimization process. Returning a
* non-zero value will cancel the optimization process.
*/
typedef int (*lbfgs_progress_t)(
void *instance,
const lbfgsfloatval_t *x,
const lbfgsfloatval_t *g,
const lbfgsfloatval_t fx,
const lbfgsfloatval_t xnorm,
const lbfgsfloatval_t gnorm,
const lbfgsfloatval_t step,
int n,
int k,
int ls
);
/*
A user must implement a function compatible with ::lbfgs_evaluate_t (evaluation
callback) and pass the pointer to the callback function to lbfgs() arguments.
Similarly, a user can implement a function compatible with ::lbfgs_progress_t
(progress callback) to obtain the current progress (e.g., variables, function
value, ||G||, etc) and to cancel the iteration process if necessary.
Implementation of a progress callback is optional: a user can pass \c NULL if
progress notification is not necessary.
In addition, a user must preserve two requirements:
- The number of variables must be multiples of 16 (this is not 4).
- The memory block of variable array ::x must be aligned to 16.
This algorithm terminates an optimization
when:
||G|| < \epsilon \cdot \max(1, ||x||) .
In this formula, ||.|| denotes the Euclidean norm.
*/
/**
* Start a L-BFGS optimization.
*
* @param n The number of variables.
* @param x The array of variables. A client program can set
* default values for the optimization and receive the
* optimization result through this array.
* @param ptr_fx The pointer to the variable that receives the final
* value of the objective function for the variables.
* This argument can be set to \c NULL if the final
* value of the objective function is unnecessary.
* @param proc_evaluate The callback function to provide function and
* gradient evaluations given a current values of
* variables. A client program must implement a
* callback function compatible with \ref
* lbfgs_evaluate_t and pass the pointer to the
* callback function.
* @param proc_progress The callback function to receive the progress
* (the number of iterations, the current value of
* the objective function) of the minimization
* process. This argument can be set to \c NULL if
* a progress report is unnecessary.
* @param instance A user data for the client program. The callback
* functions will receive the value of this argument.
* @param param The pointer to a structure representing parameters for
* L-BFGS optimization. A client program can set this
* parameter to \c NULL to use the default parameters.
* Call lbfgs_parameter_init() function to fill a
* structure with the default values.
* @retval int The status code. This function returns zero if the
* minimization process terminates without an error. A
* non-zero value indicates an error.
*/
int lbfgs(
const int n,
lbfgsfloatval_t *x,
lbfgsfloatval_t *ptr_fx,
lbfgs_evaluate_t proc_evaluate,
lbfgs_progress_t proc_progress,
void *instance,
lbfgs_parameter_t *param
);
/**
* Initialize L-BFGS parameters to the default values.
*
* Call this function to fill a parameter structure with the default values
* and overwrite parameter values if necessary.
*
* @param param The pointer to the parameter structure.
*/
void lbfgs_parameter_init(lbfgs_parameter_t *param);
/** @} */
#ifdef __cplusplus
}
#endif/*__cplusplus*/
/**
@mainpage A library of Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS)
@section intro Introduction
This library is a C port of the implementation of Limited-memory
Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method written by Jorge Nocedal.
The original FORTRAN source code is available at:
http://www.ece.northwestern.edu/~nocedal/lbfgs.html
The L-BFGS method solves the unconstrainted minimization problem,
<pre>
minimize F(x), x = (x1, x2, ..., xN),
</pre>
only if the objective function F(x) and its gradient G(x) are computable. The
well-known Newton's method requires computation of the inverse of the hessian
matrix of the objective function. However, the computational cost for the
inverse hessian matrix is expensive especially when the objective function
takes a large number of variables. The L-BFGS method iteratively find a
minimizer by approximating the inverse hessian matrix by information from last
m iterations. This innovation saves the memory storage and computational time
drastically for large-scaled problems.
Among the various ports of L-BFGS, this library provides several features:
- <b>Optimization with L1-norm (orthant-wise L-BFGS)</b>:
In addition to standard minimization problems, the library can minimize
a function F(x) combined with L1-norm |x| of the variables,
{F(x) + C |x|}, where C is a constant scalar parameter. This feature is
useful for estimating parameters of log-linear models (e.g., logistic
regression and maximum entropy) with L1-regularization.
- <b>Clean C code</b>:
Unlike C codes generated automatically by f2c (Fortran 77 into C converter),
this port includes changes based on my interpretations, improvements,
optimizations, and clean-ups so that the ported code would be well-suited
for a C code. In addition to comments inherited from the original code,
a number of comments were added through my interpretations.
- <b>Callback interface</b>:
The library receives function and gradient values via a callback interface.
The library also notifies the progress of the optimization by invoking a
callback function. In the original implementation, a user had to set
function and gradient values every time the function returns for obtaining
updated values.
- <b>Thread safe</b>:
The library is thread-safe, which is the secondary gain from the callback
interface.
- <b>Cross platform.</b> The source code can be compiled on Microsoft Visual
Studio 2005, GNU C Compiler (gcc), etc.
- <b>Configurable precision</b>: A user can choose single-precision (float)
or double-precision (double) accuracy by changing ::LBFGS_FLOAT macro.
- <b>SSE/SSE2 optimization</b>:
This library includes SSE/SSE2 optimization (written in compiler intrinsics)
for vector arithmetic operations on Intel/AMD processors. The library uses
SSE for float values and SSE2 for double values. The SSE/SSE2 optimization
routine is disabled by default; compile the library with __SSE__ symbol
defined to activate the optimization routine.
This library is used by:
- <a href="http://www.chokkan.org/software/crfsuite/">CRFsuite: A fast implementation of Conditional Random Fields (CRFs)</a>
- <a href="http://www.public.iastate.edu/~gdancik/mlegp/">mlegp: an R package for maximum likelihood estimates for Gaussian processes</a>
@section download Download
- <a href="http://www.chokkan.org/software/dist/liblbfgs-1.3.tar.gz">Source code</a>
libLBFGS is distributed under the term of the
<a href="http://opensource.org/licenses/mit-license.php">MIT license</a>.
@section changelog History
- Version 1.4 (2008-04-25):
- Configurable line search algorithms. A member variable
::lbfgs_parameter_t::linesearch was added to choose either MoreThuente
method (::LBFGS_LINESEARCH_MORETHUENTE) or backtracking algorithm
(::LBFGS_LINESEARCH_BACKTRACKING).
- Fixed a serious bug: the previous version did not compute
psuedo-gradients properly in the line search routine. This bug might
quit an iteration process too early when the orthant-wise L-BFGS routine
was activated (0 < ::lbfgs_parameter_t::orthantwise_c).
- Version 1.3 (2007-12-16):
- An API change. An argument was added to lbfgs() function to receive the
final value of the objective function. This argument can be set to
\c NULL if the final value is unnecessary.
- Fixed a null-pointer bug in the sample code (reported by Takashi Imamichi).
- Added build scripts for Microsoft Visual Studio 2005 and GCC.
- Added README file.
- Version 1.2 (2007-12-13):
- Fixed a serious bug in orthant-wise L-BFGS.
An important variable was used without initialization.
- Version 1.1 (2007-12-01):
- Implemented orthant-wise L-BFGS.
- Implemented lbfgs_parameter_init() function.
- Fixed several bugs.
- API documentation.
- Version 1.0 (2007-09-20):
- Initial release.
@section api Documentation
- @ref liblbfgs_api "libLBFGS API"
@section sample Sample code
@include sample.c
@section ack Acknowledgements
The L-BFGS algorithm is described in:
- Jorge Nocedal.
Updating Quasi-Newton Matrices with Limited Storage.
<i>Mathematics of Computation</i>, Vol. 35, No. 151, pp. 773--782, 1980.
- Dong C. Liu and Jorge Nocedal.
On the limited memory BFGS method for large scale optimization.
<i>Mathematical Programming</i> B, Vol. 45, No. 3, pp. 503-528, 1989.
The line search algorithms used in this implementation are described in:
- John E. Dennis and Robert B. Schnabel.
<i>Numerical Methods for Unconstrained Optimization and Nonlinear
Equations</i>, Englewood Cliffs, 1983.
- Jorge J. More and David J. Thuente.
Line search algorithm with guaranteed sufficient decrease.
<i>ACM Transactions on Mathematical Software (TOMS)</i>, Vol. 20, No. 3,
pp. 286-307, 1994.
This library also implements Orthant-Wise Limited-memory Quasi-Newton (OW-LQN)
method presented in:
- Galen Andrew and Jianfeng Gao.
Scalable training of L1-regularized log-linear models.
In <i>Proceedings of the 24th International Conference on Machine
Learning (ICML 2007)</i>, pp. 33-40, 2007.
Finally I would like to thank the original author, Jorge Nocedal, who has been
distributing the effieicnt and explanatory implementation in an open source
licence.
@section reference Reference
- <a href="http://www.ece.northwestern.edu/~nocedal/lbfgs.html">L-BFGS</a> by Jorge Nocedal.
- <a href="http://research.microsoft.com/research/downloads/Details/3f1840b2-dbb3-45e5-91b0-5ecd94bb73cf/Details.aspx">OWL-QN</a> by Galen Andrew.
- <a href="http://chasen.org/~taku/software/misc/lbfgs/">C port (via f2c)</a> by Taku Kudo.
- <a href="http://www.alglib.net/optimization/lbfgs.php">C#/C++/Delphi/VisualBasic6 port</a> in ALGLIB.
- <a href="http://cctbx.sourceforge.net/">Computational Crystallography Toolbox</a> includes
<a href="http://cctbx.sourceforge.net/current_cvs/c_plus_plus/namespacescitbx_1_1lbfgs.html">scitbx::lbfgs</a>.
*/
#endif/*__LBFGS_H__*/