/*
* Limited memory BFGS (L-BFGS).
*
* Copyright (c) 1990, Jorge Nocedal
* Copyright (c) 2007, 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: lbfgs.c 56 2007-12-16 08:01:47Z naoaki $ */
/*
This library is a C port of the FORTRAN 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 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.
I would like to thank the original author, Jorge Nocedal, who has been
distributing the effieicnt and explanatory implementation in an open source
licence.
*/
#ifdef HAVE_CONFIG_H
#include <config.h>
#endif/*HAVE_CONFIG_H*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <lbfgs.h>
#ifdef _MSC_VER
#define inline __inline
typedef unsigned int uint32_t;
#endif/*_MSC_VER*/
/* this block is added by laye */
#ifndef _MSC_VER
#include <stdint.h>
#endif/*_MSC_VER*/
#if defined(USE_SSE) && defined(__SSE__) && LBFGS_FLOAT == 32
/* Use SSE optimization for 32bit float precision. */
#include "arithmetic_sse_float.h"
#elif defined(USE_SSE) && defined(__SSE__) && LBFGS_FLOAT == 64
/* Use SSE2 optimization for 64bit double precision. */
#include "arithmetic_sse_double.h"
#else
/* No CPU specific optimization. */
#include "arithmetic_ansi.h"
#endif
#define min2(a, b) ((a) <= (b) ? (a) : (b))
#define max2(a, b) ((a) >= (b) ? (a) : (b))
#define max3(a, b, c) max2(max2((a), (b)), (c));
struct tag_iteration_data {
lbfgsfloatval_t alpha;
lbfgsfloatval_t *s; /* [n] */
lbfgsfloatval_t *y; /* [n] */
lbfgsfloatval_t ys; /* vecdot(y, s) */
};
typedef struct tag_iteration_data iteration_data_t;
static const lbfgs_parameter_t _defparam = {
6, 1e-5, 0, 20,
1e-20, 1e20, 1e-4, 0.9, 1.0e-16,
0.0,
};
/* Forward function declarations. */
static int line_search_backtracking(
int n,
lbfgsfloatval_t *x,
lbfgsfloatval_t *f,
lbfgsfloatval_t *g,
lbfgsfloatval_t *s,
lbfgsfloatval_t *stp,
lbfgsfloatval_t *wa,
lbfgs_evaluate_t proc_evaluate,
void *instance,
const lbfgs_parameter_t *param
);
static int line_search(
int n,
lbfgsfloatval_t *x,
lbfgsfloatval_t *f,
lbfgsfloatval_t *g,
lbfgsfloatval_t *s,
lbfgsfloatval_t *stp,
lbfgsfloatval_t *wa,
lbfgs_evaluate_t proc_evaluate,
void *instance,
const lbfgs_parameter_t *param
);
static int update_trial_interval(
lbfgsfloatval_t *x,
lbfgsfloatval_t *fx,
lbfgsfloatval_t *dx,
lbfgsfloatval_t *y,
lbfgsfloatval_t *fy,
lbfgsfloatval_t *dy,
lbfgsfloatval_t *t,
lbfgsfloatval_t *ft,
lbfgsfloatval_t *dt,
const lbfgsfloatval_t tmin,
const lbfgsfloatval_t tmax,
int *brackt
);
void lbfgs_parameter_init(lbfgs_parameter_t *param)
{
memcpy(param, &_defparam, sizeof(*param));
}
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
)
{
int ret;
int i, j, k, ls, end, bound;
lbfgsfloatval_t step;
/* Constant parameters and their default values. */
const lbfgs_parameter_t* param = (_param != NULL) ? _param : &_defparam;
const int m = param->m;
lbfgsfloatval_t *xp = NULL, *g = NULL, *gp = NULL, *d = NULL, *w = NULL;
iteration_data_t *lm = NULL, *it = NULL;
lbfgsfloatval_t ys, yy;
lbfgsfloatval_t norm, xnorm, gnorm, beta;
lbfgsfloatval_t fx = 0.;
/* Check the input parameters for errors. */
if (n <= 0) {
return LBFGSERR_INVALID_N;
}
#if defined(USE_SSE) && defined(__SSE__)
if (n % 8 != 0) {
return LBFGSERR_INVALID_N_SSE;
}
#endif/*defined(__SSE__)*/
if (param->min_step < 0.) {
return LBFGSERR_INVALID_MINSTEP;
}
if (param->max_step < param->min_step) {
return LBFGSERR_INVALID_MAXSTEP;
}
if (param->ftol < 0.) {
return LBFGSERR_INVALID_FTOL;
}
if (param->gtol < 0.) {
return LBFGSERR_INVALID_GTOL;
}
if (param->xtol < 0.) {
return LBFGSERR_INVALID_XTOL;
}
if (param->max_linesearch <= 0) {
return LBFGSERR_INVALID_MAXLINESEARCH;
}
if (param->orthantwise_c < 0.) {
return LBFGSERR_INVALID_ORTHANTWISE;
}
/* Allocate working space. */
xp = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
g = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
gp = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
d = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
w = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
if (xp == NULL || g == NULL || gp == NULL || d == NULL || w == NULL) {
ret = LBFGSERR_OUTOFMEMORY;
goto lbfgs_exit;
}
/* Allocate limited memory storage. */
lm = (iteration_data_t*)vecalloc(m * sizeof(iteration_data_t));
if (lm == NULL) {
ret = LBFGSERR_OUTOFMEMORY;
goto lbfgs_exit;
}
/* Initialize the limited memory. */
for (i = 0;i < m;++i) {
it = &lm[i];
it->alpha = 0;
it->ys = 0;
it->s = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
it->y = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
if (it->s == NULL || it->y == NULL) {
ret = LBFGSERR_OUTOFMEMORY;
goto lbfgs_exit;
}
}
/* Evaluate the function value and its gradient. */
fx = proc_evaluate(instance, x, g, n, 0);
if (0. < param->orthantwise_c) {
/* Compute L1-regularization factor and add it to the object value. */
norm = 0.;
for (i = 0;i < n;++i) {
norm += fabs(x[i]);
}
fx += norm * param->orthantwise_c;
}
/* We assume the initial hessian matrix H_0 as the identity matrix. */
if (param->orthantwise_c == 0.) {
vecncpy(d, g, n);
} else {
/* Compute the negative of psuedo-gradients. */
for (i = 0;i < n;++i) {
if (x[i] < 0.) {
/* Differentiable. */
d[i] = -g[i] + param->orthantwise_c;
} else if (0. < x[i]) {
/* Differentiable. */
d[i] = -g[i] - param->orthantwise_c;
} else {
if (g[i] < -param->orthantwise_c) {
/* Take the right partial derivative. */
d[i] = -g[i] - param->orthantwise_c;
} else if (param->orthantwise_c < g[i]) {
/* Take the left partial derivative. */
d[i] = -g[i] + param->orthantwise_c;
} else {
d[i] = 0.;
}
}
}
}
/* Compute the initial step:
step = 1.0 / sqrt(vecdot(d, d, n))
*/
vecrnorm(&step, d, n);
k = 1;
end = 0;
for (;;) {
/* Store the current position and gradient vectors. */
veccpy(xp, x, n);
veccpy(gp, g, n);
/* Search for an optimal step. */
ls = line_search(
n, x, &fx, g, d, &step, w, proc_evaluate, instance, param);
if (ls < 0) {
ret = ls;
goto lbfgs_exit;
}
/* Compute x and g norms. */
vecnorm(&gnorm, g, n);
vecnorm(&xnorm, x, n);
/* Report the progress. */
if (proc_progress) {
if (ret = proc_progress(instance, x, g, fx, xnorm, gnorm, step, n, k, ls)) {
goto lbfgs_exit;
}
}
/*
Convergence test.
The criterion is given by the following formula:
|g(x)| / \max(1, |x|) < \epsilon
*/
if (xnorm < 1.0) xnorm = 1.0;
if (gnorm / xnorm <= param->epsilon) {
/* Convergence. */
ret = 0;
break;
}
if (param->max_iterations != 0 && param->max_iterations < k+1) {
/* Maximum number of iterations. */
ret = LBFGSERR_MAXIMUMITERATION;
break;
}
/*
Update vectors s and y:
s_{k+1} = x_{k+1} - x_{k} = \step * d_{k}.
y_{k+1} = g_{k+1} - g_{k}.
*/
it = &lm[end];
vecdiff(it->s, x, xp, n);
vecdiff(it->y, g, gp, n);
/*
Compute scalars ys and yy:
ys = y^t \cdot s = 1 / \rho.
yy = y^t \cdot y.
Notice that yy is used for scaling the hessian matrix H_0 (Cholesky factor).
*/
vecdot(&ys, it->y, it->s, n);
vecdot(&yy, it->y, it->y, n);
it->ys = ys;
/*
Recursive formula to compute dir = -(H \cdot g).
This is described in page 779 of:
Jorge Nocedal.
Updating Quasi-Newton Matrices with Limited Storage.
Mathematics of Computation, Vol. 35, No. 151,
pp. 773--782, 1980.
*/
bound = (m <= k) ? m : k;
++k;
end = (end + 1) % m;
if (param->orthantwise_c == 0.) {
/* Compute the negative of gradients. */
vecncpy(d, g, n);
} else {
/* Compute the negative of psuedo-gradients. */
for (i = 0;i < n;++i) {
if (x[i] < 0.) {
/* Differentiable. */
d[i] = -g[i] + param->orthantwise_c;
} else if (0. < x[i]) {
/* Differentiable. */
d[i] = -g[i] - param->orthantwise_c;
} else {
if (g[i] < -param->orthantwise_c) {
/* Take the right partial derivative. */
d[i] = -g[i] - param->orthantwise_c;
} else if (param->orthantwise_c < g[i]) {
/* Take the left partial derivative. */
d[i] = -g[i] + param->orthantwise_c;
} else {
d[i] = 0.;
}
}
}
/* Store the steepest direction.*/
veccpy(w, d, n);
}
j = end;
for (i = 0;i < bound;++i) {
j = (j + m - 1) % m; /* if (--j == -1) j = m-1; */
it = &lm[j];
/* \alpha_{j} = \rho_{j} s^{t}_{j} \cdot q_{k+1}. */
vecdot(&it->alpha, it->s, d, n);
it->alpha /= it->ys;
/* q_{i} = q_{i+1} - \alpha_{i} y_{i}. */
vecadd(d, it->y, -it->alpha, n);
}
vecscale(d, ys / yy, n);
for (i = 0;i < bound;++i) {
it = &lm[j];
/* \beta_{j} = \rho_{j} y^t_{j} \cdot \gamma_{i}. */
vecdot(&beta, it->y, d, n);
beta /= it->ys;
/* \gamma_{i+1} = \gamma_{i} + (\alpha_{j} - \beta_{j}) s_{j}. */
vecadd(d, it->s, it->alpha - beta, n);
j = (j + 1) % m; /* if (++j == m) j = 0; */
}
/*
Constrain the search direction for orthant-wise updates.
*/
if (param->orthantwise_c != 0.) {
for (i = 0;i < n;++i) {
if (d[i] * w[i] <= 0) {
d[i] = 0;
}
}
}
/*
Now the search direction d is ready. We try step = 1 first.
*/
step = 1.0;
}
lbfgs_exit:
/* Return the final value of the objective function. */
if (ptr_fx != NULL) {
*ptr_fx = fx;
}
/* Free memory blocks used by this function. */
if (lm != NULL) {
for (i = 0;i < m;++i) {
vecfree(lm[i].s);
vecfree(lm[i].y);
}
vecfree(lm);
}
vecfree(w);
vecfree(d);
vecfree(gp);
vecfree(g);
vecfree(xp);
return ret;
}
static int line_search_backtracking(
int n,
lbfgsfloatval_t *x,
lbfgsfloatval_t *f,
lbfgsfloatval_t *g,
lbfgsfloatval_t *s,
lbfgsfloatval_t *stp,
lbfgsfloatval_t *xp,
lbfgs_evaluate_t proc_evaluate,
void *instance,
const lbfgs_parameter_t *param
)
{
int i, ret = 0, count = 0;
lbfgsfloatval_t width = 0.5, norm = 0.;
lbfgsfloatval_t finit, dginit = 0., dg, dgtest;
/* Check the input parameters for errors. */
if (*stp <= 0.) {
return LBFGSERR_INVALIDPARAMETERS;
}
/* Compute the initial gradient in the search direction. */
if (param->orthantwise_c != 0.) {
/* Use psuedo-gradients for orthant-wise updates. */
for (i = 0;i < n;++i) {
/* Notice that:
(-s[i] < 0) <==> (g[i] < -param->orthantwise_c)
(-s[i] > 0) <==> (param->orthantwise_c < g[i])
as the result of the lbfgs() function for orthant-wise updates.
*/
if (s[i] != 0.) {
if (x[i] < 0.) {
/* Differentiable. */
dginit += s[i] * (g[i] - param->orthantwise_c);
} else if (0. < x[i]) {
/* Differentiable. */
dginit += s[i] * (g[i] + param->orthantwise_c);
} else if (s[i] < 0.) {
/* Take the left partial derivative. */
dginit += s[i] * (g[i] - param->orthantwise_c);
} else if (0. < s[i]) {
/* Take the right partial derivative. */
dginit += s[i] * (g[i] + param->orthantwise_c);
}
}
}
} else {
vecdot(&dginit, g, s, n);
}
/* Make sure that s points to a descent direction. */
if (0 < dginit) {
return LBFGSERR_INCREASEGRADIENT;
}
/* The initial value of the objective function. */
finit = *f;
dgtest = param->ftol * dginit;
/* Copy the value of x to the work area. */
veccpy(xp, x, n);
for (;;) {
veccpy(x, xp, n);
vecadd(x, s, *stp, n);
if (param->orthantwise_c != 0.) {
/* The current point is projected onto the orthant of the initial one. */
for (i = 0;i < n;++i) {
if (x[i] * xp[i] < 0.) {
x[i] = 0.;
}
}
}
/* Evaluate the function and gradient values. */
*f = proc_evaluate(instance, x, g, n, *stp);
if (0. < param->orthantwise_c) {
/* Compute L1-regularization factor and add it to the object value. */
norm = 0.;
for (i = 0;i < n;++i) {
norm += fabs(x[i]);
}
*f += norm * param->orthantwise_c;
}
vecdot(&dg, g, s, n);
++count;
if (*f <= finit + *stp * dgtest) {
/* The sufficient decrease condition. */
return count;
}
if (*stp < param->min_step) {
/* The step is the minimum value. */
ret = LBFGSERR_MINIMUMSTEP;
break;
}
if (param->max_linesearch <= count) {
/* Maximum number of iteration. */
ret = LBFGSERR_MAXIMUMLINESEARCH;
break;
}
(*stp) *= width;
}
/* Revert to the previous position. */
veccpy(x, xp, n);
return ret;
}
static int line_search(
int n,
lbfgsfloatval_t *x,
lbfgsfloatval_t *f,
lbfgsfloatval_t *g,
lbfgsfloatval_t *s,
lbfgsfloatval_t *stp,
lbfgsfloatval_t *wa,
lbfgs_evaluate_t proc_evaluate,
void *instance,
const lbfgs_parameter_t *param
)
{
int i, count = 0;
int brackt, stage1, uinfo = 0;
lbfgsfloatval_t dg, norm;
lbfgsfloatval_t stx, fx, dgx;
lbfgsfloatval_t sty, fy, dgy;
lbfgsfloatval_t fxm, dgxm, fym, dgym, fm, dgm;
lbfgsfloatval_t finit, ftest1, dginit, dgtest;
lbfgsfloatval_t width, prev_width;
lbfgsfloatval_t stmin, stmax;
/* Check the input parameters for errors. */
if (*stp <= 0.) {
return LBFGSERR_INVALIDPARAMETERS;
}
/* Compute the initial gradient in the search direction. */
if (param->orthantwise_c != 0.) {
/* Use psuedo-gradients for orthant-wise updates. */
dginit = 0.;
for (i = 0;i < n;++i) {
/* Notice that:
(-s[i] < 0) <==> (g[i] < -param->orthantwise_c)
(-s[i] > 0) <==> (param->orthantwise_c < g[i])
as the result of the lbfgs() function for orthant-wise updates.
*/
if (s[i] != 0.) {
if (x[i] < 0.) {
/* Differentiable. */
dginit += s[i] * (g[i] - param->orthantwise_c);
} else if (0. < x[i]) {
/* Differentiable. */
dginit += s[i] * (g[i] + param->orthantwise_c);
} else if (s[i] < 0.) {
/* Take the left partial derivative. */
dginit += s[i] * (g[i] - param->orthantwise_c);
} else if (0. < s[i]) {
/* Take the right partial derivative. */
dginit += s[i] * (g[i] + param->orthantwise_c);
}
}
}
} else {
vecdot(&dginit, g, s, n);
}
/* Make sure that s points to a descent direction. */
if (0 < dginit) {
return LBFGSERR_INCREASEGRADIENT;
}
/* Initialize local variables. */
brackt = 0;
stage1 = 1;
finit = *f;
dgtest = param->ftol * dginit;
width = param->max_step - param->min_step;
prev_width = 2.0 * width;
/* Copy the value of x to the work area. */
veccpy(wa, x, n);
/*
The variables stx, fx, dgx contain the values of the step,
function, and directional derivative at the best step.
The variables sty, fy, dgy contain the value of the step,
function, and derivative at the other endpoint of
the interval of uncertainty.
The variables stp, f, dg contain the values of the step,
function, and derivative at the current step.
*/
stx = sty = 0.;
fx = fy = finit;
dgx = dgy = dginit;
for (;;) {
/*
Set the minimum and maximum steps to correspond to the
present interval of uncertainty.
*/
if (brackt) {
stmin = min2(stx, sty);
stmax = max2(stx, sty);
} else {
stmin = stx;
stmax = *stp + 4.0 * (*stp - stx);
}
/* Clip the step in the range of [stpmin, stpmax]. */
if (*stp < param->min_step) *stp = param->min_step;
if (param->max_step < *stp) *stp = param->max_step;
/*
If an unusual termination is to occur then let
stp be the lowest point obtained so far.
*/
if ((brackt && ((*stp <= stmin || stmax <= *stp) || param->max_linesearch <= count + 1 || uinfo != 0)) || (brackt && (stmax - stmin <= param->xtol * stmax))) {
*stp = stx;
}
/*
Compute the current value of x:
x <- x + (*stp) * s.
*/
veccpy(x, wa, n);
vecadd(x, s, *stp, n);
if (param->orthantwise_c != 0.) {
/* The current point is projected onto the orthant of the previous one. */
for (i = 0;i < n;++i) {
if (x[i] * wa[i] < 0.) {
x[i] = 0.;
}
}
}
/* Evaluate the function and gradient values. */
*f = proc_evaluate(instance, x, g, n, *stp);
if (0. < param->orthantwise_c) {
/* Compute L1-regularization factor and add it to the object value. */
norm = 0.;
for (i = 0;i < n;++i) {
norm += fabs(x[i]);
}
*f += norm * param->orthantwise_c;
}
++count;
vecdot(&dg, g, s, n);
ftest1 = finit + *stp * dgtest;
/* Test for errors and convergence. */
if (brackt && ((*stp <= stmin || stmax <= *stp) || uinfo != 0)) {
/* Rounding errors prevent further progress. */
return LBFGSERR_ROUNDING_ERROR;
}
if (*stp == param->max_step && *f <= ftest1 && dg <= dgtest) {
/* The step is the maximum value. */
return LBFGSERR_MAXIMUMSTEP;
}
if (*stp == param->min_step && (ftest1 < *f || dgtest <= dg)) {
/* The step is the minimum value. */
return LBFGSERR_MINIMUMSTEP;
}
if (brackt && (stmax - stmin) <= param->xtol * stmax) {
/* Relative width of the interval of uncertainty is at most xtol. */
return LBFGSERR_WIDTHTOOSMALL;
}
if (param->max_linesearch <= count) {
/* Maximum number of iteration. */
return LBFGSERR_MAXIMUMLINESEARCH;
}
if (*f <= ftest1 && fabs(dg) <= param->gtol * (-dginit)) {
/* The sufficient decrease condition and the directional derivative condition hold. */
return count;
}
/*
In the first stage we seek a step for which the modified
function has a nonpositive value and nonnegative derivative.
*/
if (stage1 && *f <= ftest1 && min2(param->ftol, param->gtol) * dginit <= dg) {
stage1 = 0;
}
/*
A modified function is used to predict the step only if
we have not obtained a step for which the modified
function has a nonpositive function value and nonnegative
derivative, and if a lower function value has been
obtained but the decrease is not sufficient.
*/
if (stage1 && ftest1 < *f && *f <= fx) {
/* Define the modified function and derivative values. */
fm = *f - *stp * dgtest;
fxm = fx - stx * dgtest;
fym = fy - sty * dgtest;
dgm = dg - dgtest;
dgxm = dgx - dgtest;
dgym = dgy - dgtest;
/*
Call update_trial_interval() to update the interval of
uncertainty and to compute the new step.
*/
uinfo = update_trial_interval(
&stx, &fxm, &dgxm,
&sty, &fym, &dgym,
stp, &fm, &dgm,
stmin, stmax, &brackt
);
/* Reset the function and gradient values for f. */
fx = fxm + stx * dgtest;
fy = fym + sty * dgtest;
dgx = dgxm + dgtest;
dgy = dgym + dgtest;
} else {
/*
Call update_trial_interval() to update the interval of
uncertainty and to compute the new step.
*/
uinfo = update_trial_interval(
&stx, &fx, &dgx,
&sty, &fy, &dgy,
stp, f, &dg,
stmin, stmax, &brackt
);
}
/*
Force a sufficient decrease in the interval of uncertainty.
*/
if (brackt) {
if (0.66 * prev_width <= fabs(sty - stx)) {
*stp = stx + 0.5 * (sty - stx);
}
prev_width = width;
width = fabs(sty - stx);
}
}
return LBFGSERR_LOGICERROR;
}
/**
* Define the local variables for computing minimizers.
*/
#define USES_MINIMIZER \
lbfgsfloatval_t a, d, gamma, theta, p, q, r, s;
/**
* Find a minimizer of an interpolated cubic function.
* @param cm The minimizer of the interpolated cubic.
* @param u The value of one point, u.
* @param fu The value of f(u).
* @param du The value of f'(u).
* @param v The value of another point, v.
* @param fv The value of f(v).
* @param du The value of f'(v).
*/
#define CUBIC_MINIMIZER(cm, u, fu, du, v, fv, dv) \
d = (v) - (u); \
theta = ((fu) - (fv)) * 3 / d + (du) + (dv); \
p = fabs(theta); \
q = fabs(du); \
r = fabs(dv); \
s = max3(p, q, r); \
/* gamma = s*sqrt((theta/s)**2 - (du/s) * (dv/s)) */ \
a = theta / s; \
gamma = s * sqrt(a * a - ((du) / s) * ((dv) / s)); \
if ((v) < (u)) gamma = -gamma; \
p = gamma - (du) + theta; \
q = gamma - (du) + gamma + (dv); \
r = p / q; \
(cm) = (u) + r * d;
/**
* Find a minimizer of an interpolated cubic function.
* @param cm The minimizer of the interpolated cubic.
* @param u The value of one point, u.
* @param fu The value of f(u).
* @param du The value of f'(u).
* @param v The value of another point, v.
* @param fv The value of f(v).
* @param du The value of f'(v).
* @param xmin The maximum value.
* @param xmin The minimum value.
*/
#define CUBIC_MINIMIZER2(cm, u, fu, du, v, fv, dv, xmin, xmax) \
d = (v) - (u); \
theta = ((fu) - (fv)) * 3 / d + (du) + (dv); \
p = fabs(theta); \
q = fabs(du); \
r = fabs(dv); \
s = max3(p, q, r); \
/* gamma = s*sqrt((theta/s)**2 - (du/s) * (dv/s)) */ \
a = theta / s; \
gamma = s * sqrt(max2(0, a * a - ((du) / s) * ((dv) / s))); \
if ((u) < (v)) gamma = -gamma; \
p = gamma - (dv) + theta; \
q = gamma - (dv) + gamma + (du); \
r = p / q; \
if (r < 0. && gamma != 0.) { \
(cm) = (v) - r * d; \
} else if (a < 0) { \
(cm) = (xmax); \
} else { \
(cm) = (xmin); \
}
/**
* Find a minimizer of an interpolated quadratic function.
* @param qm The minimizer of the interpolated quadratic.
* @param u The value of one point, u.
* @param fu The value of f(u).
* @param du The value of f'(u).
* @param v The value of another point, v.
* @param fv The value of f(v).
*/
#define QUARD_MINIMIZER(qm, u, fu, du, v, fv) \
a = (v) - (u); \
(qm) = (u) + (du) / (((fu) - (fv)) / a + (du)) / 2 * a;
/**
* Find a minimizer of an interpolated quadratic function.
* @param qm The minimizer of the interpolated quadratic.
* @param u The value of one point, u.
* @param du The value of f'(u).
* @param v The value of another point, v.
* @param dv The value of f'(v).
*/
#define QUARD_MINIMIZER2(qm, u, du, v, dv) \
a = (u) - (v); \
(qm) = (v) + (dv) / ((dv) - (du)) * a;
/**
* Update a safeguarded trial value and interval for line search.
*
* The parameter x represents the step with the least function value.
* The parameter t represents the current step. This function assumes
* that the derivative at the point of x in the direction of the step.
* If the bracket is set to true, the minimizer has been bracketed in
* an interval of uncertainty with endpoints between x and y.
*
* @param x The pointer to the value of one endpoint.
* @param fx The pointer to the value of f(x).
* @param dx The pointer to the value of f'(x).
* @param y The pointer to the value of another endpoint.
* @param fy The pointer to the value of f(y).
* @param dy The pointer to the value of f'(y).
* @param t The pointer to the value of the trial value, t.
* @param ft The pointer to the value of f(t).
* @param dt The pointer to the value of f'(t).
* @param tmin The minimum value for the trial value, t.
* @param tmax The maximum value for the trial value, t.
* @param brackt The pointer to the predicate if the trial value is
* bracketed.
* @retval int Status value. Zero indicates a normal termination.
*
* @see
* Jorge J. More and David J. Thuente. Line search algorithm with
* guaranteed sufficient decrease. ACM Transactions on Mathematical
* Software (TOMS), Vol 20, No 3, pp. 286-307, 1994.
*/
static int update_trial_interval(
lbfgsfloatval_t *x,
lbfgsfloatval_t *fx,
lbfgsfloatval_t *dx,
lbfgsfloatval_t *y,
lbfgsfloatval_t *fy,
lbfgsfloatval_t *dy,
lbfgsfloatval_t *t,
lbfgsfloatval_t *ft,
lbfgsfloatval_t *dt,
const lbfgsfloatval_t tmin,
const lbfgsfloatval_t tmax,
int *brackt
)
{
int bound;
int dsign = fsigndiff(dt, dx);
lbfgsfloatval_t mc; /* minimizer of an interpolated cubic. */
lbfgsfloatval_t mq; /* minimizer of an interpolated quadratic. */
lbfgsfloatval_t newt; /* new trial value. */
USES_MINIMIZER; /* for CUBIC_MINIMIZER and QUARD_MINIMIZER. */
/* Check the input parameters for errors. */
if (*brackt) {
if (*t <= min2(*x, *y) || max2(*x, *y) <= *t) {
/* The trival value t is out of the interval. */
return LBFGSERR_OUTOFINTERVAL;
}
if (0. <= *dx * (*t - *x)) {
/* The function must decrease from x. */
return LBFGSERR_INCREASEGRADIENT;
}
if (tmax < tmin) {
/* Incorrect tmin and tmax specified. */
return LBFGSERR_INCORRECT_TMINMAX;
}
}
/*
Trial value selection.
*/
if (*fx < *ft) {
/*
Case 1: a higher function value.
The minimum is brackt. If the cubic minimizer is closer
to x than the quadratic one, the cubic one is taken, else
the average of the minimizers is taken.
*/
*brackt = 1;
bound = 1;
CUBIC_MINIMIZER(mc, *x, *fx, *dx, *t, *ft, *dt);
QUARD_MINIMIZER(mq, *x, *fx, *dx, *t, *ft);
if (fabs(mc - *x) < fabs(mq - *x)) {
newt = mc;
} else {
newt = mc + 0.5 * (mq - mc);
}
} else if (dsign) {
/*
Case 2: a lower function value and derivatives of
opposite sign. The minimum is brackt. If the cubic
minimizer is closer to x than the quadratic (secant) one,
the cubic one is taken, else the quadratic one is taken.
*/
*brackt = 1;
bound = 0;
CUBIC_MINIMIZER(mc, *x, *fx, *dx, *t, *ft, *dt);
QUARD_MINIMIZER2(mq, *x, *dx, *t, *dt);
if (fabs(mc - *t) > fabs(mq - *t)) {
newt = mc;
} else {
newt = mq;
}
} else if (fabs(*dt) < fabs(*dx)) {
/*
Case 3: a lower function value, derivatives of the
same sign, and the magnitude of the derivative decreases.
The cubic minimizer is only used if the cubic tends to
infinity in the direction of the minimizer or if the minimum
of the cubic is beyond t. Otherwise the cubic minimizer is
defined to be either tmin or tmax. The quadratic (secant)
minimizer is also computed and if the minimum is brackt
then the the minimizer closest to x is taken, else the one
farthest away is taken.
*/
bound = 1;
CUBIC_MINIMIZER2(mc, *x, *fx, *dx, *t, *ft, *dt, tmin, tmax);
QUARD_MINIMIZER2(mq, *x, *dx, *t, *dt);
if (*brackt) {
if (fabs(*t - mc) < fabs(*t - mq)) {
newt = mc;
} else {
newt = mq;
}
} else {
if (fabs(*t - mc) > fabs(*t - mq)) {
newt = mc;
} else {
newt = mq;
}
}
} else {
/*
Case 4: a lower function value, derivatives of the
same sign, and the magnitude of the derivative does
not decrease. If the minimum is not brackt, the step
is either tmin or tmax, else the cubic minimizer is taken.
*/
bound = 0;
if (*brackt) {
CUBIC_MINIMIZER(newt, *t, *ft, *dt, *y, *fy, *dy);
} else if (*x < *t) {
newt = tmax;
} else {
newt = tmin;
}
}
/*
Update the interval of uncertainty. This update does not
depend on the new step or the case analysis above.
- Case a: if f(x) < f(t),
x <- x, y <- t.
- Case b: if f(t) <= f(x) && f'(t)*f'(x) > 0,
x <- t, y <- y.
- Case c: if f(t) <= f(x) && f'(t)*f'(x) < 0,
x <- t, y <- x.
*/
if (*fx < *ft) {
/* Case a */
*y = *t;
*fy = *ft;
*dy = *dt;
} else {
/* Case c */
if (dsign) {
*y = *x;
*fy = *fx;
*dy = *dx;
}
/* Cases b and c */
*x = *t;
*fx = *ft;
*dx = *dt;
}
/* Clip the new trial value in [tmin, tmax]. */
if (tmax < newt) newt = tmax;
if (newt < tmin) newt = tmin;
/*
Redefine the new trial value if it is close to the upper bound
of the interval.
*/
if (*brackt && bound) {
mq = *x + 0.66 * (*y - *x);
if (*x < *y) {
if (mq < newt) newt = mq;
} else {
if (newt < mq) newt = mq;
}
}
/* Return the new trial value. */
*t = newt;
return 0;
}