int lbfgs(
int n,
T *x,
T *ptr_fx,
typename FuncWrapper<T>::lbfgs_evaluate_t proc_evaluate,
typename FuncWrapper<T>::lbfgs_progress_t proc_progress,
void *instance,
lbfgs_parameter_t *_param
)
{
int ret;
int i, j, k, ls, end, bound;
T step;
/* Constant parameters and their default values. */
lbfgs_parameter_t param = (_param != NULL) ? (*_param) : _defparam;
const int m = param.m;
T *xp = NULL;
T *g = NULL, *gp = NULL, *pg = NULL;
T *d = NULL, *w = NULL, *pf = NULL;
iteration_data_t<T> *lm = NULL;
iteration_data_t<T>*it = NULL;
T ys, yy;
T xnorm, gnorm, beta;
T fx = 0.;
T rate = 0.;
typename LineSearchWrapper<T>::line_search_proc linesearch = line_search_morethuente;
/* Construct a callback data. */
callback_data_t<T> cd;
cd.n = n;
cd.instance = instance;
cd.proc_evaluate = proc_evaluate;
cd.proc_progress = proc_progress;
#if defined(USE_SSE) && (defined(__SSE__) || defined(__SSE2__))
/* Round out the number of variables. */
n = round_out_variables(n);
#endif/*defined(USE_SSE)*/
/* Check the input parameters for errors. */
if (n <= 0) {
return LBFGSERR_INVALID_N;
}
#if defined(USE_SSE) && (defined(__SSE__) || defined(__SSE2__))
if (n % 8 != 0) {
return LBFGSERR_INVALID_N_SSE;
}
if ((uintptr_t)(const void*)x % 16 != 0) {
return LBFGSERR_INVALID_X_SSE;
}
#endif/*defined(USE_SSE)*/
if (param.epsilon < 0.) {
return LBFGSERR_INVALID_EPSILON;
}
if (param.past < 0) {
return LBFGSERR_INVALID_TESTPERIOD;
}
if (param.delta < 0.) {
return LBFGSERR_INVALID_DELTA;
}
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.linesearch == LBFGS_LINESEARCH_BACKTRACKING_WOLFE ||
param.linesearch == LBFGS_LINESEARCH_BACKTRACKING_STRONG_WOLFE) {
if (param.wolfe <= param.ftol || 1. <= param.wolfe) {
return LBFGSERR_INVALID_WOLFE;
}
}
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;
}
if (param.orthantwise_start < 0 || n < param.orthantwise_start) {
return LBFGSERR_INVALID_ORTHANTWISE_START;
}
if (param.orthantwise_end < 0) {
param.orthantwise_end = n;
}
if (n < param.orthantwise_end) {
return LBFGSERR_INVALID_ORTHANTWISE_END;
}
if (param.orthantwise_c != 0.) {
switch (param.linesearch) {
case LBFGS_LINESEARCH_BACKTRACKING:
linesearch = line_search_backtracking_owlqn;
break;
default:
/* Only the backtracking method is available. */
return LBFGSERR_INVALID_LINESEARCH;
}
} else {
switch (param.linesearch) {
case LBFGS_LINESEARCH_MORETHUENTE:
linesearch = line_search_morethuente;
break;
case LBFGS_LINESEARCH_BACKTRACKING_ARMIJO:
case LBFGS_LINESEARCH_BACKTRACKING_WOLFE:
case LBFGS_LINESEARCH_BACKTRACKING_STRONG_WOLFE:
linesearch = line_search_backtracking;
break;
default:
return LBFGSERR_INVALID_LINESEARCH;
}
}
/* Allocate working space. */
xp = (T*)vecalloc(n * sizeof(T));
g = (T*)vecalloc(n * sizeof(T));
gp = (T*)vecalloc(n * sizeof(T));
d = (T*)vecalloc(n * sizeof(T));
w = (T*)vecalloc(n * sizeof(T));
if (xp == NULL || g == NULL || gp == NULL || d == NULL || w == NULL) {
ret = LBFGSERR_OUTOFMEMORY;
goto lbfgs_exit;
}
if (param.orthantwise_c != 0.) {
/* Allocate working space for OW-LQN. */
pg = (T*)vecalloc(n * sizeof(T));
if (pg == NULL) {
ret = LBFGSERR_OUTOFMEMORY;
goto lbfgs_exit;
}
}
/* Allocate limited memory storage. */
lm = (iteration_data_t<T>*)vecalloc(m * sizeof(iteration_data_t<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 = (T*)vecalloc(n * sizeof(T));
it->y = (T*)vecalloc(n * sizeof(T));
if (it->s == NULL || it->y == NULL) {
ret = LBFGSERR_OUTOFMEMORY;
goto lbfgs_exit;
}
}
/* Allocate an array for storing previous values of the objective function. */
if (0 < param.past) {
pf = (T*)vecalloc(param.past * sizeof(T));
}
/* Evaluate the function value and its gradient. */
fx = cd.proc_evaluate(cd.instance, x, g, cd.n, 0);
if (0. != param.orthantwise_c) {
/* Compute the L1 norm of the variable and add it to the object value. */
xnorm = owlqn_x1norm(x, param.orthantwise_start, param.orthantwise_end);
fx += xnorm * param.orthantwise_c;
owlqn_pseudo_gradient(
pg, x, g, n,
T(param.orthantwise_c), param.orthantwise_start, param.orthantwise_end
);
}
/* Store the initial value of the objective function. */
if (pf != NULL) {
pf[0] = fx;
}
/*
Compute the direction;
we assume the initial hessian matrix H_0 as the identity matrix.
*/
if (param.orthantwise_c == 0.) {
vecncpy(d, g, n);
} else {
vecncpy(d, pg, n);
}
/*
Make sure that the initial variables are not a minimizer.
*/
vec2norm(&xnorm, x, n);
if (param.orthantwise_c == 0.) {
vec2norm(&gnorm, g, n);
} else {
vec2norm(&gnorm, pg, n);
}
if (xnorm < 1.0) xnorm = 1.0;
if (gnorm / xnorm <= param.epsilon) {
ret = LBFGS_ALREADY_MINIMIZED;
goto lbfgs_exit;
}
/* Compute the initial step:
step = 1.0 / sqrt(vecdot(d, d, n))
*/
vec2norminv(&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. */
if (param.orthantwise_c == 0.) {
ls = linesearch(n, x, &fx, g, d, &step, xp, gp, w, &cd, ¶m);
} else {
ls = linesearch(n, x, &fx, g, d, &step, xp, pg, w, &cd, ¶m);
owlqn_pseudo_gradient(
pg, x, g, n,
T(param.orthantwise_c), param.orthantwise_start, param.orthantwise_end
);
}
if (ls < 0) {
/* Revert to the previous point. */
veccpy(x, xp, n);
veccpy(g, gp, n);
ret = ls;
goto lbfgs_exit;
}
/* Compute x and g norms. */
vec2norm(&xnorm, x, n);
if (param.orthantwise_c == 0.) {
vec2norm(&gnorm, g, n);
} else {
vec2norm(&gnorm, pg, n);
}
/* Report the progress. */
if (cd.proc_progress) {
if ((ret = cd.proc_progress(cd.instance, x, g, fx, xnorm, gnorm, step, cd.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 = LBFGS_SUCCESS;
break;
}
/*
Test for stopping criterion.
The criterion is given by the following formula:
(f(past_x) - f(x)) / f(x) < \delta
*/
if (pf != NULL) {
/* We don't test the stopping criterion while k < past. */
if (param.past <= k) {
/* Compute the relative improvement from the past. */
rate = (pf[k % param.past] - fx) / fx;
/* The stopping criterion. */
if (rate < param.delta) {
ret = LBFGS_STOP;
break;
}
}
/* Store the current value of the objective function. */
pf[k % param.past] = fx;
}
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;
/* Compute the steepest direction. */
if (param.orthantwise_c == 0.) {
/* Compute the negative of gradients. */
vecncpy(d, g, n);
} else {
vecncpy(d, pg, 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 = param.orthantwise_start;i < param.orthantwise_end;++i) {
if (d[i] * pg[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;
}
vecfree(pf);
/* 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(pg);
vecfree(w);
vecfree(d);
vecfree(gp);
vecfree(g);
vecfree(xp);
return ret;
}
int lbfgs(
int n,
double* x,
double* pfx,
lbfgs_evaluate_t evaluate,
lbfgs_progress_t progress,
void* instance,
const lbfgs_parameter_t* _param
) {
int ret;
int i, j, k, ls, end, bound, n_evaluate = 0;
int enalbe_owlqn;
double step;
lbfgs_parameter_t param = (_param) ? (*_param) : default_param;
const int m = param.m;
double* xp;
double* g, *gp, *pg = 0;
double* d, *w, *pf = 0;
iteration_data_t* lm = 0, *it = 0;
double ys, yy;
double xnorm, gnorm, rate, beta;
double fx;
line_search_proc_t linesearch = line_search_morethuente;
callback_data_t cd;
cd.n = n;
cd.instance = instance;
cd.evaluate = evaluate;
cd.progress = (progress) ? progress : default_lbfgs_progress;
/* Check the input parameters for errors. */
if (n <= 0) {
return LBFGSERR_INVALID_N;
}
if (param.epsilon < 0.0) {
return LBFGSERR_INVALID_EPSILON;
}
if (param.past < 0) {
return LBFGSERR_INVALID_TESTPERIOD;
}
if (param.delta < 0.0) {
return LBFGSERR_INVALID_DELTA;
}
if (param.min_step < 0.0) {
return LBFGSERR_INVALID_MINSTEP;
}
if (param.max_step < param.min_step) {
return LBFGSERR_INVALID_MAXSTEP;
}
if (param.ftol < 0.0) {
return LBFGSERR_INVALID_FTOL;
}
if (param.linesearch == LBFGS_LINESEARCH_BACKTRACKING_WOLFE ||
param.linesearch == LBFGS_LINESEARCH_BACKTRACKING_STRONG_WOLFE) {
if (param.wolfe <= param.ftol || 1. <= param.wolfe) {
return LBFGSERR_INVALID_WOLFE;
}
}
if (param.gtol < 0.0) {
return LBFGSERR_INVALID_GTOL;
}
if (param.xtol < 0.0) {
return LBFGSERR_INVALID_XTOL;
}
if (param.max_linesearch <= 0) {
return LBFGSERR_INVALID_MAXLINESEARCH;
}
if (param.orthantwise_c < 0.0) {
return LBFGSERR_INVALID_ORTHANTWISE;
}
if (param.orthantwise_start < 0 || param.orthantwise_start > n) {
return LBFGSERR_INVALID_ORTHANTWISE_START;
}
if (param.orthantwise_end < 0) {
param.orthantwise_end = n;
}
if (param.orthantwise_end > n) {
return LBFGSERR_INVALID_ORTHANTWISE_END;
}
enalbe_owlqn = (param.orthantwise_c != 0.0);
if (enalbe_owlqn) {
switch (param.linesearch) {
case LBFGS_LINESEARCH_BACKTRACKING_WOLFE:
linesearch = line_search_backtracking_owlqn;
break;
default:
/* Only the backtracking method is available. */
return LBFGSERR_INVALID_LINESEARCH;
}
} else {
switch (param.linesearch) {
case LBFGS_LINESEARCH_MORETHUENTE:
linesearch = line_search_morethuente;
break;
case LBFGS_LINESEARCH_BACKTRACKING_ARMIJO:
case LBFGS_LINESEARCH_BACKTRACKING_WOLFE:
case LBFGS_LINESEARCH_BACKTRACKING_STRONG_WOLFE:
linesearch = line_search_backtracking;
break;
default:
return LBFGSERR_INVALID_LINESEARCH;
}
}
/* Allocate working space. */
xp = vecalloc(n);
g = vecalloc(n);
gp = vecalloc(n);
d = vecalloc(n);
w = vecalloc(n);
/* Allocate pseudo gradient. */
if (enalbe_owlqn) {
pg = vecalloc(n);
}
/* Allocate and initialize the limited memory storage. */
lm = (iteration_data_t*)xalloc(m * sizeof(iteration_data_t));
for (i = 0; i < m; i++) {
it = &lm[i];
it->alpha = 0.0;
it->s = vecalloc(n);
it->y = vecalloc(n);
it->ys = 0.0;
}
/* Allocate an array for storing previous values of the objective function. */
if (param.past > 0) {
pf = vecalloc((size_t)param.past);
}
fx = cd.evaluate(cd.instance, cd.n, x, g, 0);
n_evaluate++;
if (enalbe_owlqn) {
xnorm = owlqn_x1norm(x, param.orthantwise_start, param.orthantwise_end);
fx += xnorm * param.orthantwise_c;
owlqn_pseudo_gradient(
pg, x, g, n,
param.orthantwise_c, param.orthantwise_start, param.orthantwise_end);
}
/* Store the initial value of the objective function. */
if (pf) {
pf[0] = fx;
}
/**
* Compute the direction.
* we assume the initial hessian matrix H_0 as the identity matrix.
*/
if (!enalbe_owlqn) {
vecncpy(d, g, n);
} else {
vecncpy(d, pg, n);
}
/**
* Make sure that the initial variables are not a minimizer.
*/
vec2norm(&xnorm, x, n);
if (!enalbe_owlqn) {
vec2norm(&gnorm, g, n);
} else {
vec2norm(&gnorm, pg, n);
}
if (xnorm < 1.0) {
xnorm = 1.0;
}
if (gnorm / xnorm <= param.epsilon) {
ret = LBFGS_ALREADY_MINIMIZED;
goto lbfgs_exit;
}
/**
* Compute the initial step:
* step = 1.0 / ||d||
*/
vec2norminv(&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. */
if (!enalbe_owlqn) {
ls = linesearch(n, x, &fx, g, d, &step, xp, gp, w, &cd, ¶m);
} else {
ls = linesearch(n, x, &fx, g, d, &step, xp, pg, w, &cd, ¶m);
owlqn_pseudo_gradient(
pg, x, g, n,
param.orthantwise_c, param.orthantwise_start, param.orthantwise_end
);
}
if (ls < 0) {
/* Revert to the previous point. */
veccpy(x, xp, n);
veccpy(g, gp, n);
ret = ls;
break;
}
n_evaluate += ls;
/* Compute x and g norms. */
vec2norm(&xnorm, x, n);
if (!enalbe_owlqn) {
vec2norm(&gnorm, g, n);
} else {
vec2norm(&gnorm, pg, n);
}
/* Report the progress. */
if ((ret = cd.progress(cd.instance, cd.n, x, g, fx, xnorm, gnorm, step, k, n_evaluate)) != 0) {
ret = LBFGSERR_CANCELED;
break;
}
/* Convergence test. */
if (xnorm < 1.0) {
xnorm = 1.0;
}
if (gnorm / xnorm <= param.epsilon) {
ret = LBFGS_CONVERGENCE;
break;
}
/* Stopping criterion test. */
if (pf) {
/* We don't test the stopping criterion while k < past. */
if (param.past <= k) {
/* Compute the relative improvement from the past. */
rate = (pf[k % param.past] - fx) / fx;
/* The stopping criterion. */
if (rate < param.delta) {
ret = LBFGS_CONVERGENCE_DELTA;
break;
}
}
/* Store the current value of the objective function. */
pf[k % param.past] = fx;
}
if (param.max_iterations != 0 && param.max_iterations < k + 1) {
ret = LBFGSERR_MAXIMUMITERATION;
break;
}
/**
* Update 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 s = 1 / \rho
* yy = y^t 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 d = -(H 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;
/* Compute the steepest direction. */
/* Compute the negative of (pseudo) gradient. */
if (!enalbe_owlqn) {
vecncpy(d, g, n);
} else {
vecncpy(d, pg, 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} 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} \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 (enalbe_owlqn) {
owlqn_contrain_line_search(d, pg, param.orthantwise_start, param.orthantwise_end);
}
/* 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 (pfx) {
*pfx = fx;
}
vecfree(pf);
if (lm != 0) {
for (i = 0; i < m; i++) {
vecfree(lm[i].s);
vecfree(lm[i].y);
}
xfree(lm);
}
vecfree(pg);
vecfree(w);
vecfree(d);
vecfree(gp);
vecfree(g);
vecfree(xp);
return ret;
}
