1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097
// [[file:../lbfgs.note::*header][header:1]]
//! Limited memory BFGS (L-BFGS).
//
// Copyright (c) 1990, Jorge Nocedal
// Copyright (c) 2007-2010 Naoaki Okazaki
// Copyright (c) 2018-2019 Wenping Guo
// 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.
//
// 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 (OWL-QN)
// 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.
// header:1 ends here
// [[file:../lbfgs.note::*imports][imports:1]]
use crate::core::*;
use crate::math::LbfgsMath;
use crate::line::*;
// imports:1 ends here
// [[file:../lbfgs.note::*parameters][parameters:1]]
/// L-BFGS optimization parameters.
///
/// Call lbfgs_parameter_t::default() function to initialize parameters to the
/// default values.
#[derive(Copy, Clone, Debug)]
#[repr(C)]
pub struct LbfgsParam {
/// 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 6. Values less than 3 are not
/// recommended. Large values will result in excessive computing time.
pub m: usize,
/// Epsilon for convergence test.
///
/// This parameter determines the accuracy with which the solution is to be
/// found. A minimization terminates when
///
/// ||g|| < epsilon * max(1, ||x||),
///
/// where ||.|| denotes the Euclidean (L2) norm. The default value is \c
/// 1e-5.
pub epsilon: f64,
/// Distance for delta-based convergence test.
///
/// This parameter determines the distance, in iterations, to compute the
/// rate of decrease of the objective function. If the value of this
/// parameter is zero, the library does not perform the delta-based
/// convergence test.
///
/// The default value is 0.
pub past: usize,
/// Delta for convergence test.
///
/// This parameter determines the minimum rate of decrease of the objective
/// function. The library stops iterations when the following condition is
/// met: |f' - f| / f < delta, where f' is the objective value of past
/// iterations ago, and f is the objective value of the current iteration.
/// The default value is 1e-5.
///
pub delta: f64,
/// The maximum number of LBFGS iterations.
///
/// The lbfgs optimization terminates 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.
pub max_iterations: usize,
/// The maximum allowed number of evaluations of function value and
/// gradients. This number could be larger than max_iterations since line
/// search procedure may involve one or more evaluations.
///
/// Setting this parameter to zero continues an optimization process until a
/// convergence or error. The default value is 0.
pub max_evaluations: usize,
/// The line search options.
///
/// This parameter specifies a line search algorithm to be used by the
/// L-BFGS routine.
///
pub linesearch: LineSearch,
/// Enable OWL-QN regulation or not
pub orthantwise: bool,
// FIXME: better name
pub owlqn: Orthantwise,
/// A factor for scaling initial step size.
pub initial_inverse_hessian: f64,
/// The maximum allowed step size for each optimization step, useful for
/// preventing wild step.
pub max_step_size: f64,
/// Powell damping
pub damping: bool,
}
impl Default for LbfgsParam {
/// 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.
fn default() -> Self {
LbfgsParam {
m: 6,
epsilon: 1e-5,
past: 0,
delta: 1e-5,
max_iterations: 0,
max_evaluations: 0,
orthantwise: false,
owlqn: Orthantwise::default(),
linesearch: LineSearch::default(),
initial_inverse_hessian: 1.0,
max_step_size: 1.0,
damping: false,
}
}
}
// parameters:1 ends here
// [[file:../lbfgs.note::*problem][problem:1]]
/// Represents an optimization problem.
///
/// `Problem` holds input variables `x`, gradient `gx` arrays, and function value `fx`.
pub struct Problem<'a, E>
where
E: FnMut(&[f64], &mut [f64]) -> Result<f64>,
{
/// x is an array of length n. on input it must contain the base point for
/// the line search.
pub x: &'a mut [f64],
/// `fx` is a variable. It must contain the value of problem `f` at
/// x.
pub fx: f64,
/// `gx` is an array of length n. It must contain the gradient of `f` at
/// x.
pub gx: Vec<f64>,
/// Cached position vector of previous step.
xp: Vec<f64>,
/// Cached gradient vector of previous step.
gp: Vec<f64>,
/// Pseudo gradient for OrthantWise Limited-memory Quasi-Newton (owlqn) algorithm.
pg: Vec<f64>,
/// Search direction
d: Vec<f64>,
/// Store callback function for evaluating objective function.
eval_fn: E,
/// Orthantwise operations
owlqn: Option<Orthantwise>,
/// Evaluated or not
evaluated: bool,
/// The number of evaluation.
neval: usize,
}
impl<'a, E> Problem<'a, E>
where
E: FnMut(&[f64], &mut [f64]) -> Result<f64>,
{
/// Initialize problem with array length n
pub fn new(x: &'a mut [f64], eval: E, owlqn: Option<Orthantwise>) -> Self {
let n = x.len();
Problem {
fx: 0.0,
gx: vec![0.0; n],
xp: vec![0.0; n],
gp: vec![0.0; n],
pg: vec![0.0; n],
d: vec![0.0; n],
evaluated: false,
neval: 0,
x,
eval_fn: eval,
owlqn,
}
}
/// Compute the initial gradient in the search direction.
pub fn dginit(&self) -> Result<f64> {
if self.owlqn.is_none() {
let dginit = self.gx.vecdot(&self.d);
if dginit > 0.0 {
warn!(
"The current search direction increases the objective function value. dginit = {:-0.4}",
dginit
);
}
Ok(dginit)
} else {
Ok(self.pg.vecdot(&self.d))
}
}
/// Update search direction using evaluated gradient.
pub fn update_search_direction(&mut self) {
if self.owlqn.is_some() {
self.d.vecncpy(&self.pg);
} else {
self.d.vecncpy(&self.gx);
}
}
/// Return a reference to current search direction vector
pub fn search_direction(&self) -> &[f64] {
&self.d
}
/// Return a mutable reference to current search direction vector
pub fn search_direction_mut(&mut self) -> &mut [f64] {
&mut self.d
}
/// Compute the gradient in the search direction without sign checking.
pub fn dg_unchecked(&self) -> f64 {
self.gx.vecdot(&self.d)
}
// FIXME: improve
pub fn evaluate(&mut self) -> Result<()> {
self.fx = (self.eval_fn)(&self.x, &mut self.gx)?;
// self.fx = self.eval_fn.evaluate(&self.x, &mut self.gx)?;
// Compute the L1 norm of the variables and add it to the object value.
if let Some(owlqn) = self.owlqn {
self.fx += owlqn.x1norm(&self.x)
}
// FIXME: to be better
// if self.orthantwise {
// Compute the L1 norm of the variable and add it to the object value.
// fx += self.owlqn.x1norm(x);
// self.owlqn.pseudo_gradient(&mut pg, &x, &g);
self.evaluated = true;
self.neval += 1;
Ok(())
}
/// Return total number of evaluations.
pub fn number_of_evaluation(&self) -> usize {
self.neval
}
/// Test if `Problem` has been evaluated or not
pub fn evaluated(&self) -> bool {
self.evaluated
}
/// Copies all elements from src into self.
pub fn clone_from(&mut self, src: &Problem<E>) {
self.x.clone_from_slice(&src.x);
self.gx.clone_from_slice(&src.gx);
self.fx = src.fx;
}
/// Take a line step along search direction.
///
/// Compute the current value of x: x <- x + (*step) * d.
///
pub fn take_line_step(&mut self, step: f64) {
self.x.veccpy(&self.xp);
self.x.vecadd(&self.d, step);
// Choose the orthant for the new point.
// The current point is projected onto the orthant.
if let Some(owlqn) = self.owlqn {
owlqn.project(&mut self.x, &self.xp, &self.gp);
}
}
/// Return gradient vector norm: ||gx||
pub fn gnorm(&self) -> f64 {
if self.owlqn.is_some() {
self.pg.vec2norm()
} else {
self.gx.vec2norm()
}
}
/// Return position vector norm: ||x||
pub fn xnorm(&self) -> f64 {
self.x.vec2norm()
}
pub fn orthantwise(&self) -> bool {
self.owlqn.is_some()
}
/// Revert to previous step
pub fn revert(&mut self) {
self.x.veccpy(&self.xp);
self.gx.veccpy(&self.gp);
}
/// Store the current position and gradient vectors.
pub fn save_state(&mut self) {
self.xp.veccpy(&self.x);
self.gp.veccpy(&self.gx);
}
/// Constrain the search direction for orthant-wise updates.
pub fn constrain_search_direction(&mut self) {
if let Some(owlqn) = self.owlqn {
owlqn.constrain(&mut self.d, &self.pg);
}
}
// FIXME
pub fn update_owlqn_gradient(&mut self) {
if let Some(owlqn) = self.owlqn {
owlqn.pseudo_gradient(&mut self.pg, &self.x, &self.gx);
}
}
}
// problem:1 ends here
// [[file:../lbfgs.note::*progress][progress:1]]
/// Store optimization progress data, for progress monitor
#[repr(C)]
#[derive(Debug, Clone)]
pub struct Progress<'a> {
/// The current values of variables
pub x: &'a [f64],
/// The current gradient values of variables.
pub gx: &'a [f64],
/// The current value of the objective function.
pub fx: f64,
/// The Euclidean norm of the variables
pub xnorm: f64,
/// The Euclidean norm of the gradients.
pub gnorm: f64,
/// The line-search step used for this iteration.
pub step: f64,
/// The iteration count.
pub niter: usize,
/// The total number of evaluations.
pub neval: usize,
/// The number of function evaluation calls in line search procedure
pub ncall: usize,
}
impl<'a> Progress<'a> {
fn new<E>(prb: &'a Problem<E>, niter: usize, ncall: usize, step: f64) -> Self
where
E: FnMut(&[f64], &mut [f64]) -> Result<f64>,
{
Progress {
x: &prb.x,
gx: &prb.gx,
fx: prb.fx,
xnorm: prb.xnorm(),
gnorm: prb.gnorm(),
neval: prb.number_of_evaluation(),
ncall,
step,
niter,
}
}
}
pub struct Report {
/// The current value of the objective function.
pub fx: f64,
/// The Euclidean norm of the variables
pub xnorm: f64,
/// The Euclidean norm of the gradients.
pub gnorm: f64,
/// The total number of evaluations.
pub neval: usize,
}
impl Report {
fn new<E>(prb: &Problem<E>) -> Self
where
E: FnMut(&[f64], &mut [f64]) -> Result<f64>,
{
Self {
fx: prb.fx,
xnorm: prb.xnorm(),
gnorm: prb.gnorm(),
neval: prb.number_of_evaluation(),
}
}
}
// progress:1 ends here
// [[file:../lbfgs.note::*orthantwise][orthantwise:1]]
use crate::base::Orthantwise;
// orthantwise:1 ends here
// [[file:../lbfgs.note::*builder][builder:1]]
#[derive(Default, Debug, Clone)]
pub struct Lbfgs {
param: LbfgsParam,
}
/// Create lbfgs optimizer with epsilon convergence
impl Lbfgs {
/// Set scaled gradient norm for converence test
///
/// This parameter determines the accuracy with which the solution is to be
/// found. A minimization terminates when
///
/// ||g|| < epsilon * max(1, ||x||),
///
/// where ||.|| denotes the Euclidean (L2) norm. The default value is 1e-5.
pub fn with_epsilon(mut self, epsilon: f64) -> Self {
assert!(epsilon.is_sign_positive(), "Invalid parameter epsilon specified.");
self.param.epsilon = epsilon;
self
}
/// Set initial step size for optimization. The default value is 1.0.
pub fn with_initial_step_size(mut self, b: f64) -> Self {
assert!(
b.is_sign_positive(),
"Invalid beta parameter for scaling the initial step size."
);
self.param.initial_inverse_hessian = b;
self
}
/// Set the maximum allowed step size for optimization. The default value is 1.0.
pub fn with_max_step_size(mut self, s: f64) -> Self {
assert!(s.is_sign_positive(), "Invalid max_step_size parameter.");
self.param.max_step_size = s;
self
}
/// Enable Powell damping.
pub fn with_damping(mut self, damped: bool) -> Self {
self.param.damping = damped;
self
}
/// Set orthantwise parameters
pub fn with_orthantwise(mut self, c: f64, start: usize, end: usize) -> Self {
assert!(
c.is_sign_positive(),
"Invalid parameter orthantwise c parameter specified."
);
warn!("Only the backtracking line search is available for OWL-QN algorithm.");
self.param.orthantwise = true;
self.param.owlqn.c = c;
self.param.owlqn.start = start as i32;
self.param.owlqn.end = end as i32;
self
}
/// 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.
pub fn with_linesearch_ftol(mut self, ftol: f64) -> Self {
assert!(ftol >= 0.0, "Invalid parameter ftol specified.");
self.param.linesearch.ftol = ftol;
self
}
/// 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.
pub fn with_linesearch_gtol(mut self, gtol: f64) -> Self {
assert!(
gtol >= 0.0 && gtol < 1.0 && gtol > self.param.linesearch.ftol,
"Invalid parameter gtol specified."
);
self.param.linesearch.gtol = gtol;
self
}
/// Try to follow gradient only during optimization, by allowing object
/// value rises, which removes the sufficient decrease condition constrain
/// in line search. This option also implies Powell damping and
/// BacktrackingStrongWolfe line search for improving robustness.
pub fn with_gradient_only(mut self) -> Self {
self.param.linesearch.gradient_only = true;
self.param.damping = true;
self.param.linesearch.algorithm = LineSearchAlgorithm::BacktrackingStrongWolfe;
self
}
/// Set the max number of iterations for line search.
pub fn with_max_linesearch(mut self, n: usize) -> Self {
self.param.linesearch.max_linesearch = n;
self
}
/// xtol is a nonnegative input variable. termination occurs when the
/// relative width of the interval of uncertainty is at most 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.
pub fn with_linesearch_xtol(mut self, xtol: f64) -> Self {
assert!(xtol >= 0.0, "Invalid parameter xtol specified.");
self.param.linesearch.xtol = xtol;
self
}
/// 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).
pub fn with_linesearch_min_step(mut self, min_step: f64) -> Self {
assert!(min_step >= 0.0, "Invalid parameter min_step specified.");
self.param.linesearch.min_step = min_step;
self
}
/// Set the maximum number of iterations.
///
/// The lbfgs optimization terminates 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.
pub fn with_max_iterations(mut self, niter: usize) -> Self {
self.param.max_iterations = niter;
self
}
/// The maximum allowed number of evaluations of function value and
/// gradients. This number could be larger than max_iterations since line
/// search procedure may involve one or more evaluations.
///
/// Setting this parameter to zero continues an optimization process until a
/// convergence or error. The default value is 0.
pub fn with_max_evaluations(mut self, neval: usize) -> Self {
self.param.max_evaluations = neval;
self
}
/// This parameter determines the minimum rate of decrease of the objective
/// function. The library stops iterations when the following condition is
/// met: |f' - f| / f < delta, where f' is the objective value of past
/// iterations ago, and f is the objective value of the current iteration.
///
/// If `past` is zero, the library does not perform the delta-based
/// convergence test.
///
/// The default value of delta is 1e-5.
///
pub fn with_fx_delta(mut self, delta: f64, past: usize) -> Self {
assert!(delta >= 0.0, "Invalid parameter delta specified.");
self.param.past = past;
self.param.delta = delta;
self
}
/// Select line search algorithm
///
/// The default is "MoreThuente" line search algorithm.
pub fn with_linesearch_algorithm(mut self, algo: &str) -> Self {
match algo {
"MoreThuente" => self.param.linesearch.algorithm = LineSearchAlgorithm::MoreThuente,
"BacktrackingArmijo" => self.param.linesearch.algorithm = LineSearchAlgorithm::BacktrackingArmijo,
"BacktrackingStrongWolfe" => self.param.linesearch.algorithm = LineSearchAlgorithm::BacktrackingStrongWolfe,
"BacktrackingWolfe" | "Backtracking" => {
self.param.linesearch.algorithm = LineSearchAlgorithm::BacktrackingWolfe
}
_ => unimplemented!(),
}
self
}
}
// builder:1 ends here
// [[file:../lbfgs.note::*hack][hack:1]]
impl Lbfgs {
/// Start the L-BFGS optimization; this will invoke the callback functions evaluate
/// and progress.
///
/// # Parameters
///
/// * x : The array of input variables.
/// * evaluate: A closure for evaluating function value and gradient
/// * progress: A closure for monitor progress or defining stopping condition
///
/// # Return
///
/// * on success, return final evaluated `Problem`.
pub fn minimize<E, G>(self, x: &mut [f64], eval_fn: E, mut prgr_fn: G) -> Result<Report>
where
E: FnMut(&[f64], &mut [f64]) -> Result<f64>,
G: FnMut(&Progress) -> bool,
{
let mut state = self.build(x, eval_fn)?;
info!("start lbfgs loop...");
for _ in 0.. {
if state.is_converged() {
break;
}
let prgr = state.get_progress();
let cancel = prgr_fn(&prgr);
if cancel {
info!("The minimization process has been canceled.");
break;
}
state.propagate()?;
}
// Return the final value of the objective function.
Ok(state.report())
}
}
// hack:1 ends here
// [[file:../lbfgs.note::*state][state:1]]
/// LBFGS optimization state allowing iterative propagation
pub struct LbfgsState<'a, E>
where
E: FnMut(&[f64], &mut [f64]) -> Result<f64>,
{
/// LBFGS parameters
vars: LbfgsParam,
/// Define how to evaluate gradient and value
prbl: Option<Problem<'a, E>>,
end: usize,
step: f64,
k: usize,
lm_arr: Vec<IterationData>,
pf: Vec<f64>,
ncall: usize,
}
// state:1 ends here
// [[file:../lbfgs.note::*build][build:1]]
impl Lbfgs {
/// Build LBFGS state struct for iteration.
pub fn build<'a, E>(self, x: &'a mut [f64], eval_fn: E) -> Result<LbfgsState<'a, E>>
where
E: FnMut(&[f64], &mut [f64]) -> Result<f64>,
{
// Initialize the limited memory.
let param = &self.param;
let lm_arr = (0..param.m).map(|_| IterationData::new(x.len())).collect();
// Allocate working space for LBFGS optimization
let owlqn = if param.orthantwise {
Some(param.owlqn.clone())
} else {
None
};
let mut problem = Problem::new(x, eval_fn, owlqn);
// Evaluate the function value and its gradient.
problem.evaluate()?;
// Compute the L1 norm of the variable and add it to the object value.
problem.update_owlqn_gradient();
// Compute the search direction with current gradient.
problem.update_search_direction();
// Compute the initial step:
let h0 = param.initial_inverse_hessian;
let step = problem.search_direction().vec2norminv() * h0;
// Apply Powell damping or not
let damping = param.damping;
if damping {
info!("Powell damping Enabled.");
}
let state = LbfgsState {
vars: self.param.clone(),
prbl: Some(problem),
end: 0,
step,
k: 0,
lm_arr,
pf: vec![],
ncall: 0,
};
Ok(state)
}
}
// build:1 ends here
// [[file:../lbfgs.note::*propagate][propagate:1]]
impl<'a, E> LbfgsState<'a, E>
where
E: FnMut(&[f64], &mut [f64]) -> Result<f64>,
{
/// Check if stopping critera met. Panics if not initialized.
pub fn is_converged(&mut self) -> bool {
// Monitor the progress.
let prgr = self.get_progress();
let converged = satisfying_stop_conditions(&self.vars, prgr);
converged
}
/// Report minimization progress. Panics if not initialized yet.
pub fn report(&self) -> Report {
Report::new(self.prbl.as_ref().expect("problem for report"))
}
/// Propagate in next LBFGS step. Return optimization progress on success.
/// Panics if not initialized.
pub fn propagate(&mut self) -> Result<Progress> {
self.k += 1;
// special case: already converged at the first point
if self.k == 1 {
let progress = self.get_progress();
return Ok(progress);
}
// Store the current position and gradient vectors.
let problem = self.prbl.as_mut().expect("problem for propagate");
problem.save_state();
// Search for an optimal step.
self.ncall = self
.vars
.linesearch
.find(problem, &mut self.step)
.context("Failure during line search")?;
problem.update_owlqn_gradient();
// Update LBFGS iteration data.
let it = &mut self.lm_arr[self.end];
let gamma = it.update(
&problem.x,
&problem.xp,
&problem.gx,
&problem.gp,
self.step,
self.vars.damping,
);
// Compute the steepest direction
problem.update_search_direction();
let d = problem.search_direction_mut();
// Apply LBFGS recursion procedure.
self.end = lbfgs_two_loop_recursion(&mut self.lm_arr, d, gamma, self.vars.m, self.k - 1, self.end);
// Now the search direction d is ready. Constrains the step size to
// prevent wild steps.
let dnorm = d.vec2norm();
self.step = self.vars.max_step_size.min(dnorm) / dnorm;
// Constrain the search direction for orthant-wise updates.
problem.constrain_search_direction();
let progress = self.get_progress();
Ok(progress)
}
fn get_progress(&self) -> Progress {
let problem = self.prbl.as_ref().expect("problem for progress");
Progress::new(&problem, self.k, self.ncall, self.step)
}
}
// propagate:1 ends here
// [[file:../lbfgs.note::*recursion][recursion:1]]
/// Algorithm 7.4, in Nocedal, J.; Wright, S. Numerical Optimization; Springer Science & Business Media, 2006.
fn lbfgs_two_loop_recursion(
lm_arr: &mut [IterationData],
d: &mut [f64], // search direction
gamma: f64, // H_k^{0} = \gamma I
m: usize,
k: usize,
end: usize,
) -> usize {
let end = (end + 1) % m;
let mut j = end;
let bound = m.min(k);
// L-BFGS two-loop recursion, part1
for _ in 0..bound {
j = (j + m - 1) % m;
let it = &mut lm_arr[j as usize];
// \alpha_{j} = \rho_{j} s^{t}_{j} \cdot q_{k+1}.
it.alpha = it.s.vecdot(&d) / it.ys;
// q_{i} = q_{i+1} - \alpha_{i} y_{i}.
d.vecadd(&it.y, -it.alpha);
}
d.vecscale(gamma);
// L-BFGS two-loop recursion, part2
for _ in 0..bound {
let it = &mut lm_arr[j as usize];
// \beta_{j} = \rho_{j} y^t_{j} \cdot \gamma_{i}.
let beta = it.y.vecdot(d) / it.ys;
// \gamma_{i+1} = \gamma_{i} + (\alpha_{j} - \beta_{j}) s_{j}.
d.vecadd(&it.s, it.alpha - beta);
j = (j + 1) % m;
}
end
}
// recursion:1 ends here
// [[file:../lbfgs.note::*iteration data][iteration data:1]]
/// Internal iternation data for L-BFGS
#[derive(Clone)]
struct IterationData {
alpha: f64,
s: Vec<f64>,
y: Vec<f64>,
/// vecdot(y, s)
ys: f64,
}
impl IterationData {
fn new(n: usize) -> Self {
IterationData {
alpha: 0.0,
ys: 0.0,
s: vec![0.0; n],
y: vec![0.0; n],
}
}
/// Updates L-BFGS correction pairs, returns Cholesky factor \gamma for
/// scaling the initial inverse Hessian matrix $H_k^0$
///
/// # Arguments
///
/// * x, xp: current position, and previous position
/// * gx, gp: current gradient and previous gradient
/// * step: step size along search direction
/// * damping: applying Powell damping to the gradient difference `y` helps
/// stabilize L-BFGS from numerical noise in function value and gradient
///
fn update(&mut self, x: &[f64], xp: &[f64], gx: &[f64], gp: &[f64], step: f64, damping: bool) -> f64 {
// Update vectors s and y:
// s_{k} = x_{k+1} - x_{k} = \alpha * d_{k}.
// y_{k} = g_{k+1} - g_{k}.
self.s.vecdiff(x, xp);
self.y.vecdiff(gx, gp);
// 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 intial inverse hessian matrix H_0 (Cholesky factor).
let ys = self.y.vecdot(&self.s);
let yy = self.y.vecdot(&self.y);
self.ys = ys;
// Al-Baali2014JOTA: Damped Techniques for the Limited Memory BFGS
// Method for Large-Scale Optimization. J. Optim. Theory Appl. 2014,
// 161 (2), 688–699.
//
// Nocedal suggests an equivalent value of 0.8 for sigma2 (Damped BFGS
// updating)
let sigma2 = 0.6;
let sigma3 = 3.0;
if damping {
debug!("Applying Powell damping, sigma2 = {}, sigma3 = {}", sigma2, sigma3);
// B_k * Sk = B_k * (x_k + step*d_k - x_k) = B_k * step * d_k = -g_k * step
let mut bs = gp.to_vec();
bs.vecscale(-step);
// s_k^T * B_k * s_k
let sbs = self.s.vecdot(&bs);
if ys < (1.0 - sigma2) * sbs {
trace!("damping case1");
let theta = sigma2 * sbs / (sbs - ys);
bs.vecscale(1.0 - theta);
bs.vecadd(&self.y, theta);
self.y.veccpy(&bs);
} else if ys > (1.0 + sigma3) * sbs {
trace!("damping case2");
let theta = sigma3 * sbs / (ys - sbs);
bs.vecscale(1.0 - theta);
bs.vecadd(&self.y, theta);
} else {
trace!("damping case3");
// for theta = 1.0, yk = yk, so do nothing here.
}
}
ys / yy
}
}
// iteration data:1 ends here
// [[file:../lbfgs.note::*stopping conditions][stopping conditions:1]]
/// test if progress satisfying stop condition
#[inline]
fn satisfying_stop_conditions(param: &LbfgsParam, prgr: Progress) -> bool {
// Buildin tests for stopping conditions
if satisfying_max_iterations(&prgr, param.max_iterations)
|| satisfying_max_evaluations(&prgr, param.max_evaluations)
|| satisfying_scaled_gnorm(&prgr, param.epsilon)
// || satisfying_delta(&prgr, pf, param.delta)
// || satisfying_max_gnorm(&prgr, self.param.max_gnorm)
{
return true;
}
false
}
/// The criterion is given by the following formula:
/// |g(x)| / \max(1, |x|) < \epsilon
#[inline]
fn satisfying_scaled_gnorm(prgr: &Progress, epsilon: f64) -> bool {
if prgr.gnorm / prgr.xnorm.max(1.0) <= epsilon {
// Convergence.
info!("L-BFGS reaches convergence.");
true
} else {
false
}
}
/// Maximum number of lbfgs iterations.
#[inline]
fn satisfying_max_iterations(prgr: &Progress, max_iterations: usize) -> bool {
if max_iterations == 0 {
false
} else if prgr.niter >= max_iterations {
warn!("max iterations reached!");
true
} else {
false
}
}
/// Maximum number of function evaluations
#[inline]
fn satisfying_max_evaluations(prgr: &Progress, max_evaluations: usize) -> bool {
if max_evaluations == 0 {
false
} else if prgr.neval >= max_evaluations {
warn!("Max allowed evaluations reached!");
true
} else {
false
}
}
#[inline]
fn satisfying_max_gnorm(prgr: &Progress, max_gnorm: f64) -> bool {
prgr.gx.vec2norm() <= max_gnorm
}
/// Functiona value (fx) delta based stopping criterion
///
/// Test for stopping criterion.
/// The criterion is given by the following formula:
/// |f(past_x) - f(x)| / f(x) < delta
///
/// # Parameters
///
/// * pf: an array for storing previous values of the objective function.
/// * delta: max fx delta allowed
///
#[inline]
fn satisfying_delta<'a>(prgr: &Progress, pf: &'a mut [f64], delta: f64) -> bool {
let k = prgr.niter;
let fx = prgr.fx;
let past = pf.len();
if past < 1 {
return false;
}
// We don't test the stopping criterion while k < past.
if past <= k {
// Compute the relative improvement from the past.
let rate = (pf[(k % past) as usize] - fx).abs() / fx;
// The stopping criterion.
if rate < delta {
info!("The stopping criterion.");
return true;
}
}
// Store the current value of the objective function.
pf[(k % past) as usize] = fx;
false
}
// stopping conditions:1 ends here