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|
@menu
* Introduction to lbfgs::
* Functions and Variables for lbfgs::
@end menu
@node Introduction to lbfgs, Functions and Variables for lbfgs, Top, Top
@section Introduction to lbfgs
@code{lbfgs} is an implementation of the L-BFGS algorithm [1]
to solve unconstrained minimization problems via a limited-memory quasi-Newton (BFGS) algorithm.
It is called a limited-memory method because a low-rank approximation of the
Hessian matrix inverse is stored instead of the entire Hessian inverse.
The program was originally written in Fortran [2] by Jorge Nocedal,
incorporating some functions originally written by Jorge J. Mor@'{e} and David J. Thuente,
and translated into Lisp automatically via the program @code{f2cl}.
The Maxima package @code{lbfgs} comprises the translated code plus
an interface function which manages some details.
References:
[1] D. Liu and J. Nocedal. "On the limited memory BFGS method for large
scale optimization". @i{Mathematical Programming B} 45:503--528 (1989)
[2] @url{http://netlib.org/opt/lbfgs_um.shar}
@opencatbox
@category{Numerical methods} @category{Share packages} @category{Package lbfgs}
@closecatbox
@node Functions and Variables for lbfgs, , Introduction to lbfgs, Top
@section Functions and Variables for lbfgs
@deffn {Function} lbfgs (@var{FOM}, @var{X}, @var{X0}, @var{epsilon}, @var{iprint})
@deffnx {Function} lbfgs ([@var{FOM}, @var{grad}] @var{X}, @var{X0}, @var{epsilon}, @var{iprint})
Finds an approximate solution of the unconstrained minimization of the figure of merit @var{FOM}
over the list of variables @var{X},
starting from initial estimates @var{X0},
such that @math{norm(grad(FOM)) < epsilon*max(1, norm(X))}.
@var{grad}, if present, is the gradient of @var{FOM} with respect to the variables @var{X}.
@var{grad} is a list, with one element for each element of @var{X}.
If not present, the gradient is computed automatically by symbolic differentiation.
The algorithm applied is a limited-memory quasi-Newton (BFGS) algorithm [1].
It is called a limited-memory method because a low-rank approximation of the
Hessian matrix inverse is stored instead of the entire Hessian inverse.
Each iteration of the algorithm is a line search, that is,
a search along a ray in the variables @var{X},
with the search direction computed from the approximate Hessian inverse.
The FOM is always decreased by a successful line search.
Usually (but not always) the norm of the gradient of FOM also decreases.
@var{iprint} controls progress messages printed by @code{lbfgs}.
@table @code
@item iprint[1]
@code{@var{iprint}[1]} controls the frequency of progress messages.
@table @code
@item iprint[1] < 0
No progress messages.
@item iprint[1] = 0
Messages at the first and last iterations.
@item iprint[1] > 0
Print a message every @code{@var{iprint}[1]} iterations.
@end table
@item iprint[2]
@code{@var{iprint}[2]} controls the verbosity of progress messages.
@table @code
@item iprint[2] = 0
Print out iteration count, number of evaluations of @var{FOM}, value of @var{FOM},
norm of the gradient of @var{FOM}, and step length.
@item iprint[2] = 1
Same as @code{@var{iprint}[2] = 0}, plus @var{X0} and the gradient of @var{FOM} evaluated at @var{X0}.
@item iprint[2] = 2
Same as @code{@var{iprint}[2] = 1}, plus values of @var{X} at each iteration.
@item iprint[2] = 3
Same as @code{@var{iprint}[2] = 2}, plus the gradient of @var{FOM} at each iteration.
@end table
@end table
The columns printed by @code{lbfgs} are the following.
@table @code
@item I
Number of iterations. It is incremented for each line search.
@item NFN
Number of evaluations of the figure of merit.
@item FUNC
Value of the figure of merit at the end of the most recent line search.
@item GNORM
Norm of the gradient of the figure of merit at the end of the most recent line search.
@item STEPLENGTH
An internal parameter of the search algorithm.
@end table
Additional information concerning details of the algorithm are found in the
comments of the original Fortran code [2].
See also @code{lbfgs_nfeval_max} and @code{lbfgs_ncorrections}.
References:
[1] D. Liu and J. Nocedal. "On the limited memory BFGS method for large
scale optimization". @i{Mathematical Programming B} 45:503--528 (1989)
[2] @url{http://netlib.org/opt/lbfgs_um.shar}
Examples:
The same FOM as computed by FGCOMPUTE in the program sdrive.f in the LBFGS package from Netlib.
Note that the variables in question are subscripted variables.
The FOM has an exact minimum equal to zero at @math{u[k] = 1} for @math{k = 1, ..., 8}.
@c ===beg===
@c load (lbfgs);
@c t1[j] := 1 - u[j];
@c t2[j] := 10*(u[j + 1] - u[j]^2);
@c n : 8;
@c FOM : sum (t1[2*j - 1]^2 + t2[2*j - 1]^2, j, 1, n/2);
@c lbfgs (FOM, '[u[1], u[2], u[3], u[4], u[5], u[6], u[7], u[8]],
@c [-1.2, 1, -1.2, 1, -1.2, 1, -1.2, 1], 1e-3, [1, 0]);
@c ===end===
@example
(%i1) load (lbfgs);
(%o1) /usr/share/maxima/5.10.0cvs/share/lbfgs/lbfgs.mac
(%i2) t1[j] := 1 - u[j];
(%o2) t1 := 1 - u
j j
(%i3) t2[j] := 10*(u[j + 1] - u[j]^2);
2
(%o3) t2 := 10 (u - u )
j j + 1 j
(%i4) n : 8;
(%o4) 8
(%i5) FOM : sum (t1[2*j - 1]^2 + t2[2*j - 1]^2, j, 1, n/2);
2 2 2 2 2 2
(%o5) 100 (u - u ) + (1 - u ) + 100 (u - u ) + (1 - u )
8 7 7 6 5 5
2 2 2 2 2 2
+ 100 (u - u ) + (1 - u ) + 100 (u - u ) + (1 - u )
4 3 3 2 1 1
(%i6) lbfgs (FOM, '[u[1],u[2],u[3],u[4],u[5],u[6],u[7],u[8]],
[-1.2, 1, -1.2, 1, -1.2, 1, -1.2, 1], 1e-3, [1, 0]);
*************************************************
N= 8 NUMBER OF CORRECTIONS=25
INITIAL VALUES
F= 9.680000000000000D+01 GNORM= 4.657353755084532D+02
*************************************************
@end example
@ifnottex
@example
I NFN FUNC GNORM STEPLENGTH
1 3 1.651479526340304D+01 4.324359291335977D+00 7.926153934390631D-04
2 4 1.650209316638371D+01 3.575788161060007D+00 1.000000000000000D+00
3 5 1.645461701312851D+01 6.230869903601577D+00 1.000000000000000D+00
4 6 1.636867301275588D+01 1.177589920974980D+01 1.000000000000000D+00
5 7 1.612153014409201D+01 2.292797147151288D+01 1.000000000000000D+00
6 8 1.569118407390628D+01 3.687447158775571D+01 1.000000000000000D+00
7 9 1.510361958398942D+01 4.501931728123680D+01 1.000000000000000D+00
8 10 1.391077875774294D+01 4.526061463810632D+01 1.000000000000000D+00
9 11 1.165625686278198D+01 2.748348965356917D+01 1.000000000000000D+00
10 12 9.859422687859137D+00 2.111494974231644D+01 1.000000000000000D+00
11 13 7.815442521732281D+00 6.110762325766556D+00 1.000000000000000D+00
12 15 7.346380905773160D+00 2.165281166714631D+01 1.285316401779533D-01
13 16 6.330460634066370D+00 1.401220851762050D+01 1.000000000000000D+00
14 17 5.238763939851439D+00 1.702473787613255D+01 1.000000000000000D+00
15 18 3.754016790406701D+00 7.981845727704576D+00 1.000000000000000D+00
16 20 3.001238402309352D+00 3.925482944716691D+00 2.333129631296807D-01
17 22 2.794390709718290D+00 8.243329982546473D+00 2.503577283782332D-01
18 23 2.563783562918759D+00 1.035413426521790D+01 1.000000000000000D+00
19 24 2.019429976377856D+00 1.065187312346769D+01 1.000000000000000D+00
20 25 1.428003167670903D+00 2.475962450826961D+00 1.000000000000000D+00
21 27 1.197874264861340D+00 8.441707983493810D+00 4.303451060808756D-01
22 28 9.023848941942773D-01 1.113189216635162D+01 1.000000000000000D+00
23 29 5.508226405863770D-01 2.380830600326308D+00 1.000000000000000D+00
24 31 3.902893258815567D-01 5.625595816584421D+00 4.834988416524465D-01
25 32 3.207542206990315D-01 1.149444645416472D+01 1.000000000000000D+00
26 33 1.874468266362791D-01 3.632482152880997D+00 1.000000000000000D+00
27 34 9.575763380706598D-02 4.816497446154354D+00 1.000000000000000D+00
28 35 4.085145107543406D-02 2.087009350166495D+00 1.000000000000000D+00
29 36 1.931106001379290D-02 3.886818608498966D+00 1.000000000000000D+00
30 37 6.894000721499670D-03 3.198505796342214D+00 1.000000000000000D+00
31 38 1.443296033051864D-03 1.590265471025043D+00 1.000000000000000D+00
32 39 1.571766603154336D-04 3.098257063980634D-01 1.000000000000000D+00
33 40 1.288011776581970D-05 1.207784183577257D-02 1.000000000000000D+00
34 41 1.806140173752971D-06 4.587890233385193D-02 1.000000000000000D+00
35 42 1.769004645459358D-07 1.790537375052208D-02 1.000000000000000D+00
36 43 3.312164100763217D-10 6.782068426119681D-04 1.000000000000000D+00
@end example
@end ifnottex
@tex
\halign{\hfil\tt#&\quad\hfil\tt#\quad&\tt#\hfil\quad&\tt#\hfil\quad&\tt#\hfil\cr
I& NFN& FUNC& GNORM& STEPLENGTH\cr
&&&&\cr
1& 3& 1.651479526340304D+01& 4.324359291335977D+00& 7.926153934390631D-04\cr
2& 4& 1.650209316638371D+01& 3.575788161060007D+00& 1.000000000000000D+00\cr
3& 5& 1.645461701312851D+01& 6.230869903601577D+00& 1.000000000000000D+00\cr
4& 6& 1.636867301275588D+01& 1.177589920974980D+01& 1.000000000000000D+00\cr
5& 7& 1.612153014409201D+01& 2.292797147151288D+01& 1.000000000000000D+00\cr
6& 8& 1.569118407390628D+01& 3.687447158775571D+01& 1.000000000000000D+00\cr
7& 9& 1.510361958398942D+01& 4.501931728123680D+01& 1.000000000000000D+00\cr
8&10& 1.391077875774294D+01& 4.526061463810632D+01& 1.000000000000000D+00\cr
9&11& 1.165625686278198D+01& 2.748348965356917D+01& 1.000000000000000D+00\cr
10&12& 9.859422687859137D+00& 2.111494974231644D+01& 1.000000000000000D+00\cr
11&13& 7.815442521732281D+00& 6.110762325766556D+00& 1.000000000000000D+00\cr
12&15& 7.346380905773160D+00& 2.165281166714631D+01& 1.285316401779533D-01\cr
13&16& 6.330460634066370D+00& 1.401220851762050D+01& 1.000000000000000D+00\cr
14&17& 5.238763939851439D+00& 1.702473787613255D+01& 1.000000000000000D+00\cr
15&18& 3.754016790406701D+00& 7.981845727704576D+00& 1.000000000000000D+00\cr
16&20& 3.001238402309352D+00& 3.925482944716691D+00& 2.333129631296807D-01\cr
17&22& 2.794390709718290D+00& 8.243329982546473D+00& 2.503577283782332D-01\cr
18&23& 2.563783562918759D+00& 1.035413426521790D+01& 1.000000000000000D+00\cr
19&24& 2.019429976377856D+00& 1.065187312346769D+01& 1.000000000000000D+00\cr
20&25& 1.428003167670903D+00& 2.475962450826961D+00& 1.000000000000000D+00\cr
21&27& 1.197874264861340D+00& 8.441707983493810D+00& 4.303451060808756D-01\cr
22&28& 9.023848941942773D-01& 1.113189216635162D+01& 1.000000000000000D+00\cr
23&29& 5.508226405863770D-01& 2.380830600326308D+00& 1.000000000000000D+00\cr
24&31& 3.902893258815567D-01& 5.625595816584421D+00& 4.834988416524465D-01\cr
25&32& 3.207542206990315D-01& 1.149444645416472D+01& 1.000000000000000D+00\cr
26&33& 1.874468266362791D-01& 3.632482152880997D+00& 1.000000000000000D+00\cr
27&34& 9.575763380706598D-02& 4.816497446154354D+00& 1.000000000000000D+00\cr
28&35& 4.085145107543406D-02& 2.087009350166495D+00& 1.000000000000000D+00\cr
29&36& 1.931106001379290D-02& 3.886818608498966D+00& 1.000000000000000D+00\cr
30&37& 6.894000721499670D-03& 3.198505796342214D+00& 1.000000000000000D+00\cr
31&38& 1.443296033051864D-03& 1.590265471025043D+00& 1.000000000000000D+00\cr
32&39& 1.571766603154336D-04& 3.098257063980634D-01& 1.000000000000000D+00\cr
33&40& 1.288011776581970D-05& 1.207784183577257D-02& 1.000000000000000D+00\cr
34&41& 1.806140173752971D-06& 4.587890233385193D-02& 1.000000000000000D+00\cr
35&42& 1.769004645459358D-07& 1.790537375052208D-02& 1.000000000000000D+00\cr
36&43& 3.312164100763217D-10& 6.782068426119681D-04& 1.000000000000000D+00\cr
}
@end tex
@example
THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
IFLAG = 0
(%o6) [u = 1.000005339815974, u = 1.000009942839805,
1 2
u = 1.000005339815974, u = 1.000009942839805,
3 4
u = 1.000005339815974, u = 1.000009942839805,
5 6
u = 1.000005339815974, u = 1.000009942839805]
7 8
@end example
A regression problem.
The FOM is the mean square difference between the predicted value @math{F(X[i])}
and the observed value @math{Y[i]}.
The function @math{F} is a bounded monotone function (a so-called "sigmoidal" function).
In this example, @code{lbfgs} computes approximate values for the parameters of @math{F}
and @code{plot2d} displays a comparison of @math{F} with the observed data.
@c ===beg===
@c load (lbfgs);
@c FOM : '((1/length(X))*sum((F(X[i]) - Y[i])^2, i, 1,
@c length(X)));
@c X : [1, 2, 3, 4, 5];
@c Y : [0, 0.5, 1, 1.25, 1.5];
@c F(x) := A/(1 + exp(-B*(x - C)));
@c ''FOM;
@c estimates : lbfgs (FOM, '[A, B, C], [1, 1, 1], 1e-4, [1, 0]);
@c plot2d ([F(x), [discrete, X, Y]], [x, -1, 6]), ''estimates;
@c ===end===
@example
(%i1) load (lbfgs);
(%o1) /usr/share/maxima/5.10.0cvs/share/lbfgs/lbfgs.mac
(%i2) FOM : '((1/length(X))*sum((F(X[i]) - Y[i])^2, i, 1,
length(X)));
2
sum((F(X ) - Y ) , i, 1, length(X))
i i
(%o2) -----------------------------------
length(X)
(%i3) X : [1, 2, 3, 4, 5];
(%o3) [1, 2, 3, 4, 5]
(%i4) Y : [0, 0.5, 1, 1.25, 1.5];
(%o4) [0, 0.5, 1, 1.25, 1.5]
(%i5) F(x) := A/(1 + exp(-B*(x - C)));
A
(%o5) F(x) := ----------------------
1 + exp((- B) (x - C))
(%i6) ''FOM;
A 2 A 2
(%o6) ((----------------- - 1.5) + (----------------- - 1.25)
- B (5 - C) - B (4 - C)
%e + 1 %e + 1
A 2 A 2
+ (----------------- - 1) + (----------------- - 0.5)
- B (3 - C) - B (2 - C)
%e + 1 %e + 1
2
A
+ --------------------)/5
- B (1 - C) 2
(%e + 1)
(%i7) estimates : lbfgs (FOM, '[A, B, C], [1, 1, 1], 1e-4, [1, 0]);
*************************************************
N= 3 NUMBER OF CORRECTIONS=25
INITIAL VALUES
F= 1.348738534246918D-01 GNORM= 2.000215531936760D-01
*************************************************
@end example
@ifnottex
@example
I NFN FUNC GNORM STEPLENGTH
1 3 1.177820636622582D-01 9.893138394953992D-02 8.554435968992371D-01
2 6 2.302653892214013D-02 1.180098521565904D-01 2.100000000000000D+01
3 8 1.496348495303005D-02 9.611201567691633D-02 5.257340567840707D-01
4 9 7.900460841091139D-03 1.325041647391314D-02 1.000000000000000D+00
5 10 7.314495451266917D-03 1.510670810312237D-02 1.000000000000000D+00
6 11 6.750147275936680D-03 1.914964958023047D-02 1.000000000000000D+00
7 12 5.850716021108205D-03 1.028089194579363D-02 1.000000000000000D+00
8 13 5.778664230657791D-03 3.676866074530332D-04 1.000000000000000D+00
9 14 5.777818823650782D-03 3.010740179797255D-04 1.000000000000000D+00
@end example
@end ifnottex
@tex
\halign{\hfil\tt#&\quad\hfil\tt#\quad&\tt#\hfil\quad&\tt#\hfil\quad&\tt#\hfil\cr
I& NFN& FUNC& GNORM& STEPLENGTH\cr
&&&&\cr
1& 3&1.177820636622582D-01& 9.893138394953992D-02& 8.554435968992371D-01\cr
2& 6&2.302653892214013D-02& 1.180098521565904D-01& 2.100000000000000D+01\cr
3& 8&1.496348495303005D-02& 9.611201567691633D-02& 5.257340567840707D-01\cr
4& 9&7.900460841091139D-03& 1.325041647391314D-02& 1.000000000000000D+00\cr
5& 10&7.314495451266917D-03& 1.510670810312237D-02& 1.000000000000000D+00\cr
6& 11&6.750147275936680D-03& 1.914964958023047D-02& 1.000000000000000D+00\cr
7& 12&5.850716021108205D-03& 1.028089194579363D-02& 1.000000000000000D+00\cr
8& 13&5.778664230657791D-03& 3.676866074530332D-04& 1.000000000000000D+00\cr
9& 14&5.777818823650782D-03& 3.010740179797255D-04& 1.000000000000000D+00\cr
}
@end tex
@example
THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
IFLAG = 0
(%o7) [A = 1.461933911464101, B = 1.601593973254802,
C = 2.528933072164854]
(%i8) plot2d ([F(x), [discrete, X, Y]], [x, -1, 6]), ''estimates;
(%o8)
@end example
Gradient of FOM is specified (instead of computing it automatically).
@c ===beg===
@c load (lbfgs)$
@c F(a, b, c) := (a - 5)^2 + (b - 3)^4 + (c - 2)^6;
@c F_grad : map (lambda ([x], diff (F(a, b, c), x)), [a, b, c]);
@c estimates : lbfgs ([F(a, b, c), F_grad], [a, b, c], [0, 0, 0], 1e-4, [1, 0]);
@c ===end===
@example
(%i1) load (lbfgs)$
(%i2) F(a, b, c) := (a - 5)^2 + (b - 3)^4 + (c - 2)^6;
2 4 6
(%o2) F(a, b, c) := (a - 5) + (b - 3) + (c - 2)
(%i3) F_grad : map (lambda ([x], diff (F(a, b, c), x)), [a, b, c]);
3 5
(%o3) [2 (a - 5), 4 (b - 3) , 6 (c - 2) ]
(%i4) estimates : lbfgs ([F(a, b, c), F_grad], [a, b, c], [0, 0, 0], 1e-4, [1, 0]);
*************************************************
N= 3 NUMBER OF CORRECTIONS=25
INITIAL VALUES
F= 1.700000000000000D+02 GNORM= 2.205175729958953D+02
*************************************************
@end example
@ifnottex
@example
I NFN FUNC GNORM STEPLENGTH
1 2 6.632967565917638D+01 6.498411132518770D+01 4.534785987412505D-03
2 3 4.368890936228036D+01 3.784147651974131D+01 1.000000000000000D+00
3 4 2.685298972775190D+01 1.640262125898521D+01 1.000000000000000D+00
4 5 1.909064767659852D+01 9.733664001790506D+00 1.000000000000000D+00
5 6 1.006493272061515D+01 6.344808151880209D+00 1.000000000000000D+00
6 7 1.215263596054294D+00 2.204727876126879D+00 1.000000000000000D+00
7 8 1.080252896385334D-02 1.431637116951849D-01 1.000000000000000D+00
8 9 8.407195124830908D-03 1.126344579730013D-01 1.000000000000000D+00
9 10 5.022091686198527D-03 7.750731829225274D-02 1.000000000000000D+00
10 11 2.277152808939775D-03 5.032810859286795D-02 1.000000000000000D+00
11 12 6.489384688303218D-04 1.932007150271008D-02 1.000000000000000D+00
12 13 2.075791943844548D-04 6.964319310814364D-03 1.000000000000000D+00
13 14 7.349472666162257D-05 4.017449067849554D-03 1.000000000000000D+00
14 15 2.293617477985237D-05 1.334590390856715D-03 1.000000000000000D+00
15 16 7.683645404048675D-06 6.011057038099201D-04 1.000000000000000D+00
@end example
@end ifnottex
@tex
\halign{\hfil\tt#&\quad\hfil\tt#\quad&\tt#\hfil\quad&\tt#\hfil\quad&\tt#\hfil\cr
I& NFN& FUNC& GNORM& STEPLENGTH\cr
&&&&\cr
1& 2& 6.632967565917638D+01& 6.498411132518770D+01& 4.534785987412505D-03\cr
2& 3& 4.368890936228036D+01& 3.784147651974131D+01& 1.000000000000000D+00\cr
3& 4& 2.685298972775190D+01& 1.640262125898521D+01& 1.000000000000000D+00\cr
4& 5& 1.909064767659852D+01& 9.733664001790506D+00& 1.000000000000000D+00\cr
5& 6& 1.006493272061515D+01& 6.344808151880209D+00& 1.000000000000000D+00\cr
6& 7& 1.215263596054294D+00& 2.204727876126879D+00& 1.000000000000000D+00\cr
7& 8& 1.080252896385334D-02& 1.431637116951849D-01& 1.000000000000000D+00\cr
8& 9& 8.407195124830908D-03& 1.126344579730013D-01& 1.000000000000000D+00\cr
9& 10& 5.022091686198527D-03& 7.750731829225274D-02& 1.000000000000000D+00\cr
10& 11& 2.277152808939775D-03& 5.032810859286795D-02& 1.000000000000000D+00\cr
11& 12& 6.489384688303218D-04& 1.932007150271008D-02& 1.000000000000000D+00\cr
12& 13& 2.075791943844548D-04& 6.964319310814364D-03& 1.000000000000000D+00\cr
13& 14& 7.349472666162257D-05& 4.017449067849554D-03& 1.000000000000000D+00\cr
14& 15& 2.293617477985237D-05& 1.334590390856715D-03& 1.000000000000000D+00\cr
15& 16& 7.683645404048675D-06& 6.011057038099201D-04& 1.000000000000000D+00\cr
}
@end tex
@example
THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
IFLAG = 0
(%o4) [a = 5.000086823042934, b = 3.05239542970518,
c = 1.927980629919583]
@end example
@opencatbox
@category{Package lbfgs}
@closecatbox
@end deffn
@defvr {Variable} lbfgs_nfeval_max
Default value: 100
@code{lbfgs_nfeval_max} is the maximum number of evaluations of the figure of merit (FOM) in @code{lbfgs}.
When @code{lbfgs_nfeval_max} is reached,
@code{lbfgs} returns the result of the last successful line search.
@opencatbox
@category{Package lbfgs}
@closecatbox
@end defvr
@defvr {Variable} lbfgs_ncorrections
Default value: 25
@code{lbfgs_ncorrections} is the number of corrections applied
to the approximate inverse Hessian matrix which is maintained by @code{lbfgs}.
@opencatbox
@category{Package lbfgs}
@closecatbox
@end defvr
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