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56. lbfgs


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56.1 Introduction to lbfgs

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. Moré and David J. Thuente, and translated into Lisp automatically via the program f2cl. The Maxima package 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". Mathematical Programming B 45:503-528 (1989)

[2] http://netlib.org/opt/lbfgs_um.shar


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56.2 Functions and Variables for lbfgs

Function: lbfgs (FOM, X, X0, epsilon, iprint)
Function: lbfgs ([FOM, grad] X, X0, epsilon, iprint)

Finds an approximate solution of the unconstrained minimization of the figure of merit FOM over the list of variables X, starting from initial estimates X0, such that norm(grad(FOM)) < epsilon*max(1, norm(X)).

grad, if present, is the gradient of FOM with respect to the variables X. grad is a list, with one element for each element of 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 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.

iprint controls progress messages printed by lbfgs.

iprint[1]

iprint[1] controls the frequency of progress messages.

iprint[1] < 0

No progress messages.

iprint[1] = 0

Messages at the first and last iterations.

iprint[1] > 0

Print a message every iprint[1] iterations.

iprint[2]

iprint[2] controls the verbosity of progress messages.

iprint[2] = 0

Print out iteration count, number of evaluations of FOM, value of FOM, norm of the gradient of FOM, and step length.

iprint[2] = 1

Same as iprint[2] = 0, plus X0 and the gradient of FOM evaluated at X0.

iprint[2] = 2

Same as iprint[2] = 1, plus values of X at each iteration.

iprint[2] = 3

Same as iprint[2] = 2, plus the gradient of FOM at each iteration.

The columns printed by lbfgs are the following.

I

Number of iterations. It is incremented for each line search.

NFN

Number of evaluations of the figure of merit.

FUNC

Value of the figure of merit at the end of the most recent line search.

GNORM

Norm of the gradient of the figure of merit at the end of the most recent line search.

STEPLENGTH

An internal parameter of the search algorithm.

Additional information concerning details of the algorithm are found in the comments of the original Fortran code [2].

See also lbfgs_nfeval_max and lbfgs_ncorrections.

References:

[1] D. Liu and J. Nocedal. "On the limited memory BFGS method for large scale optimization". Mathematical Programming B 45:503-528 (1989)

[2] 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 u[k] = 1 for k = 1, ..., 8.

(%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
*************************************************
 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
 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

A regression problem. The FOM is the mean square difference between the predicted value F(X[i]) and the observed value Y[i]. The function F is a bounded monotone function (a so-called "sigmoidal" function). In this example, lbfgs computes approximate values for the parameters of F and plot2d displays a comparison of F with the observed data.

(%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
*************************************************

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  
 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) 

Gradient of FOM is specified (instead of computing it automatically).

(%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
*************************************************

   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  
 THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
 IFLAG = 0
(%o4) [a = 5.000086823042934, b = 3.05239542970518, 
                                           c = 1.927980629919583]
Variable: lbfgs_nfeval_max

Default value: 100

lbfgs_nfeval_max is the maximum number of evaluations of the figure of merit (FOM) in lbfgs. When lbfgs_nfeval_max is reached, lbfgs returns the result of the last successful line search.

Variable: lbfgs_ncorrections

Default value: 25

lbfgs_ncorrections is the number of corrections applied to the approximate inverse Hessian matrix which is maintained by lbfgs.


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