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
|
// Copyright (C) 2016-2019 Yixuan Qiu <yixuan.qiu@cos.name>
// Under MIT license
#ifndef LBFGS_H
#define LBFGS_H
#include <Eigen/Core>
#include "LBFGS/Param.h"
#include "LBFGS/LineSearchBacktracking.h"
#include "LBFGS/LineSearchBracketing.h"
#include "LBFGS/LineSearchNocedalWright.h"
namespace LBFGSpp {
///
/// LBFGS solver for unconstrained numerical optimization
///
template < typename Scalar,
template<class> class LineSearch = LineSearchBacktracking >
class LBFGSSolver
{
private:
typedef Eigen::Matrix<Scalar, Eigen::Dynamic, 1> Vector;
typedef Eigen::Matrix<Scalar, Eigen::Dynamic, Eigen::Dynamic> Matrix;
typedef Eigen::Map<Vector> MapVec;
const LBFGSParam<Scalar>& m_param; // Parameters to control the LBFGS algorithm
Matrix m_s; // History of the s vectors
Matrix m_y; // History of the y vectors
Vector m_ys; // History of the s'y values
Vector m_alpha; // History of the step lengths
Vector m_fx; // History of the objective function values
Vector m_xp; // Old x
Vector m_grad; // New gradient
Vector m_gradp; // Old gradient
Vector m_drt; // Moving direction
inline void reset(int n)
{
const int m = m_param.m;
m_s.resize(n, m);
m_y.resize(n, m);
m_ys.resize(m);
m_alpha.resize(m);
m_xp.resize(n);
m_grad.resize(n);
m_gradp.resize(n);
m_drt.resize(n);
if(m_param.past > 0)
m_fx.resize(m_param.past);
}
public:
///
/// Constructor for LBFGS solver.
///
/// \param param An object of \ref LBFGSParam to store parameters for the
/// algorithm
///
LBFGSSolver(const LBFGSParam<Scalar>& param) :
m_param(param)
{
m_param.check_param();
}
///
/// Minimizing a multivariate function using LBFGS algorithm.
/// Exceptions will be thrown if error occurs.
///
/// \param f A function object such that `f(x, grad)` returns the
/// objective function value at `x`, and overwrites `grad` with
/// the gradient.
/// \param x In: An initial guess of the optimal point. Out: The best point
/// found.
/// \param fx Out: The objective function value at `x`.
///
/// \return Number of iterations used.
///
template <typename Foo>
inline int minimize(Foo& f, Vector& x, Scalar& fx)
{
const int n = x.size();
const int fpast = m_param.past;
reset(n);
// Evaluate function and compute gradient
fx = f(x, m_grad);
Scalar xnorm = x.norm();
Scalar gnorm = m_grad.norm();
if(fpast > 0)
m_fx[0] = fx;
// Early exit if the initial x is already a minimizer
if(gnorm <= m_param.epsilon * std::max(xnorm, Scalar(1.0)))
{
return 1;
}
// Initial direction
m_drt.noalias() = -m_grad;
// Initial step
Scalar step = Scalar(1.0) / m_drt.norm();
int k = 1;
int end = 0;
for( ; ; )
{
// Save the curent x and gradient
m_xp.noalias() = x;
m_gradp.noalias() = m_grad;
// Line search to update x, fx and gradient
LineSearch<Scalar>::LineSearch(f, fx, x, m_grad, step, m_drt, m_xp, m_param);
// New x norm and gradient norm
xnorm = x.norm();
gnorm = m_grad.norm();
// Convergence test -- gradient
if(gnorm <= m_param.epsilon * std::max(xnorm, Scalar(1.0)))
{
return k;
}
// Convergence test -- objective function value
if(fpast > 0)
{
if(k >= fpast && std::abs((m_fx[k % fpast] - fx) / fx) < m_param.delta)
return k;
m_fx[k % fpast] = fx;
}
// Maximum number of iterations
if(m_param.max_iterations != 0 && k >= m_param.max_iterations)
{
return k;
}
// Update s and y
// s_{k+1} = x_{k+1} - x_k
// y_{k+1} = g_{k+1} - g_k
MapVec svec(&m_s(0, end), n);
MapVec yvec(&m_y(0, end), n);
svec.noalias() = x - m_xp;
yvec.noalias() = m_grad - m_gradp;
// ys = y's = 1/rho
// yy = y'y
Scalar ys = yvec.dot(svec);
Scalar yy = yvec.squaredNorm();
m_ys[end] = ys;
// Recursive formula to compute d = -H * g
m_drt.noalias() = -m_grad;
int bound = std::min(m_param.m, k);
end = (end + 1) % m_param.m;
int j = end;
for(int i = 0; i < bound; i++)
{
j = (j + m_param.m - 1) % m_param.m;
MapVec sj(&m_s(0, j), n);
MapVec yj(&m_y(0, j), n);
m_alpha[j] = sj.dot(m_drt) / m_ys[j];
m_drt.noalias() -= m_alpha[j] * yj;
}
m_drt *= (ys / yy);
for(int i = 0; i < bound; i++)
{
MapVec sj(&m_s(0, j), n);
MapVec yj(&m_y(0, j), n);
Scalar beta = yj.dot(m_drt) / m_ys[j];
m_drt.noalias() += (m_alpha[j] - beta) * sj;
j = (j + 1) % m_param.m;
}
// step = 1.0 as initial guess
step = Scalar(1.0);
k++;
}
return k;
}
};
} // namespace LBFGSpp
#endif // LBFGS_H
|
