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
|
"""
Tests for line search routines
"""
from __future__ import division, print_function, absolute_import
import warnings
from numpy.testing import assert_, assert_equal, \
assert_array_almost_equal, assert_array_almost_equal_nulp
import scipy.optimize.linesearch as ls
from scipy.optimize.linesearch import LineSearchWarning
import numpy as np
def assert_wolfe(s, phi, derphi, c1=1e-4, c2=0.9, err_msg=""):
"""
Check that strong Wolfe conditions apply
"""
phi1 = phi(s)
phi0 = phi(0)
derphi0 = derphi(0)
derphi1 = derphi(s)
msg = "s = %s; phi(0) = %s; phi(s) = %s; phi'(0) = %s; phi'(s) = %s; %s" % (
s, phi0, phi1, derphi0, derphi1, err_msg)
assert_(phi1 <= phi0 + c1*s*derphi0, "Wolfe 1 failed: " + msg)
assert_(abs(derphi1) <= abs(c2*derphi0), "Wolfe 2 failed: " + msg)
def assert_armijo(s, phi, c1=1e-4, err_msg=""):
"""
Check that Armijo condition applies
"""
phi1 = phi(s)
phi0 = phi(0)
msg = "s = %s; phi(0) = %s; phi(s) = %s; %s" % (s, phi0, phi1, err_msg)
assert_(phi1 <= (1 - c1*s)*phi0, msg)
def assert_line_wolfe(x, p, s, f, fprime, **kw):
assert_wolfe(s, phi=lambda sp: f(x + p*sp),
derphi=lambda sp: np.dot(fprime(x + p*sp), p), **kw)
def assert_line_armijo(x, p, s, f, **kw):
assert_armijo(s, phi=lambda sp: f(x + p*sp), **kw)
def assert_fp_equal(x, y, err_msg="", nulp=50):
"""Assert two arrays are equal, up to some floating-point rounding error"""
try:
assert_array_almost_equal_nulp(x, y, nulp)
except AssertionError as e:
raise AssertionError("%s\n%s" % (e, err_msg))
class TestLineSearch(object):
# -- scalar functions; must have dphi(0.) < 0
def _scalar_func_1(self, s):
self.fcount += 1
p = -s - s**3 + s**4
dp = -1 - 3*s**2 + 4*s**3
return p, dp
def _scalar_func_2(self, s):
self.fcount += 1
p = np.exp(-4*s) + s**2
dp = -4*np.exp(-4*s) + 2*s
return p, dp
def _scalar_func_3(self, s):
self.fcount += 1
p = -np.sin(10*s)
dp = -10*np.cos(10*s)
return p, dp
# -- n-d functions
def _line_func_1(self, x):
self.fcount += 1
f = np.dot(x, x)
df = 2*x
return f, df
def _line_func_2(self, x):
self.fcount += 1
f = np.dot(x, np.dot(self.A, x)) + 1
df = np.dot(self.A + self.A.T, x)
return f, df
# --
def __init__(self):
self.scalar_funcs = []
self.line_funcs = []
self.N = 20
self.fcount = 0
def bind_index(func, idx):
# Remember Python's closure semantics!
return lambda *a, **kw: func(*a, **kw)[idx]
for name in sorted(dir(self)):
if name.startswith('_scalar_func_'):
value = getattr(self, name)
self.scalar_funcs.append(
(name, bind_index(value, 0), bind_index(value, 1)))
elif name.startswith('_line_func_'):
value = getattr(self, name)
self.line_funcs.append(
(name, bind_index(value, 0), bind_index(value, 1)))
def setUp(self):
np.random.seed(1234)
self.A = np.random.randn(self.N, self.N)
def scalar_iter(self):
for name, phi, derphi in self.scalar_funcs:
for old_phi0 in np.random.randn(3):
yield name, phi, derphi, old_phi0
def line_iter(self):
for name, f, fprime in self.line_funcs:
k = 0
while k < 9:
x = np.random.randn(self.N)
p = np.random.randn(self.N)
if np.dot(p, fprime(x)) >= 0:
# always pick a descent direction
continue
k += 1
old_fv = float(np.random.randn())
yield name, f, fprime, x, p, old_fv
# -- Generic scalar searches
def test_scalar_search_wolfe1(self):
c = 0
for name, phi, derphi, old_phi0 in self.scalar_iter():
c += 1
s, phi1, phi0 = ls.scalar_search_wolfe1(phi, derphi, phi(0),
old_phi0, derphi(0))
assert_fp_equal(phi0, phi(0), name)
assert_fp_equal(phi1, phi(s), name)
assert_wolfe(s, phi, derphi, err_msg=name)
assert_(c > 3) # check that the iterator really works...
def test_scalar_search_wolfe2(self):
for name, phi, derphi, old_phi0 in self.scalar_iter():
s, phi1, phi0, derphi1 = ls.scalar_search_wolfe2(
phi, derphi, phi(0), old_phi0, derphi(0))
assert_fp_equal(phi0, phi(0), name)
assert_fp_equal(phi1, phi(s), name)
if derphi1 is not None:
assert_fp_equal(derphi1, derphi(s), name)
assert_wolfe(s, phi, derphi, err_msg="%s %g" % (name, old_phi0))
def test_scalar_search_armijo(self):
for name, phi, derphi, old_phi0 in self.scalar_iter():
s, phi1 = ls.scalar_search_armijo(phi, phi(0), derphi(0))
assert_fp_equal(phi1, phi(s), name)
assert_armijo(s, phi, err_msg="%s %g" % (name, old_phi0))
# -- Generic line searches
def test_line_search_wolfe1(self):
c = 0
smax = 100
for name, f, fprime, x, p, old_f in self.line_iter():
f0 = f(x)
g0 = fprime(x)
self.fcount = 0
s, fc, gc, fv, ofv, gv = ls.line_search_wolfe1(f, fprime, x, p,
g0, f0, old_f,
amax=smax)
assert_equal(self.fcount, fc+gc)
assert_fp_equal(ofv, f(x))
if s is None:
continue
assert_fp_equal(fv, f(x + s*p))
assert_array_almost_equal(gv, fprime(x + s*p), decimal=14)
if s < smax:
c += 1
assert_line_wolfe(x, p, s, f, fprime, err_msg=name)
assert_(c > 3) # check that the iterator really works...
def test_line_search_wolfe2(self):
c = 0
smax = 100
for name, f, fprime, x, p, old_f in self.line_iter():
f0 = f(x)
g0 = fprime(x)
self.fcount = 0
with warnings.catch_warnings():
warnings.simplefilter('ignore', LineSearchWarning)
s, fc, gc, fv, ofv, gv = ls.line_search_wolfe2(f, fprime, x, p,
g0, f0, old_f,
amax=smax)
assert_equal(self.fcount, fc+gc)
assert_fp_equal(ofv, f(x))
assert_fp_equal(fv, f(x + s*p))
if gv is not None:
assert_array_almost_equal(gv, fprime(x + s*p), decimal=14)
if s < smax:
c += 1
assert_line_wolfe(x, p, s, f, fprime, err_msg=name)
assert_(c > 3) # check that the iterator really works...
def test_line_search_armijo(self):
c = 0
for name, f, fprime, x, p, old_f in self.line_iter():
f0 = f(x)
g0 = fprime(x)
self.fcount = 0
s, fc, fv = ls.line_search_armijo(f, x, p, g0, f0)
c += 1
assert_equal(self.fcount, fc)
assert_fp_equal(fv, f(x + s*p))
assert_line_armijo(x, p, s, f, err_msg=name)
assert_(c >= 9)
# -- More specific tests
def test_armijo_terminate_1(self):
# Armijo should evaluate the function only once if the trial step
# is already suitable
count = [0]
def phi(s):
count[0] += 1
return -s + 0.01*s**2
s, phi1 = ls.scalar_search_armijo(phi, phi(0), -1, alpha0=1)
assert_equal(s, 1)
assert_equal(count[0], 2)
assert_armijo(s, phi)
def test_wolfe_terminate(self):
# wolfe1 and wolfe2 should also evaluate the function only a few
# times if the trial step is already suitable
def phi(s):
count[0] += 1
return -s + 0.05*s**2
def derphi(s):
count[0] += 1
return -1 + 0.05*2*s
for func in [ls.scalar_search_wolfe1, ls.scalar_search_wolfe2]:
count = [0]
r = func(phi, derphi, phi(0), None, derphi(0))
assert_(r[0] is not None, (r, func))
assert_(count[0] <= 2 + 2, (count, func))
assert_wolfe(r[0], phi, derphi, err_msg=str(func))
|
