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
|
# ----------------------------------------------------------------------------
# - Open3D: www.open3d.org -
# ----------------------------------------------------------------------------
# The MIT License (MIT)
#
# Copyright (c) 2018-2021 www.open3d.org
#
# 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.
# ----------------------------------------------------------------------------
import numpy as np
from collections import OrderedDict
def compute_jacobian_finite_differences(x0, fn, epsilon):
"""Computes the Jacobian using finite differences
x0: The positions at which to compute J.
fn: A function of the form fn(x) which returns a single numpy array.
epsilon: A scalar or an array that can be broadcasted to the same
shape as x0.
"""
dtype = x0.dtype
y0 = fn(x0)
h = np.zeros_like(x0)
J = np.zeros((x0.size, y0.size), dtype=dtype)
epsilon_arr = np.broadcast_to(epsilon, x0.shape)
for i in range(x0.size):
eps = epsilon_arr.flat[i]
h.flat[i] = eps
J[i, :] = ((fn(x0 + h) - y0) / eps).flat
h.flat[i] = 0
return J
def compute_jacobian_analytical(x0, y_shape, fn_grad, y_bp=None):
"""Computes the analytical Jacobian
x0: The position at which to compute J.
y_shape: The shape of the backpropagated value, i.e. the shape of
the output of the corresponding function 'fn'.
fn_grad: The gradient of the original function with the form
x_grad = fn_grad(y_bp, x0) where 'y_bp' is the backpropagated
value and 'x0' is the original input to 'fn'. The output of
the function is the gradient of x wrt to y.
y_bp: Optional array with custom values for individually scaling
the gradients.
"""
dtype = x0.dtype
y_size = 1
for k in y_shape:
y_size *= k
J = np.zeros((x0.size, y_size), dtype=dtype)
y = np.zeros(y_shape, dtype=dtype)
y_bp_arr = np.broadcast_to(y_bp, y_shape) if not y_bp is None else np.ones(
y_shape, dtype=dtype)
for j in range(y_size):
y.flat[j] = y_bp_arr.flat[j]
J[:, j] = fn_grad(y, x0).flat
y.flat[j] = 0
return J
def check_gradients(x0,
fn,
fn_grad,
epsilon=1e-6,
rtol=1e-3,
atol=1e-5,
debug_outputs=OrderedDict()):
"""Checks if the numerical and analytical gradients are compatible for a function 'fn'
x0: The position at which to compute the gradients.
fn: A function of the form fn(x) which returns a single numpy array.
fn_grad: The gradient of the original function with the form
x_grad = fn_grad(y_bp, x0) where 'y_bp' is the backpropagated
value and 'x0' is the original input to 'fn'. The output of
the function is the gradient of x wrt to y.
epsilon: A scalar or an array that can be broadcasted to the same
shape as x0. This is used for computing the numerical Jacobian
rtol: The relative tolerance parameter used in numpy.allclose()
atol: The absolute tolerance parameter used in numpy.allclose()
debug_outputs: Output variable which stores additional outputs useful for
debugging in a dictionary.
"""
dtype = x0.dtype
y = fn(x0) # compute y to get the shape
grad = fn_grad(np.zeros(y.shape, dtype=dtype), x0)
grad_shape_correct = x0.shape == grad.shape
if not grad_shape_correct:
print(
'The shape of the gradient [{0}] does not match the shape of "x0" [{1}].'
.format(grad.shape, x0.shape))
zero_grad = np.count_nonzero(grad) == 0
if not zero_grad:
print('The gradient is not zero for a zero backprop vector.')
ana_J = compute_jacobian_analytical(x0, y.shape, fn_grad)
ana_J2 = compute_jacobian_analytical(x0, y.shape, fn_grad,
2 * np.ones(y.shape, dtype=x0.dtype))
num_J = compute_jacobian_finite_differences(x0, fn, epsilon)
does_scale = np.allclose(0.5 * ana_J2, ana_J, rtol, atol)
isclose = np.allclose(ana_J, num_J, rtol, atol)
ana_J_iszero = np.all(ana_J == 0)
if ana_J_iszero and not np.allclose(num_J, np.zeros_like(num_J), rtol,
atol):
print(
'The values of the analytical Jacobian are all zero but the values of the numerical Jacobian are not.'
)
elif not does_scale:
print(
'The gradients do not scale with respect to the backpropagated values.'
)
if not isclose:
print('The gradients are not close to the numerical Jacobian.')
debug_outputs.update(
OrderedDict([
('isclose', isclose),
('does_scale', does_scale),
('ana_J_iszero', ana_J_iszero),
('grad_shape_correct', grad_shape_correct),
('zero_grad', zero_grad),
('ana_J', ana_J),
('num_J', num_J),
('absdiff', np.abs(ana_J - num_J)),
]))
result = isclose and does_scale
return result
|
