Note
Click here to download the full example code
Neural Networks¶
Neural networks can be constructed using the borch.nn package.
Now that you’ve had a glimpse of autograd, nn depends on autograd
to define models and differentiate them. An nn.Module contains layers,
and a method forward(input) that returns the output.
For example, look at this network that classifies digit images:
convnet¶
It is a simple feed-forward network. It takes an input, feeds it through several layers one after the other, and then finally gives the output.
A typical training procedure for a neural network is as follows:
Define a network that has some learnable parameters and/or randomVariables
For each batch in a dataset, do:
Process the input data through the network
Compute the loss (how far is the output from being correct?)
Propagate gradients back into the network’s parameters
Update the weights of the network, typically using a simple update rule:
weight = weight - learning_rate * gradient
Define the network¶
Let’s define this network:
import torch
import torch.nn.functional as F
import borch
from borch import distributions, posterior, nn, infer
class Net(nn.Module):
def __init__(self):
super(Net, self).__init__(posterior=posterior.Automatic())
# 1 input image channel, 6 output channels, 5x5 convolution kernel
self.conv1 = nn.Conv2d(1, 6, 5)
# 6 input channels, 16 output channels, 5x5 convolution kernel
self.conv2 = nn.Conv2d(6, 16, 5)
# An affine operation: y = Wx + b
# NB after two convolutional operations with 5x5 kernels and no padding,
# the spatial dimension of an image with intial dimension 32x32 is
# 5x5 (with 16 channels)
self.fc1 = nn.Linear(16 * 5 * 5, 120)
self.fc2 = nn.Linear(120, 10)
def forward(self, x):
# Max pooling over a (2, 2) window
x = F.max_pool2d(F.relu(self.conv1(x)), (2, 2))
# If the size is a square you can only specify a single number
x = F.max_pool2d(F.relu(self.conv2(x)), 2)
x = x.view(1, self.num_flat_features(x))
x = F.relu(self.fc1(x))
x = self.fc2(x)
# Specifying the likelihood function
self.classification = distributions.Categorical(logits=x)
return x
def num_flat_features(self, x):
size = x.size()[1:] # all dimensions except the batch dimension
num_features = 1
for s in size:
num_features *= s
return num_features
net = Net()
print(net)
Out:
Net(
(posterior): Automatic()
(prior): Module()
(observed): Observed()
(conv1): Conv2d(
1, 6, kernel_size=(5, 5), stride=(1, 1)
(posterior): Normal(
(weight): Normal:
posterior: Automatic()
prior: Module()
observed: Observed()
scale: Transform:
tensor([[[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]]],
[[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]]],
[[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]]],
[[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]]],
[[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]]],
[[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]]]], grad_fn=<ExpBackward0>)
loc: Parameter containing:
tensor([[[[ 0.1680, 0.1718, 0.0098, -0.0428, 0.0831],
[ 0.0618, 0.1618, -0.0325, -0.1448, -0.1646],
[-0.0362, -0.0103, -0.1824, 0.1346, 0.0135],
[-0.0389, 0.0558, -0.0680, -0.1515, 0.1680],
[-0.1748, 0.1496, -0.1459, 0.0181, -0.1115]]],
[[[ 0.1081, -0.0935, -0.1567, 0.1787, 0.0230],
[-0.1890, -0.0129, 0.0133, 0.0472, 0.0424],
[-0.0680, -0.0616, -0.0199, 0.0837, 0.1960],
[ 0.0917, 0.0248, 0.0921, -0.0139, 0.0422],
[-0.0997, -0.0637, 0.1865, -0.1982, 0.1430]]],
[[[-0.1463, -0.0137, 0.1728, -0.0899, -0.0427],
[-0.0654, -0.0375, -0.1964, -0.1289, -0.0483],
[ 0.0495, 0.0236, 0.1467, 0.1145, -0.0054],
[-0.0240, -0.0058, -0.0281, 0.1004, -0.0038],
[ 0.0126, 0.0019, -0.1089, -0.1572, -0.1472]]],
[[[-0.0682, -0.0487, 0.0365, -0.1739, -0.0511],
[-0.0746, 0.1603, 0.0149, 0.1684, 0.0557],
[-0.1502, -0.1176, -0.0681, 0.1512, -0.0974],
[-0.0197, -0.1146, 0.0302, -0.0089, -0.0336],
[ 0.1591, 0.1478, -0.1726, -0.0903, 0.0923]]],
[[[-0.1594, 0.1311, 0.0003, -0.0176, -0.1664],
[ 0.0809, 0.1795, 0.1855, 0.1246, -0.0944],
[-0.1488, -0.0044, 0.0512, -0.1854, 0.0012],
[-0.0820, 0.1209, -0.0562, 0.0859, -0.1175],
[ 0.0652, 0.1324, -0.1412, 0.1655, 0.1654]]],
[[[-0.0019, -0.0864, 0.1593, 0.1857, -0.1915],
[ 0.1527, -0.0412, -0.0173, 0.0072, -0.0933],
[-0.1378, -0.1514, 0.1343, -0.0660, -0.0785],
[ 0.1723, 0.0924, 0.1815, -0.0553, 0.1033],
[ 0.0797, 0.0611, 0.0834, -0.1489, 0.1146]]]], requires_grad=True)
tensor: tensor([[[[ 0.1549, 0.1702, 0.0258, -0.0510, 0.1557],
[ 0.0103, 0.1476, 0.0522, -0.2040, -0.1602],
[-0.0236, 0.0437, -0.2519, 0.0928, -0.0489],
[-0.0862, 0.1147, -0.0815, -0.1318, 0.2126],
[-0.2679, 0.1714, -0.1526, 0.0422, -0.1586]]],
[[[ 0.1015, -0.1919, -0.1668, 0.2028, -0.0101],
[-0.1396, -0.0280, 0.0852, 0.0446, 0.0493],
[-0.0921, -0.0273, -0.0788, 0.0889, 0.2471],
[ 0.0824, 0.0427, 0.1677, -0.0809, -0.0029],
[-0.0677, -0.1122, 0.1575, -0.2243, 0.1292]]],
[[[-0.1460, -0.0464, 0.1325, -0.0497, -0.0142],
[-0.0287, 0.0196, -0.2510, -0.1181, 0.0341],
[ 0.0990, 0.0839, 0.1291, 0.1391, -0.0971],
[-0.0203, 0.0117, 0.0436, 0.1213, -0.0132],
[ 0.0082, -0.0122, -0.1402, -0.2093, -0.1193]]],
[[[-0.0340, -0.0688, 0.0707, -0.1087, -0.0648],
[-0.0878, 0.2453, 0.0498, 0.1437, 0.0903],
[-0.1033, -0.1412, -0.0248, 0.1735, -0.0668],
[ 0.0400, -0.0714, 0.0677, 0.0780, -0.0020],
[ 0.1382, 0.1072, -0.1638, -0.0979, 0.0906]]],
[[[-0.2221, 0.1444, -0.0130, -0.0024, -0.1279],
[ 0.0786, 0.2368, 0.2252, 0.1285, -0.0604],
[-0.0719, -0.0685, 0.0014, -0.2441, -0.0323],
[-0.0953, 0.1052, 0.0052, 0.0528, -0.1838],
[ 0.1155, 0.1910, -0.2823, 0.1724, 0.1009]]],
[[[-0.0248, -0.0756, 0.0753, 0.1658, -0.1588],
[ 0.0979, -0.0104, -0.0249, 0.0274, -0.1429],
[-0.1627, -0.1360, 0.1719, -0.1276, -0.0804],
[ 0.1876, 0.0270, 0.1819, -0.0160, 0.1198],
[ 0.1410, 0.0349, 0.1013, -0.1741, 0.2346]]]],
grad_fn=<AddBackward0>)
(bias): Normal:
posterior: Automatic()
prior: Module()
observed: Observed()
scale: Transform:
tensor([0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
grad_fn=<ExpBackward0>)
loc: Parameter containing:
tensor([ 0.0894, 0.1877, -0.0535, 0.0308, 0.1593, 0.0810],
requires_grad=True)
tensor: tensor([ 0.1004, 0.2790, -0.0837, -0.0029, 0.1159, 0.0025],
grad_fn=<AddBackward0>)
)
(prior): Module(
(weight): Normal:
loc: tensor([[[[0., 0., 0., -0., 0.],
[0., 0., -0., -0., -0.],
[-0., -0., -0., 0., 0.],
[-0., 0., -0., -0., 0.],
[-0., 0., -0., 0., -0.]]],
[[[0., -0., -0., 0., 0.],
[-0., -0., 0., 0., 0.],
[-0., -0., -0., 0., 0.],
[0., 0., 0., -0., 0.],
[-0., -0., 0., -0., 0.]]],
[[[-0., -0., 0., -0., -0.],
[-0., -0., -0., -0., -0.],
[0., 0., 0., 0., -0.],
[-0., -0., -0., 0., -0.],
[0., 0., -0., -0., -0.]]],
[[[-0., -0., 0., -0., -0.],
[-0., 0., 0., 0., 0.],
[-0., -0., -0., 0., -0.],
[-0., -0., 0., -0., -0.],
[0., 0., -0., -0., 0.]]],
[[[-0., 0., 0., -0., -0.],
[0., 0., 0., 0., -0.],
[-0., -0., 0., -0., 0.],
[-0., 0., -0., 0., -0.],
[0., 0., -0., 0., 0.]]],
[[[-0., -0., 0., 0., -0.],
[0., -0., -0., 0., -0.],
[-0., -0., 0., -0., -0.],
[0., 0., 0., -0., 0.],
[0., 0., 0., -0., 0.]]]])
scale: tensor([[[[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000]]],
[[[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000]]],
[[[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000]]],
[[[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000]]],
[[[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000]]],
[[[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000],
[0.2000, 0.2000, 0.2000, 0.2000, 0.2000]]]])
posterior: Automatic()
prior: Module()
observed: Observed()
tensor: tensor([[[[ 0.1680, 0.1718, 0.0098, -0.0428, 0.0831],
[ 0.0618, 0.1618, -0.0325, -0.1448, -0.1646],
[-0.0362, -0.0103, -0.1824, 0.1346, 0.0135],
[-0.0389, 0.0558, -0.0680, -0.1515, 0.1680],
[-0.1748, 0.1496, -0.1459, 0.0181, -0.1115]]],
[[[ 0.1081, -0.0935, -0.1567, 0.1787, 0.0230],
[-0.1890, -0.0129, 0.0133, 0.0472, 0.0424],
[-0.0680, -0.0616, -0.0199, 0.0837, 0.1960],
[ 0.0917, 0.0248, 0.0921, -0.0139, 0.0422],
[-0.0997, -0.0637, 0.1865, -0.1982, 0.1430]]],
[[[-0.1463, -0.0137, 0.1728, -0.0899, -0.0427],
[-0.0654, -0.0375, -0.1964, -0.1289, -0.0483],
[ 0.0495, 0.0236, 0.1467, 0.1145, -0.0054],
[-0.0240, -0.0058, -0.0281, 0.1004, -0.0038],
[ 0.0126, 0.0019, -0.1089, -0.1572, -0.1472]]],
[[[-0.0682, -0.0487, 0.0365, -0.1739, -0.0511],
[-0.0746, 0.1603, 0.0149, 0.1684, 0.0557],
[-0.1502, -0.1176, -0.0681, 0.1512, -0.0974],
[-0.0197, -0.1146, 0.0302, -0.0089, -0.0336],
[ 0.1591, 0.1478, -0.1726, -0.0903, 0.0923]]],
[[[-0.1594, 0.1311, 0.0003, -0.0176, -0.1664],
[ 0.0809, 0.1795, 0.1855, 0.1246, -0.0944],
[-0.1488, -0.0044, 0.0512, -0.1854, 0.0012],
[-0.0820, 0.1209, -0.0562, 0.0859, -0.1175],
[ 0.0652, 0.1324, -0.1412, 0.1655, 0.1654]]],
[[[-0.0019, -0.0864, 0.1593, 0.1857, -0.1915],
[ 0.1527, -0.0412, -0.0173, 0.0072, -0.0933],
[-0.1378, -0.1514, 0.1343, -0.0660, -0.0785],
[ 0.1723, 0.0924, 0.1815, -0.0553, 0.1033],
[ 0.0797, 0.0611, 0.0834, -0.1489, 0.1146]]]])
(bias): Normal:
loc: tensor([0., 0., -0., 0., 0., 0.])
scale: tensor([0.4082, 0.4082, 0.4082, 0.4082, 0.4082, 0.4082])
posterior: Automatic()
prior: Module()
observed: Observed()
tensor: tensor([ 0.0894, 0.1877, -0.0535, 0.0308, 0.1593, 0.0810])
)
(observed): Observed()
)
(conv2): Conv2d(
6, 16, kernel_size=(5, 5), stride=(1, 1)
(posterior): Normal(
(weight): Normal:
posterior: Automatic()
prior: Module()
observed: Observed()
scale: Transform:
tensor([[[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]],
[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]],
[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]],
[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]],
[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]],
[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]]],
[[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]],
[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]],
[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]],
[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]],
[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]],
[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]]],
[[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]],
[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]],
[[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, 0.0498, 0.0498]],
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loc: Parameter containing:
tensor([[[[ 4.2191e-02, 7.6840e-02, -7.4705e-02, -2.1865e-02, 6.1931e-02],
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grad_fn=<AddBackward0>)
(bias): Normal:
posterior: Automatic()
prior: Module()
observed: Observed()
scale: Transform:
tensor([0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498],
grad_fn=<ExpBackward0>)
loc: Parameter containing:
tensor([-0.0365, 0.0072, 0.0093, 0.0025, 0.0342, 0.0735, 0.0308, 0.0723,
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requires_grad=True)
tensor: tensor([ 0.0190, -0.0501, 0.0052, -0.1015, -0.0062, 0.0528, 0.0634, 0.0589,
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grad_fn=<AddBackward0>)
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(prior): Module(
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posterior: Automatic()
prior: Module()
observed: Observed()
tensor: tensor([[[[ 4.2191e-02, 7.6840e-02, -7.4705e-02, -2.1865e-02, 6.1931e-02],
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[-5.6072e-02, -6.5883e-02, 3.5538e-02, -7.8521e-02, -6.9720e-02],
[-1.9393e-02, -7.3491e-02, 8.1700e-03, -1.9778e-03, 2.8227e-02],
[-1.6722e-02, 5.5947e-02, 6.2404e-03, -5.1478e-03, -2.5255e-02]],
[[-7.8407e-02, 7.2449e-02, -6.2823e-02, -6.6556e-03, -1.6768e-02],
[-3.4428e-02, -1.3495e-02, 6.7206e-04, 2.5644e-02, -3.1432e-02],
[-7.0893e-02, 2.9846e-02, -5.0699e-03, 2.9505e-02, -1.8095e-02],
[ 6.7203e-02, 5.6675e-03, -3.5811e-02, 1.9892e-02, 5.2260e-02],
[-6.8506e-02, -7.1065e-02, 3.3085e-02, 4.3750e-02, 1.6927e-02]],
[[-6.5512e-02, -5.6648e-02, 4.6444e-02, 7.4754e-02, 1.1064e-02],
[ 6.3824e-02, 6.5449e-02, -4.9581e-02, -1.4683e-02, -4.2090e-02],
[-4.7219e-02, -3.5248e-02, 3.6657e-04, -2.6233e-02, 7.6344e-02],
[-6.2184e-02, -6.0576e-02, 8.0155e-02, -4.5559e-02, 5.8270e-02],
[ 3.0984e-02, 5.9146e-02, 5.2023e-02, -8.1066e-02, 1.9684e-02]]],
[[[ 7.3765e-02, 3.5141e-03, -2.1752e-03, 7.4100e-02, 7.4641e-02],
[ 4.8591e-02, 7.6598e-02, 2.9565e-02, 2.1044e-02, -5.0159e-02],
[-6.1025e-02, -2.8593e-02, 2.4891e-02, -6.0556e-02, 6.2090e-02],
[-1.5755e-02, 4.0377e-02, 3.5295e-03, 5.3803e-02, -6.6197e-02],
[ 4.4328e-02, -1.8031e-02, -7.4217e-02, -9.6550e-03, -6.4711e-03]],
[[-7.8584e-02, -4.6375e-02, -3.3496e-02, -6.2696e-02, 3.7588e-02],
[-1.4592e-02, -2.4390e-02, -5.0144e-02, -5.0049e-02, 3.2448e-02],
[ 7.5607e-02, -6.8160e-02, -1.8825e-02, 6.0839e-02, 2.1488e-02],
[-6.6120e-02, 6.7261e-02, -5.8349e-02, 4.2199e-02, -5.2458e-02],
[-4.8494e-02, -5.4509e-02, 9.2273e-05, -3.4371e-02, 5.9357e-04]],
[[-7.8414e-02, 7.0503e-02, -3.8403e-02, -4.4648e-02, 7.5146e-02],
[-5.6907e-02, 3.4213e-05, -6.8368e-02, -5.0295e-02, -4.3787e-02],
[-5.8034e-02, -6.0247e-02, -5.8353e-02, -2.5280e-02, 6.9298e-02],
[-1.4764e-03, 4.3781e-02, -2.5240e-02, -4.3813e-02, 2.8583e-02],
[-4.1418e-02, -4.6101e-02, 2.7372e-03, -3.2267e-02, 5.5665e-02]],
[[-7.9642e-02, 5.8555e-02, 5.8883e-02, -6.3063e-02, -3.0692e-02],
[ 5.0561e-02, -6.8027e-02, 4.7893e-02, -6.2822e-02, 1.0364e-02],
[-6.9147e-02, -7.2420e-02, 7.8318e-02, 6.8844e-02, 5.5427e-03],
[ 6.0117e-02, -3.5221e-02, 7.4160e-02, -1.2965e-02, 6.2557e-02],
[ 4.4865e-04, -6.0832e-02, 7.6485e-03, 6.0057e-02, 2.3943e-02]],
[[-2.4230e-02, 6.7864e-02, 7.2121e-02, 3.4157e-02, -7.9538e-02],
[-3.3560e-02, 1.1223e-02, -1.7517e-02, -5.4311e-02, 6.5203e-02],
[-7.7116e-02, -6.3548e-02, 3.0147e-02, -5.6751e-02, -4.5155e-02],
[ 7.9671e-02, -6.0086e-02, -6.9947e-02, 1.2981e-02, -2.5239e-02],
[-2.4339e-03, 7.9699e-02, 2.4683e-02, -4.5459e-02, 9.2444e-03]],
[[ 4.2156e-02, -5.2864e-02, 7.0858e-02, 7.3412e-02, -1.2266e-02],
[ 4.9417e-02, 4.1968e-02, -6.1616e-02, -1.7818e-02, -2.4427e-02],
[ 2.1473e-02, -1.1306e-02, 4.7061e-02, -2.3690e-02, -6.6106e-02],
[ 8.0653e-02, -6.1394e-02, -9.6741e-03, 3.2104e-02, -3.5300e-02],
[-3.7291e-02, -1.6968e-02, 1.3973e-02, 5.5290e-02, 4.0622e-03]]],
[[[ 2.2983e-02, 1.0195e-02, 4.0193e-02, -7.5025e-02, -1.5421e-03],
[ 6.0510e-02, 1.8363e-02, -3.8094e-02, 4.5445e-02, -2.9622e-03],
[-5.3075e-02, 3.4295e-02, 5.0654e-02, -1.9342e-02, 3.2361e-02],
[ 7.3380e-02, -1.0884e-02, -5.3324e-02, -5.0394e-02, 3.1872e-02],
[-4.9773e-02, -2.8900e-02, -2.6879e-02, -2.8097e-02, 6.6398e-02]],
[[-2.5804e-02, -4.6905e-02, -4.2523e-02, -5.0381e-02, -2.0208e-02],
[-4.8815e-02, 2.2532e-02, 6.9881e-02, 2.1225e-02, 4.3858e-02],
[ 3.8482e-02, 7.5890e-03, 1.4969e-02, 1.8850e-02, -2.9226e-02],
[ 3.2008e-02, 2.3823e-02, 7.4640e-02, -3.2508e-02, -5.9983e-02],
[ 2.0176e-02, -2.4570e-02, -4.5569e-02, 6.8979e-02, -4.4682e-02]],
[[-4.6043e-02, -3.2025e-02, 2.8151e-02, -4.8214e-02, 7.1796e-02],
[ 6.4546e-02, -2.4392e-03, 6.5753e-02, 1.0233e-02, 3.1131e-02],
[-2.9636e-02, 7.3704e-02, -7.7781e-02, 8.0873e-02, -8.5714e-04],
[ 3.0164e-02, -6.3718e-02, 8.1516e-02, 1.4934e-02, 6.1368e-02],
[ 4.9876e-02, 9.5079e-03, 7.7987e-02, -3.8619e-02, 5.9578e-02]],
[[-4.2667e-03, -4.3751e-02, -2.8233e-02, -4.1592e-02, -6.0769e-02],
[ 4.8450e-02, -1.2896e-02, -4.6601e-03, -6.4206e-03, -1.1431e-02],
[-7.7503e-02, -1.0773e-02, -8.0490e-03, -3.1181e-03, 1.3285e-02],
[ 6.8160e-02, 6.7861e-02, 1.4774e-02, -1.3599e-02, -2.4826e-02],
[ 4.7705e-02, -7.3378e-02, -1.3825e-02, -3.1077e-02, -3.7972e-02]],
[[ 5.6202e-02, 4.1837e-02, -2.8773e-02, 7.2132e-02, 2.5515e-02],
[-3.3901e-02, 7.0663e-03, 6.8551e-02, -9.9631e-03, 3.0268e-02],
[-7.2736e-02, 2.3806e-02, 2.9459e-02, 6.0356e-02, -4.3899e-03],
[-2.8373e-02, -1.1198e-02, -1.2081e-02, 1.7582e-02, -6.8985e-02],
[-7.2981e-02, -5.7756e-02, -2.6205e-02, 5.1297e-02, -5.9565e-02]],
[[ 6.5016e-03, 6.3833e-02, 1.0038e-02, -7.4870e-02, -8.0244e-02],
[-1.2594e-02, 2.0456e-02, -4.8056e-02, 1.1776e-02, -7.6218e-02],
[ 5.1599e-02, -9.6947e-03, -1.6420e-02, -5.6519e-02, -7.0490e-02],
[-5.8097e-02, 7.7906e-03, -3.3099e-03, 1.3623e-02, -5.2314e-02],
[-1.3419e-02, 6.9299e-02, 7.2203e-02, 4.8437e-02, -4.1129e-02]]]])
(bias): Normal:
loc: tensor([-0., 0., 0., 0., 0., 0., 0., 0., -0., 0., 0., 0., -0., 0., 0., -0.])
scale: tensor([0.2500, 0.2500, 0.2500, 0.2500, 0.2500, 0.2500, 0.2500, 0.2500, 0.2500,
0.2500, 0.2500, 0.2500, 0.2500, 0.2500, 0.2500, 0.2500])
posterior: Automatic()
prior: Module()
observed: Observed()
tensor: tensor([-0.0365, 0.0072, 0.0093, 0.0025, 0.0342, 0.0735, 0.0308, 0.0723,
-0.0758, 0.0133, 0.0423, 0.0314, -0.0622, 0.0366, 0.0547, -0.0157])
)
(observed): Observed()
)
(fc1): Linear(
in_features=400, out_features=120, bias=True
(posterior): Normal(
(weight): Normal:
posterior: Automatic()
prior: Module()
observed: Observed()
scale: Transform:
tensor([[0.0498, 0.0498, 0.0498, ..., 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, ..., 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, ..., 0.0498, 0.0498, 0.0498],
...,
[0.0498, 0.0498, 0.0498, ..., 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, ..., 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, ..., 0.0498, 0.0498, 0.0498]],
grad_fn=<ExpBackward0>)
loc: Parameter containing:
tensor([[ 0.0290, 0.0450, 0.0432, ..., -0.0188, 0.0460, 0.0460],
[-0.0440, 0.0107, -0.0302, ..., -0.0457, 0.0447, -0.0215],
[-0.0185, 0.0321, -0.0076, ..., -0.0138, 0.0309, 0.0219],
...,
[ 0.0011, 0.0479, 0.0438, ..., 0.0297, -0.0310, 0.0051],
[ 0.0461, -0.0489, 0.0107, ..., -0.0152, 0.0484, -0.0418],
[ 0.0300, -0.0080, 0.0244, ..., -0.0223, 0.0077, -0.0147]],
requires_grad=True)
tensor: tensor([[-0.0771, 0.0682, 0.0952, ..., -0.0025, 0.0747, 0.0905],
[-0.0600, -0.0432, -0.0064, ..., -0.0666, 0.0176, 0.0168],
[ 0.0137, 0.0625, 0.0067, ..., 0.0625, 0.0137, 0.1347],
...,
[ 0.0379, -0.0104, 0.0276, ..., 0.0230, -0.0285, 0.0612],
[ 0.0619, -0.0173, 0.0673, ..., 0.0148, 0.0073, 0.0058],
[ 0.1118, -0.0446, -0.0601, ..., 0.0168, -0.0052, -0.0763]],
grad_fn=<AddBackward0>)
(bias): Normal:
posterior: Automatic()
prior: Module()
observed: Observed()
scale: Transform:
tensor([0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498, 0.0498, 0.0498], grad_fn=<ExpBackward0>)
loc: Parameter containing:
tensor([ 3.4268e-02, -4.9700e-02, 1.3390e-02, -2.0985e-02, -1.6808e-02,
2.4641e-02, -4.8857e-02, 3.6003e-02, 1.0424e-02, -2.8776e-02,
4.2286e-02, -2.7017e-02, 4.5771e-03, 2.5405e-02, 1.3040e-02,
4.0760e-02, 3.4614e-02, 9.6058e-03, -1.0389e-02, 1.7939e-02,
2.7261e-02, 1.0442e-02, -4.0359e-02, 6.0608e-04, -2.8432e-02,
-2.3294e-02, 2.5094e-02, -2.7444e-02, -3.9087e-02, 4.8333e-02,
-3.8067e-02, 4.3731e-02, -1.1337e-02, 6.5628e-03, -3.9661e-02,
-3.0479e-02, 4.2353e-02, -2.1659e-02, 1.7937e-02, 4.8377e-02,
-7.3253e-03, -2.8690e-02, -2.0960e-02, -2.0081e-02, -4.1321e-02,
-2.5133e-02, -3.6166e-02, 3.5816e-02, 3.0482e-03, -4.1686e-02,
3.5028e-02, 3.1139e-02, -2.5572e-03, -8.7303e-03, 4.4109e-02,
1.0638e-02, 3.5676e-02, 1.9173e-02, 2.6669e-02, -3.7786e-02,
-1.8936e-02, -1.0854e-05, 3.6708e-02, -2.0134e-02, -4.2009e-02,
3.9515e-02, -3.7057e-02, 2.5732e-02, -2.7906e-02, 3.4639e-02,
4.3312e-03, 1.9998e-02, -2.2635e-02, 2.4380e-02, -1.2083e-02,
3.3281e-02, 4.6888e-02, 3.9643e-02, 1.1001e-02, 4.8635e-02,
-1.2259e-03, 3.2456e-02, -1.0144e-02, -6.7254e-03, 1.5385e-03,
1.6456e-02, -3.1296e-02, -1.5280e-02, 3.8949e-02, -1.9992e-02,
-2.8284e-02, 4.0517e-02, -9.7160e-03, 7.7505e-04, 1.6642e-02,
4.0081e-02, 9.7422e-04, 3.7670e-02, 4.7919e-02, 4.4317e-02,
1.0160e-02, 1.9730e-02, -8.4861e-03, 3.3960e-02, -2.3660e-02,
-4.4850e-02, -3.0455e-02, -1.5874e-03, 1.4935e-02, 2.1578e-02,
4.7099e-02, -3.9308e-02, 1.2687e-02, -3.1501e-02, 2.6750e-02,
1.3101e-02, -4.7801e-02, 4.5735e-02, 2.1969e-02, 1.0239e-02],
requires_grad=True)
tensor: tensor([ 0.0461, -0.0686, -0.0018, -0.0174, -0.1039, 0.0644, 0.0129, -0.0319,
0.0540, -0.0665, 0.0386, -0.0862, -0.0050, -0.0043, 0.0145, 0.0309,
0.1006, 0.0050, -0.0034, 0.0625, -0.0306, 0.0265, -0.1264, -0.0380,
0.0116, -0.0690, 0.0944, -0.0293, -0.0990, 0.1073, 0.0308, 0.0658,
-0.0371, 0.0542, -0.0284, -0.0286, -0.0326, -0.0041, 0.0036, 0.0200,
-0.0654, -0.0131, -0.0922, -0.0521, -0.0524, 0.0307, 0.0191, 0.0887,
0.0523, -0.0082, 0.0263, 0.0368, -0.0741, 0.0028, 0.0159, 0.0315,
-0.0049, 0.0991, 0.0387, -0.0024, -0.0229, 0.1443, -0.0206, -0.0440,
0.0354, 0.0254, -0.0601, 0.0427, -0.0854, 0.0387, 0.0435, 0.0373,
0.0068, 0.0281, 0.0257, -0.0699, -0.0028, 0.0847, 0.0240, -0.0674,
-0.0045, 0.1192, 0.0438, -0.0321, 0.0262, -0.0189, -0.0056, -0.0746,
0.0501, 0.0481, -0.1232, -0.0209, 0.0074, -0.0143, -0.0851, 0.1050,
0.0118, 0.0893, 0.0022, -0.0285, -0.0378, -0.0167, -0.0383, 0.0616,
-0.0673, -0.0795, -0.0137, 0.0920, 0.0490, 0.0294, 0.1110, 0.0153,
0.0588, -0.0804, 0.0791, 0.1213, 0.0279, 0.0537, 0.0901, -0.0692],
grad_fn=<AddBackward0>)
)
(prior): Module(
(weight): Normal:
loc: tensor([[0., 0., 0., ..., -0., 0., 0.],
[-0., 0., -0., ..., -0., 0., -0.],
[-0., 0., -0., ..., -0., 0., 0.],
...,
[0., 0., 0., ..., 0., -0., 0.],
[0., -0., 0., ..., -0., 0., -0.],
[0., -0., 0., ..., -0., 0., -0.]])
scale: tensor([[0.0500, 0.0500, 0.0500, ..., 0.0500, 0.0500, 0.0500],
[0.0500, 0.0500, 0.0500, ..., 0.0500, 0.0500, 0.0500],
[0.0500, 0.0500, 0.0500, ..., 0.0500, 0.0500, 0.0500],
...,
[0.0500, 0.0500, 0.0500, ..., 0.0500, 0.0500, 0.0500],
[0.0500, 0.0500, 0.0500, ..., 0.0500, 0.0500, 0.0500],
[0.0500, 0.0500, 0.0500, ..., 0.0500, 0.0500, 0.0500]])
posterior: Automatic()
prior: Module()
observed: Observed()
tensor: tensor([[ 0.0290, 0.0450, 0.0432, ..., -0.0188, 0.0460, 0.0460],
[-0.0440, 0.0107, -0.0302, ..., -0.0457, 0.0447, -0.0215],
[-0.0185, 0.0321, -0.0076, ..., -0.0138, 0.0309, 0.0219],
...,
[ 0.0011, 0.0479, 0.0438, ..., 0.0297, -0.0310, 0.0051],
[ 0.0461, -0.0489, 0.0107, ..., -0.0152, 0.0484, -0.0418],
[ 0.0300, -0.0080, 0.0244, ..., -0.0223, 0.0077, -0.0147]])
(bias): Normal:
loc: tensor([0., -0., 0., -0., -0., 0., -0., 0., 0., -0., 0., -0., 0., 0., 0., 0., 0., 0., -0., 0., 0., 0., -0., 0.,
-0., -0., 0., -0., -0., 0., -0., 0., -0., 0., -0., -0., 0., -0., 0., 0., -0., -0., -0., -0., -0., -0., -0., 0.,
0., -0., 0., 0., -0., -0., 0., 0., 0., 0., 0., -0., -0., -0., 0., -0., -0., 0., -0., 0., -0., 0., 0., 0.,
-0., 0., -0., 0., 0., 0., 0., 0., -0., 0., -0., -0., 0., 0., -0., -0., 0., -0., -0., 0., -0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., -0., 0., -0., -0., -0., -0., 0., 0., 0., -0., 0., -0., 0., 0., -0., 0., 0., 0.])
scale: tensor([0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913,
0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913,
0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913,
0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913,
0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913,
0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913,
0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913,
0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913,
0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913,
0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913,
0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913,
0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913,
0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913, 0.0913,
0.0913, 0.0913, 0.0913])
posterior: Automatic()
prior: Module()
observed: Observed()
tensor: tensor([ 3.4268e-02, -4.9700e-02, 1.3390e-02, -2.0985e-02, -1.6808e-02,
2.4641e-02, -4.8857e-02, 3.6003e-02, 1.0424e-02, -2.8776e-02,
4.2286e-02, -2.7017e-02, 4.5771e-03, 2.5405e-02, 1.3040e-02,
4.0760e-02, 3.4614e-02, 9.6058e-03, -1.0389e-02, 1.7939e-02,
2.7261e-02, 1.0442e-02, -4.0359e-02, 6.0608e-04, -2.8432e-02,
-2.3294e-02, 2.5094e-02, -2.7444e-02, -3.9087e-02, 4.8333e-02,
-3.8067e-02, 4.3731e-02, -1.1337e-02, 6.5628e-03, -3.9661e-02,
-3.0479e-02, 4.2353e-02, -2.1659e-02, 1.7937e-02, 4.8377e-02,
-7.3253e-03, -2.8690e-02, -2.0960e-02, -2.0081e-02, -4.1321e-02,
-2.5133e-02, -3.6166e-02, 3.5816e-02, 3.0482e-03, -4.1686e-02,
3.5028e-02, 3.1139e-02, -2.5572e-03, -8.7303e-03, 4.4109e-02,
1.0638e-02, 3.5676e-02, 1.9173e-02, 2.6669e-02, -3.7786e-02,
-1.8936e-02, -1.0854e-05, 3.6708e-02, -2.0134e-02, -4.2009e-02,
3.9515e-02, -3.7057e-02, 2.5732e-02, -2.7906e-02, 3.4639e-02,
4.3312e-03, 1.9998e-02, -2.2635e-02, 2.4380e-02, -1.2083e-02,
3.3281e-02, 4.6888e-02, 3.9643e-02, 1.1001e-02, 4.8635e-02,
-1.2259e-03, 3.2456e-02, -1.0144e-02, -6.7254e-03, 1.5385e-03,
1.6456e-02, -3.1296e-02, -1.5280e-02, 3.8949e-02, -1.9992e-02,
-2.8284e-02, 4.0517e-02, -9.7160e-03, 7.7505e-04, 1.6642e-02,
4.0081e-02, 9.7422e-04, 3.7670e-02, 4.7919e-02, 4.4317e-02,
1.0160e-02, 1.9730e-02, -8.4861e-03, 3.3960e-02, -2.3660e-02,
-4.4850e-02, -3.0455e-02, -1.5874e-03, 1.4935e-02, 2.1578e-02,
4.7099e-02, -3.9308e-02, 1.2687e-02, -3.1501e-02, 2.6750e-02,
1.3101e-02, -4.7801e-02, 4.5735e-02, 2.1969e-02, 1.0239e-02])
)
(observed): Observed()
)
(fc2): Linear(
in_features=120, out_features=10, bias=True
(posterior): Normal(
(weight): Normal:
posterior: Automatic()
prior: Module()
observed: Observed()
scale: Transform:
tensor([[0.0498, 0.0498, 0.0498, ..., 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, ..., 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, ..., 0.0498, 0.0498, 0.0498],
...,
[0.0498, 0.0498, 0.0498, ..., 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, ..., 0.0498, 0.0498, 0.0498],
[0.0498, 0.0498, 0.0498, ..., 0.0498, 0.0498, 0.0498]],
grad_fn=<ExpBackward0>)
loc: Parameter containing:
tensor([[ 0.0853, -0.0135, 0.0683, ..., 0.0101, -0.0386, 0.0218],
[ 0.0366, -0.0514, 0.0796, ..., -0.0416, -0.0514, 0.0824],
[-0.0907, 0.0680, 0.0275, ..., -0.0242, 0.0592, -0.0864],
...,
[-0.0862, 0.0863, -0.0316, ..., 0.0718, 0.0438, 0.0558],
[-0.0711, -0.0183, -0.0767, ..., 0.0480, 0.0798, -0.0622],
[ 0.0360, -0.0011, -0.0885, ..., 0.0526, 0.0213, -0.0500]],
requires_grad=True)
tensor: tensor([[ 0.0446, -0.0119, 0.0613, ..., 0.0736, -0.1029, -0.0459],
[ 0.0366, -0.1579, 0.0943, ..., 0.0078, -0.0376, 0.0493],
[-0.1436, 0.0732, 0.0015, ..., -0.0362, -0.0015, -0.0596],
...,
[-0.0071, 0.0492, 0.0063, ..., 0.0985, 0.0816, 0.0473],
[-0.1276, 0.0231, -0.0409, ..., 0.0153, 0.0586, -0.0618],
[ 0.0311, -0.0013, -0.0697, ..., 0.0642, -0.0082, -0.0596]],
grad_fn=<AddBackward0>)
(bias): Normal:
posterior: Automatic()
prior: Module()
observed: Observed()
scale: Transform:
tensor([0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498, 0.0498,
0.0498], grad_fn=<ExpBackward0>)
loc: Parameter containing:
tensor([ 0.0895, 0.0306, -0.0073, 0.0383, 0.0428, -0.0883, 0.0808, -0.0065,
0.0228, 0.0389], requires_grad=True)
tensor: tensor([-0.0167, 0.0773, 0.0665, 0.0994, 0.0772, -0.0356, -0.0222, -0.0223,
-0.0587, 0.0343], grad_fn=<AddBackward0>)
)
(prior): Module(
(weight): Normal:
loc: tensor([[0., -0., 0., ..., 0., -0., 0.],
[0., -0., 0., ..., -0., -0., 0.],
[-0., 0., 0., ..., -0., 0., -0.],
...,
[-0., 0., -0., ..., 0., 0., 0.],
[-0., -0., -0., ..., 0., 0., -0.],
[0., -0., -0., ..., 0., 0., -0.]])
scale: tensor([[0.0913, 0.0913, 0.0913, ..., 0.0913, 0.0913, 0.0913],
[0.0913, 0.0913, 0.0913, ..., 0.0913, 0.0913, 0.0913],
[0.0913, 0.0913, 0.0913, ..., 0.0913, 0.0913, 0.0913],
...,
[0.0913, 0.0913, 0.0913, ..., 0.0913, 0.0913, 0.0913],
[0.0913, 0.0913, 0.0913, ..., 0.0913, 0.0913, 0.0913],
[0.0913, 0.0913, 0.0913, ..., 0.0913, 0.0913, 0.0913]])
posterior: Automatic()
prior: Module()
observed: Observed()
tensor: tensor([[ 0.0853, -0.0135, 0.0683, ..., 0.0101, -0.0386, 0.0218],
[ 0.0366, -0.0514, 0.0796, ..., -0.0416, -0.0514, 0.0824],
[-0.0907, 0.0680, 0.0275, ..., -0.0242, 0.0592, -0.0864],
...,
[-0.0862, 0.0863, -0.0316, ..., 0.0718, 0.0438, 0.0558],
[-0.0711, -0.0183, -0.0767, ..., 0.0480, 0.0798, -0.0622],
[ 0.0360, -0.0011, -0.0885, ..., 0.0526, 0.0213, -0.0500]])
(bias): Normal:
loc: tensor([0., 0., -0., 0., 0., -0., 0., -0., 0., 0.])
scale: tensor([0.3162, 0.3162, 0.3162, 0.3162, 0.3162, 0.3162, 0.3162, 0.3162, 0.3162,
0.3162])
posterior: Automatic()
prior: Module()
observed: Observed()
tensor: tensor([ 0.0895, 0.0306, -0.0073, 0.0383, 0.0428, -0.0883, 0.0808, -0.0065,
0.0228, 0.0389])
)
(observed): Observed()
)
)
You just have to define the forward function, and the backward
function (where gradients are computed) is automatically defined for you
using autograd.
You can use any of the Tensor operations in the forward function.
The learnable parameters of a model are returned by net.parameters()
params = list(net.parameters())
print(len(params))
print(params[0].size())
Out:
16
torch.Size([6, 1, 5, 5])
Let try a random 32x32 input
input = torch.randn(1, 1, 32, 32)
out = net(input)
print(out)
Out:
tensor([[ 0.4980, 0.6893, 0.0096, -0.4008, -0.1188, -0.2803, 0.3551, 0.4623,
-0.2278, -0.0520]], grad_fn=<AddmmBackward0>)
Zero the gradient buffers of all parameters and backprops with random gradients:
net.zero_grad()
out.backward(torch.randn(1, 10))
Note
borch.nn only supports mini-batches. The entire borch.nn
package only supports inputs that are a mini-batch of samples, and not
a single sample.
For example, nn.Conv2d will take in a 4D Tensor of
nSamples x nChannels x Height x Width.
If you have a single sample, just use input.unsqueeze(0) to add
a fake batch dimension.
Before proceeding further, let’s recap all the classes you’ve seen so far.
- Recap:
torch.Tensor- A multi-dimensional array with support for autograd operations likebackward(). Also holds the gradient w.r.t. the tensor.nn.Module- Neural network module. Convenient way of encapsulating parameters, with helpers for moving them to GPU, exporting, loading, etc.nn.Parameter- A kind of Tensor, that is automatically registered as a parameter when assigned as an attribute to aModule.autograd.Function- Implements forward and backward definitions of an autograd operation. EveryTensoroperation, creates at least a singleFunctionnode, that connects to functions that created aTensorand encodes its history.
- At this point, we covered:
Defining a neural network
Processing inputs and calling backward
- Still Left:
Computing the loss
Updating the weights of the network
Loss Function¶
A loss function takes the (output, target) pair of inputs, and computes a value that estimates how far away the output is from the target.
There are several different
loss functions under the
nn package .
A simple loss is: nn.MSELoss which computes the mean-squared error
between the input and the target. They are how ever only equivalent to an maximum
likelihood approach in deep learning.
In order to infer the posterior of the weights and thus capture the uncertainty
of the weights as well, we have to use the infer package. In this example we
will use infer.vi_loss function that automatically creates the best loss function
for variational inference given the latent variables in your model.
Similar to how it’s done for random varibles, we can also observe on the module using keyword arguments matching the names of the random variables we want to observe. This will add those random variables to the likelihood term and we will not infer the distribution over it. For example:
target = torch.randint(10, (1,)) # a dummy target, for example
net.observe(classification=target)
borch.sample(net)
output = net(input)
loss = infer.vi_loss(**borch.pq_to_infer(net))
print(loss)
Out:
tensor(9549.6875, grad_fn=<AddBackward0>)
Now, if you would follow loss in the backward direction you will see a graph of
computations that looks like this:
input -> conv2d -> relu -> maxpool2d -> conv2d -> relu -> maxpool2d
-> view -> linear -> relu -> linear ->
-> loss
So, when we call loss.backward(), the whole graph is differentiated
w.r.t. the loss, and all Tensors in the graph that has requires_grad=True
will have their .grad Tensor accumulated with the gradient.
Backprop¶
To backpropagate the error all we have to do is to loss.backward().
You need to clear the existing gradients though, else gradients will be
accumulated to existing gradients.
Now we shall call loss.backward(), and have a look at conv1’s bias
gradients before and after the backward.
net.zero_grad() # zeroes the gradient buffers of all parameters
The value for the loc paramater of the approximating distribution of
conv1.bias zeroing the gradients is
print(net.conv1.posterior.bias.loc.grad)
loss.backward()
Out:
tensor([0., 0., 0., 0., 0., 0.])
after calling backward the value is
print(net.conv1.posterior.bias.loc.grad)
Out:
tensor([ 0.6376, 1.3013, -0.0847, 0.4150, 1.0615, 0.2077])
The only thing left to learn is:
Updating the weights of the network
Update the weights¶
The simplest update rule used in practice is the Stochastic Gradient Descent (SGD):
weight = weight - learning_rate * gradient
We can implement this using simple python code:
learning_rate = 0.01
for f in net.parameters():
f.data.sub_(f.grad.data * learning_rate)
However, as you use neural networks, you want to use various different
update rules such as SGD, Nesterov-SGD, Adam, RMSProp, etc.
To enable this, torch built a small package: torch.optim that
implements all these methods. Using it is very simple:
import torch.optim as optim
# create your optimizer
optimizer = optim.SGD(net.parameters(), lr=0.01)
# in your training loop:
n_batch_epoch = 10 # number of batches per epoch usually len(dataloader)
optimizer.zero_grad() # zero the gradient buffers
borch.sample(net)
output = net(input)
loss = infer.vi_loss(**borch.pq_to_infer(net), kl_scaling=1 / n_batch_epoch)
loss.backward()
optimizer.step() # Does the update
Exercises¶
The neural network package contains various modules and loss functions that form the building blocks of deep neural networks. Have a look at the documentation to see what is available.
Try designing yor own feed forward networks with two different types of non lineareties ex. relu
Total running time of the script: ( 0 minutes 0.126 seconds)