# 15. Pytorch¶

Pytorch is a framework for doing differentition, optimization, and in particular optimization of neural networks (therefore, sometimes just called a deep learning framework) using the native imperative language style of Python.

## 15.1. Some optional information about Pytorch and how it differs from its competitors¶

Most deep learning frameworks like TensorFlow and Theanos work with symbolic differentian and therefore it is necessary to declare the model structure before actually supplying any data and then ask the framework to compile the model (having the gradients calculated symbolically), after which point, the model will be good to go as long it does not change.

While those have advantages like possible algebraic simplifications when working out the derivates, they also come at some price: they impose a necessary "recompilation" of the model if it changes, they make code less intuitive to writting and more difficult to debug, are potentially more restrictive on model characteristics.

On the other hand, the framework of choice for this section, called PyTorch works with reverse-mode automatic differentiation which consists of creating the chain of differentiation (on a "tape") on-the-fly. That is, after the last operation is done (in our case, that’s the calculation of the loss), the chain of operations is back-propagated (i.e.: running the "tape" backwards) and the gradients of the parameters of interest are calculated using chain rule.

Frameworks which works using symbolic differention are often called static while the ones that use automatic differentiation are called dynamic. Regardless of this, most (if not all) of those deep learning framework have two common characteristic that are worth emphasizing: they allow one to use its differentiation facilities to work with problems other than deep learning, neural networks or optimization (e.g.: Markov Chain Monte Carlo and Bayesian inference) and they natively support GPU acceleration (generally, using Nvidia CUDA).

The reason GPU acceleration is a common denominator over the deep learning frameworks is due to the fact that neural networks are strongly parallelizable problems and this make them well suited for GPU acceleration. This thus explain, at least in part, their recent surge in popularity given the scalability properties of such methods to big datasets, which on the other hand, are getting increasingly common and important.

## 15.2. Now getting started¶

First of all, let’s start by importing some basic stuff

```
[1]:
```

```
import numpy as np
import scipy.stats as stats
import pandas as pd
import matplotlib.pyplot as plt
import torch
import torch.nn as nn
import torch.optim as optim
from sklearn.externals import joblib
```

## 15.3. Tensors and GPU¶

Now, let me present you with the basic storage type of Pytorch: the tensors. They work in very similar fashion to the numpy arrays.

```
[2]:
```

```
x = torch.tensor([.45, .53])
y = x**2
y[0] = .3
a = torch.ones((2,4))
a = a * 2
a = a + 1
a[0,0] = .4
b = torch.zeros((4,2))
b = (b + 3) / 2
b[0,1] = .2
bt = b.transpose(0,1) # transpose
a + bt
torch.mm(a,b) # matrix multiplication
```

```
[2]:
```

```
tensor([[14.1000, 13.5800],
[18.0000, 14.1000]])
```

But Pytoch tensors have a special caveat: they can live on the GPU (if you have one proper installed… if you don’t, then chances are that your computer might explode if you try to run this)!

```
[3]:
```

```
a = torch.rand((3, 5)) # some random numbers
a = a.cuda()
b = torch.ones((5, 3))
b = b.cuda()
b = (b + 3) / 2
b[0,1] = .2
torch.mm(a, b) # matrix multiplication
```

```
[3]:
```

```
tensor([[4.9632, 3.9542, 4.9632],
[2.3442, 2.1493, 2.3442],
[5.6313, 4.4108, 5.6313]], device='cuda:0')
```

## 15.4. Float data type¶

Note that the default data type of Pytorch floats is 32 bit precision type:

```
[4]:
```

```
torch.rand((3, 5)).dtype
```

```
[4]:
```

```
torch.float32
```

While on numpy the default is 64 bit precision type:

```
[5]:
```

```
np.arange(4).dtype
```

```
[5]:
```

```
dtype('int64')
```

You can pass a custom data type `dtype`

function parameter or with special constructors:

```
[6]:
```

```
print(torch.rand((3, 5), dtype=torch.float64).dtype)
print(torch.FloatTensor([.5]).dtype)
print(torch.HalfTensor([.5]).dtype)
print(torch.DoubleTensor([.5]).dtype)
```

```
torch.float64
torch.float32
torch.float16
torch.float64
```

But it’s recommended to use the default float32 for most deep learning applications due to the speed up that it can give on vectorized operations, specially on GPUs.

## 15.5. Differentiation¶

We can use Pytorch to do numerical differention using the code below:

```
[7]:
```

```
x = torch.tensor([.45], requires_grad=True)
y = x**2
y.backward()
x.grad
```

```
[7]:
```

```
tensor([0.9000])
```

`x.grad`

gives us the gradient of `y`

with respect to `x`

. Now pay attetion at (and play with) this other example:

```
[8]:
```

```
x = torch.tensor([.5,.3,.6], requires_grad=True)
y = x**2
z = y.sum()
z.backward()
x.grad
```

```
[8]:
```

```
tensor([1.0000, 0.6000, 1.2000])
```

## 15.6. Optimization¶

Now let’s try using this integration framework for optimization:

```
[9]:
```

```
x = torch.tensor([.45], requires_grad=True)
# "declares" that x is the variable being optimized by the Adam optimization algorith
optimizer = optim.Adam([x])
y = 2 * x**2 - 7 * x
y.backward() # "declares" that y is value being minimized
optimizer.step() # i.e. find the x that minimizes y
x
```

```
[9]:
```

```
tensor([0.4510], requires_grad=True)
```

Here `optimizer.step()`

moved `x`

in the direction of its gradient; i.e. it moved `x`

in the direction of that minimizes `y`

.

(Note that `requires_grad=True`

is necessary in order to Pytorch to now that it must keep track of the operations done with `x`

to backpropagate it back in the future to get the gradient of `x`

, it requires you to do so manually because otherwise it could save computational resources by not creating this structures).

However, this is just a little step towards optimization, we must repeat this many time to get there:

```
[10]:
```

```
x = torch.tensor([-2.45], requires_grad=True)
optimizer = optim.Adam([x], lr=0.05)
for _ in range(1000):
optimizer.zero_grad()
y = 2 * x**2 - 7 * x
y.backward()
optimizer.step()
print("Numerical optimization solution:", x)
print("Analytic optimization solution:", 7/4)
```

```
Numerical optimization solution: tensor([1.7500], requires_grad=True)
Analytic optimization solution: 1.75
```

Great, we did it! Now let’s try something more difficult, given i.i.d. Gaussian samples, let’s try to \(\hat{\mu}\) such that it minimize the sum of the squared errors of those samples and \(\hat{\mu}\). We know from Statistical theory that the analytical solutional for this is the sample average.

```
[11]:
```

```
mu_hat = torch.tensor([0.1], requires_grad=True)
mu_true = 1.5
x = stats.norm.rvs(size=2000, loc=mu_true, scale=3, random_state=0)
x = torch.as_tensor(x, dtype=torch.float32)
optimizer = optim.Adam([mu_hat], lr=0.05)
criterion = nn.MSELoss()
for _ in range(1000):
optimizer.zero_grad()
loss = criterion(x, mu_hat)
loss.backward()
optimizer.step()
print("Numerical optimization solution:", mu_hat)
print("Analytic optimization solution:", x.mean())
```

```
Numerical optimization solution: tensor([1.4525], requires_grad=True)
Analytic optimization solution: tensor(1.4525)
```

And voila! It worked again!

## 15.7. Neural networks with Pytorch¶

Now probably the most expected part, your first neural network with Pytorch:

```
[12]:
```

```
# Declares the structure of our neural network
class Net(nn.Module):
def __init__(self):
# this is strictly necessary!
super(Net, self).__init__()
# fully connected layer with input of size 10 and output of size 120
self.fc1 = nn.Linear(10, 120)
# fully connected layer with input of size 10 and output of size 120
self.fc2 = nn.Linear(120, 84)
# fully connected layer with input of size 10 and output of size 120
self.fc3 = nn.Linear(84, 1)
self.relu = nn.ReLU()
def forward(self, x):
x = self.relu(self.fc1(x))
x = self.relu(self.fc2(x))
x = self.fc3(x)
return x
net = Net() # Construct the neural network object
# Creates some data using a linear regression
beta = torch.rand(10, 1)
inputv = torch.randn(70, 10)
target = torch.mm(inputv, beta)
target = target + torch.randn(70, 1)
# If a GPU is available, move the network parameters and data into it
if torch.cuda.is_available():
net.cuda()
inputv = inputv.cuda()
target = target.cuda()
criterion = nn.MSELoss()
optimizer = optim.SGD(net.parameters(), lr=0.01)
for epoch in range(1000):
optimizer.zero_grad()
output = net(inputv)
loss = criterion(output, target)
print('Loss', np.round(loss.item(), 2), 'in epoch', epoch + 1)
loss.backward()
optimizer.step()
```

```
Loss 3.46 in epoch 1
Loss 3.42 in epoch 2
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Loss 0.09 in epoch 581
Loss 0.09 in epoch 582
Loss 0.09 in epoch 583
Loss 0.09 in epoch 584
Loss 0.09 in epoch 585
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Loss 0.09 in epoch 587
Loss 0.09 in epoch 588
Loss 0.09 in epoch 589
Loss 0.09 in epoch 590
Loss 0.09 in epoch 591
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Loss 0.09 in epoch 594
Loss 0.09 in epoch 595
Loss 0.09 in epoch 596
Loss 0.09 in epoch 597
Loss 0.09 in epoch 598
Loss 0.09 in epoch 599
Loss 0.09 in epoch 600
Loss 0.09 in epoch 601
Loss 0.08 in epoch 602
Loss 0.08 in epoch 603
Loss 0.08 in epoch 604
Loss 0.08 in epoch 605
Loss 0.08 in epoch 606
Loss 0.08 in epoch 607
Loss 0.08 in epoch 608
Loss 0.08 in epoch 609
Loss 0.08 in epoch 610
Loss 0.08 in epoch 611
Loss 0.08 in epoch 612
Loss 0.08 in epoch 613
Loss 0.08 in epoch 614
Loss 0.08 in epoch 615
Loss 0.08 in epoch 616
Loss 0.08 in epoch 617
Loss 0.08 in epoch 618
Loss 0.08 in epoch 619
Loss 0.08 in epoch 620
Loss 0.08 in epoch 621
Loss 0.08 in epoch 622
Loss 0.08 in epoch 623
Loss 0.08 in epoch 624
Loss 0.08 in epoch 625
Loss 0.08 in epoch 626
Loss 0.08 in epoch 627
Loss 0.08 in epoch 628
Loss 0.08 in epoch 629
Loss 0.08 in epoch 630
Loss 0.07 in epoch 631
Loss 0.07 in epoch 632
Loss 0.07 in epoch 633
Loss 0.07 in epoch 634
Loss 0.07 in epoch 635
Loss 0.07 in epoch 636
Loss 0.07 in epoch 637
Loss 0.07 in epoch 638
Loss 0.07 in epoch 639
Loss 0.07 in epoch 640
Loss 0.07 in epoch 641
Loss 0.07 in epoch 642
Loss 0.07 in epoch 643
Loss 0.07 in epoch 644
Loss 0.07 in epoch 645
Loss 0.07 in epoch 646
Loss 0.07 in epoch 647
Loss 0.07 in epoch 648
Loss 0.07 in epoch 649
Loss 0.07 in epoch 650
Loss 0.07 in epoch 651
Loss 0.07 in epoch 652
Loss 0.07 in epoch 653
Loss 0.07 in epoch 654
Loss 0.07 in epoch 655
Loss 0.07 in epoch 656
Loss 0.07 in epoch 657
Loss 0.07 in epoch 658
Loss 0.07 in epoch 659
Loss 0.07 in epoch 660
Loss 0.07 in epoch 661
Loss 0.07 in epoch 662
Loss 0.07 in epoch 663
Loss 0.07 in epoch 664
Loss 0.06 in epoch 665
Loss 0.06 in epoch 666
Loss 0.06 in epoch 667
Loss 0.06 in epoch 668
Loss 0.06 in epoch 669
Loss 0.06 in epoch 670
Loss 0.06 in epoch 671
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Loss 0.06 in epoch 675
Loss 0.06 in epoch 676
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Loss 0.06 in epoch 678
Loss 0.06 in epoch 679
Loss 0.06 in epoch 680
Loss 0.06 in epoch 681
Loss 0.06 in epoch 682
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Loss 0.06 in epoch 684
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Loss 0.06 in epoch 686
Loss 0.06 in epoch 687
Loss 0.06 in epoch 688
Loss 0.06 in epoch 689
Loss 0.06 in epoch 690
Loss 0.06 in epoch 691
Loss 0.06 in epoch 692
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Loss 0.06 in epoch 694
Loss 0.06 in epoch 695
Loss 0.06 in epoch 696
Loss 0.06 in epoch 697
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Loss 0.06 in epoch 699
Loss 0.06 in epoch 700
Loss 0.06 in epoch 701
Loss 0.06 in epoch 702
Loss 0.06 in epoch 703
Loss 0.06 in epoch 704
Loss 0.05 in epoch 705
Loss 0.05 in epoch 706
Loss 0.05 in epoch 707
Loss 0.05 in epoch 708
Loss 0.05 in epoch 709
Loss 0.05 in epoch 710
Loss 0.05 in epoch 711
Loss 0.05 in epoch 712
Loss 0.05 in epoch 713
Loss 0.05 in epoch 714
Loss 0.05 in epoch 715
Loss 0.05 in epoch 716
Loss 0.05 in epoch 717
Loss 0.05 in epoch 718
Loss 0.05 in epoch 719
Loss 0.05 in epoch 720
Loss 0.05 in epoch 721
Loss 0.05 in epoch 722
Loss 0.05 in epoch 723
Loss 0.05 in epoch 724
Loss 0.05 in epoch 725
Loss 0.05 in epoch 726
Loss 0.05 in epoch 727
Loss 0.05 in epoch 728
Loss 0.05 in epoch 729
Loss 0.05 in epoch 730
Loss 0.05 in epoch 731
Loss 0.05 in epoch 732
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Loss 0.05 in epoch 734
Loss 0.05 in epoch 735
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Loss 0.05 in epoch 737
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Loss 0.05 in epoch 739
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Loss 0.05 in epoch 741
Loss 0.05 in epoch 742
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Loss 0.05 in epoch 746
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Loss 0.05 in epoch 748
Loss 0.05 in epoch 749
Loss 0.05 in epoch 750
Loss 0.05 in epoch 751
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Loss 0.04 in epoch 761
Loss 0.04 in epoch 762
Loss 0.04 in epoch 763
Loss 0.04 in epoch 764
Loss 0.04 in epoch 765
Loss 0.04 in epoch 766
Loss 0.04 in epoch 767
Loss 0.04 in epoch 768
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Loss 0.04 in epoch 770
Loss 0.04 in epoch 771
Loss 0.04 in epoch 772
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Loss 0.04 in epoch 774
Loss 0.04 in epoch 775
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Loss 0.04 in epoch 777
Loss 0.04 in epoch 778
Loss 0.04 in epoch 779
Loss 0.04 in epoch 780
Loss 0.04 in epoch 781
Loss 0.04 in epoch 782
Loss 0.04 in epoch 783
Loss 0.04 in epoch 784
Loss 0.04 in epoch 785
Loss 0.04 in epoch 786
Loss 0.04 in epoch 787
Loss 0.04 in epoch 788
Loss 0.04 in epoch 789
Loss 0.04 in epoch 790
Loss 0.04 in epoch 791
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Loss 0.04 in epoch 793
Loss 0.04 in epoch 794
Loss 0.04 in epoch 795
Loss 0.04 in epoch 796
Loss 0.04 in epoch 797
Loss 0.04 in epoch 798
Loss 0.04 in epoch 799
Loss 0.04 in epoch 800
Loss 0.04 in epoch 801
Loss 0.04 in epoch 802
Loss 0.04 in epoch 803
Loss 0.04 in epoch 804
Loss 0.04 in epoch 805
Loss 0.04 in epoch 806
Loss 0.04 in epoch 807
Loss 0.04 in epoch 808
Loss 0.04 in epoch 809
Loss 0.04 in epoch 810
Loss 0.04 in epoch 811
Loss 0.04 in epoch 812
Loss 0.04 in epoch 813
Loss 0.04 in epoch 814
Loss 0.04 in epoch 815
Loss 0.04 in epoch 816
Loss 0.03 in epoch 817
Loss 0.03 in epoch 818
Loss 0.03 in epoch 819
Loss 0.03 in epoch 820
Loss 0.03 in epoch 821
Loss 0.03 in epoch 822
Loss 0.03 in epoch 823
Loss 0.03 in epoch 824
Loss 0.03 in epoch 825
Loss 0.03 in epoch 826
Loss 0.03 in epoch 827
Loss 0.03 in epoch 828
Loss 0.03 in epoch 829
Loss 0.03 in epoch 830
Loss 0.03 in epoch 831
Loss 0.03 in epoch 832
Loss 0.03 in epoch 833
Loss 0.03 in epoch 834
Loss 0.03 in epoch 835
Loss 0.03 in epoch 836
Loss 0.03 in epoch 837
Loss 0.03 in epoch 838
Loss 0.03 in epoch 839
Loss 0.03 in epoch 840
Loss 0.03 in epoch 841
Loss 0.03 in epoch 842
Loss 0.03 in epoch 843
Loss 0.03 in epoch 844
Loss 0.03 in epoch 845
Loss 0.03 in epoch 846
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Loss 0.03 in epoch 848
Loss 0.03 in epoch 849
Loss 0.03 in epoch 850
Loss 0.03 in epoch 851
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Loss 0.03 in epoch 853
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Loss 0.03 in epoch 856
Loss 0.03 in epoch 857
Loss 0.03 in epoch 858
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Loss 0.03 in epoch 860
Loss 0.03 in epoch 861
Loss 0.03 in epoch 862
Loss 0.03 in epoch 863
Loss 0.03 in epoch 864
Loss 0.03 in epoch 865
Loss 0.03 in epoch 866
Loss 0.03 in epoch 867
Loss 0.03 in epoch 868
Loss 0.03 in epoch 869
Loss 0.03 in epoch 870
Loss 0.03 in epoch 871
Loss 0.03 in epoch 872
Loss 0.03 in epoch 873
Loss 0.03 in epoch 874
Loss 0.03 in epoch 875
Loss 0.03 in epoch 876
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Loss 0.03 in epoch 878
Loss 0.03 in epoch 879
Loss 0.03 in epoch 880
Loss 0.03 in epoch 881
Loss 0.03 in epoch 882
Loss 0.03 in epoch 883
Loss 0.03 in epoch 884
Loss 0.03 in epoch 885
Loss 0.03 in epoch 886
Loss 0.03 in epoch 887
Loss 0.03 in epoch 888
Loss 0.03 in epoch 889
Loss 0.03 in epoch 890
Loss 0.03 in epoch 891
Loss 0.03 in epoch 892
Loss 0.03 in epoch 893
Loss 0.03 in epoch 894
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Loss 0.03 in epoch 896
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Loss 0.03 in epoch 899
Loss 0.03 in epoch 900
Loss 0.03 in epoch 901
Loss 0.02 in epoch 902
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Loss 0.02 in epoch 910
Loss 0.02 in epoch 911
Loss 0.02 in epoch 912
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Loss 0.02 in epoch 916
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Loss 0.02 in epoch 920
Loss 0.02 in epoch 921
Loss 0.02 in epoch 922
Loss 0.02 in epoch 923
Loss 0.02 in epoch 924
Loss 0.02 in epoch 925
Loss 0.02 in epoch 926
Loss 0.02 in epoch 927
Loss 0.02 in epoch 928
Loss 0.02 in epoch 929
Loss 0.02 in epoch 930
Loss 0.02 in epoch 931
Loss 0.02 in epoch 932
Loss 0.02 in epoch 933
Loss 0.02 in epoch 934
Loss 0.02 in epoch 935
Loss 0.02 in epoch 936
Loss 0.02 in epoch 937
Loss 0.02 in epoch 938
Loss 0.02 in epoch 939
Loss 0.02 in epoch 940
Loss 0.02 in epoch 941
Loss 0.02 in epoch 942
Loss 0.02 in epoch 943
Loss 0.02 in epoch 944
Loss 0.02 in epoch 945
Loss 0.02 in epoch 946
Loss 0.02 in epoch 947
Loss 0.02 in epoch 948
Loss 0.02 in epoch 949
Loss 0.02 in epoch 950
Loss 0.02 in epoch 951
Loss 0.02 in epoch 952
Loss 0.02 in epoch 953
Loss 0.02 in epoch 954
Loss 0.02 in epoch 955
Loss 0.02 in epoch 956
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Loss 0.02 in epoch 958
Loss 0.02 in epoch 959
Loss 0.02 in epoch 960
Loss 0.02 in epoch 961
Loss 0.02 in epoch 962
Loss 0.02 in epoch 963
Loss 0.02 in epoch 964
Loss 0.02 in epoch 965
Loss 0.02 in epoch 966
Loss 0.02 in epoch 967
Loss 0.02 in epoch 968
Loss 0.02 in epoch 969
Loss 0.02 in epoch 970
Loss 0.02 in epoch 971
Loss 0.02 in epoch 972
Loss 0.02 in epoch 973
Loss 0.02 in epoch 974
Loss 0.02 in epoch 975
Loss 0.02 in epoch 976
Loss 0.02 in epoch 977
Loss 0.02 in epoch 978
Loss 0.02 in epoch 979
Loss 0.02 in epoch 980
Loss 0.02 in epoch 981
Loss 0.02 in epoch 982
Loss 0.02 in epoch 983
Loss 0.02 in epoch 984
Loss 0.02 in epoch 985
Loss 0.02 in epoch 986
Loss 0.02 in epoch 987
Loss 0.02 in epoch 988
Loss 0.02 in epoch 989
Loss 0.02 in epoch 990
Loss 0.02 in epoch 991
Loss 0.02 in epoch 992
Loss 0.02 in epoch 993
Loss 0.02 in epoch 994
Loss 0.02 in epoch 995
Loss 0.02 in epoch 996
Loss 0.02 in epoch 997
Loss 0.02 in epoch 998
Loss 0.02 in epoch 999
Loss 0.02 in epoch 1000
```

Now let’s create more data from the same linear regression and see how well our network is able to predict it:

```
[13]:
```

```
# Moves the network back to the CPU if it was on a GPU.
net.cpu()
# Since we are not training the network anymore, let's put in
# evaluation mode which is faster. In case you need to train it
# again, call net.train()
net.eval()
# Creates some data using a linear regression
inputv = torch.randn(5, 10)
correct_values = torch.mm(inputv, beta)
predicted_values = net(inputv)
criterion(correct_values, predicted_values).item()
```

```
[13]:
```

```
0.4604862332344055
```

Exercise: try decreasing and increasing the amount of training data and see if this error goes down! Also try to change the number of features to see how it affects the error.