---
title: Neural Networks Lecture
categories: lecture
geometry: margin=2cm
---
# Briefing
## What is a neural network
+ The single Neuron
+ Weighted Input
+ Activation
+ The network model
+ Input/Output
+ Weights
+ Activation Function
+ The Tensor Model
## Output and Loss Function
+ Classification versus Regression
+ Two-class classification (0 or 1)
+ Regression $y=f(x)\in \mathbb{R}$
+ Multiclass $y=(y_1,y_2,\ldots,y_n)$
+ $y_i=1$ means membership in class 1
+ Soft-decision: $y$ is a continuous variable
+ higher values are more probably one
+ Loss functiion
+ Mean-squared error (common for regression)
$$L = (x-y)^2$$
+ Cross Entropy (common for classification)
$$L = \log \frac{ \exp x_{y} } { \sum \exp x_i }$$
+ There are others
## Training
+ Optimisation problem
+ tune the weights to minimise the loss function
+ if the activation function is differentiable, the entire system is
+ different optimisation algorithms;
trust the API or do a more advanced module
## Activation Functions
+ Threshold functions
+ Approximations to the threshold function
+ Logistic: $f(x) = \frac1{1+e^{-\beta x}}$
+ ReLU: $f(x)=\max(x,0)$
- not differentiable
## Tools
Two main contenders.
+ TensorFlow
+ PyTorch
+ A replacement for NumPy to use the power of GPUs and other accelerators.
+ An automatic differentiation library that is useful to implement neural networks.
Note that PyTorch replaces NumPy; i.e. it is primarily a python tool,
and operaes in the object oriented framework of python.
The reason for using PyTorch in these examples is primarily that I have
lately been working off some code created by some final year students
this Spring, and they happened to choose PyTorch.
The choice of TensorFlow or PyTorch is otherwise arbitrary.
## Sample Program
### Training
```python
model = Inception3(num_outputs=nparams)
criterion = nn.MSELoss()
optimizer = torch.optim.Adam(model.parameters(),
model.train()
for epoch in range(num_epochs):
tloss = 0.0
for i, (images, params) in enumerate(trainloader):
optimizer.zero_grad()
output = model(images)
loss = criterion(output, params)
loss.backward()
optimizer.step()
tloss += loss.item() * len(images)
print( f"Epoch {epoch+1}: Loss = {tloss}" )
```
### Testing
```python
total_loss = 0
model.eval()
with torch.no_grad():
for images, params in testloader:
output = model(images)
loss = criterion(output, params)
total_loss += loss * len(images)
```
## Loss Functions and Evaluation
+ Accuracy: ratio of correctly classified items
+ What is the difference between a rate and a probability?
+ Statistics
+ Standard deviation
+ Hypothesis Tests
+ Confidence Interval
+ Other heuristics
## Computational Power
+ Neural Networks are Computationally Expensive
+ GPU or CPU - what's the difference?
+ what resources do you have?
+ Remedies
+ Reduce image resolution
+ Reduce number of images
+ Reduce number of epochs
+ In particular, it is necessary to sacrifice accuracy during
development and testing.
+ In the final stages you may need big datasets to achieve satisfactory
results, and then you may need more computing power.