---
title: Neural Networks Lecture
categories: lecture
---

# 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.