## Neural Network Unit

A real-valued vector as input, each multiplied by some weights, summed, and squashed by some non-linear transform.

\begin{equation} z = w\cdot x + b \end{equation}

and then, we will squash this using it as an “activation”

\begin{equation} y = \sigmoid(z) \end{equation}

One common activation is sigmoid. So, one common formulation would be:

\begin{equation} y = \frac{1}{1+\exp (- (w \cdot x + b))} \end{equation}

## Tanh

\begin{equation} y(z) = \frac{e^{z} - e^{-z}}{e^{z}+e^{-z}} \end{equation}

This causes “saturation”—meaning derivatives to be \(0\) at high values

## relu

\begin{equation} y(z) = \max(z,0) \end{equation}

## multi-layer networks

Single computing units can’t compute XOR. Consider a perceptron:

\begin{equation} w_1x_1 + w_2x_2 + b = 0 \end{equation}

meaning:

\begin{equation} x_2 = \qty(\frac{-w_1}{w_2})x_1 + \qty(\frac{-b}{w_2}) \end{equation}

meaning, obtain a line that acts as a **decision boundary**—we obtain 0 if the input is on one side of the line, and 1 if on the other. XOR, unfortunately, does not have a single linear boundary, its not **linearly seperable**.

logistic regression, for instance, can’t compute XOR because it is linear until squashing.

## feed-forward network

we can think about logistic regression as a one layer network, generalizing over sigmoid:

\begin{equation} \text{softmax} = \frac{\exp(z_{i})}{\sum_{j=1}^{k} \exp(z_{j})} \end{equation}

and a multinomial logistic regression which uses the above. This is considered a “layer” in the feed-forward network.

notation:

- \(W^{(j)}\), weight matrix for layer \(j\)
- \(b^{(j)}\), the bias vector for layer \(j\)
- \(g^{(j)}\), the activation function at \(j\)
- and \(z^{(i)}\), the output at \(i\) (before activation function)
- \(a^{(i)}\), the activation at \(i\)

instead of bias, we sometimes add a dummy node \(a_{0}\), we will force a value \(1\) at \(a_{0}\) and use its weights as bias.

### embeddings

We use vector-space model to feed words into networks: converting each word first into embeddings, then feeding it into the network

Fix length problems:

- sentence embedding (mean of all the embeddings)
- element wise max of all the word embeddings to create sentence embedding
- use the max length + pad

For Language Models, we can use a “sliding window”; that is:

\begin{equation} P(w_{t}|w_{1 \dots t-1}) \approx P(w_{t} | w_{t-N+1 \dots t-1}) \end{equation}

## Training

For every tuple \((x,y)\), we run a forward pass to obtain \(\hat{y}\). Then, we run the network backwards to update the weights.

A loss function calculates the negative of the probability of the correct labels.