Feed Forward Networks: Introduction

CSE 849: Deep Learning

Vishnu Boddeti

Today

Artificial Neuron
Activation Functions
Capacity of Neural Networks
Biological Motivation

Simplest Neural Network

Artificial Neuron

Neuron pre-activation (or input activation)
$a(\mathbf{x}) = b + \sum_i w_ix_i = b + \mathbf{w}^T\mathbf{x}$
Neuron (output) activation
$h(\mathbf{x}) = g(a(\mathbf{x})) = g\left(b+\sum_iw_ix_i\right)$
$\mathbf{w}$ are the connection weights
$b$ is the neuron bias
$g(\cdot)$ is called activation function

Artificial Neuron

range determined by $g(\cdot)$
bias $b$ only changes the position of the riff

Linear Activation

$g(x)=x$

Performs no input squashing
Quite a boring function...

Sigmoid Activation

$g(x)=\frac{1}{1+e^{-x}}$

Squashes the neuron's pre-activation between 0 and 1
Always positive
Bounded
Strictly increasing

Tanh Activation

$g(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$

Squashes the neuron's pre-activation between -1 and 1
Can be positive or negative
Bounded
Strictly increasing

Rectified Linear Unit Activation

$g(x)=max(0,x)$

Bounded below by 0 (always non-negative)
Not upper bounded
Strictly increasing
Tends to yeild neurons with sparse activities

Capcity of Neural Networks

Single Neuron

Could do binary classification:

with sigmoid, can interpret neuron as estimating $p(y=1|\mathbf{x})$
also known as logistic regression classifier
if greater than 0.5, predict class 1
otherwise, predict class 0
similar idea can be used with Tanh

decision boundary is linear

Capacity of a Single Neuron

Can solve linearly seperable problems

Capacity of a Single Neuron

Cannot solve non-linearly separable problems....

...unless the input is transformed in a better representation

Neural Network with Hidden Layer

Hidden layer pre-activation:

$\mathbf{a}(\mathbf{x}) = \mathbf{b}_1 + \mathbf{W}_1\mathbf{x}$

$\left(a(\mathbf{x})^i = \mathbf{b}^i_1 + \sum_{j}W^{i,j}_1x^j\right)$

Hidden layer activation:

$\mathbf{h}(\mathbf{x}) = \mathbf{g}(\mathbf{a}(\mathbf{x}))$

Output layer activation:

$f(\mathbf{x}) = o(b_2 + \mathbf{w}_2^T\mathbf{h}(\mathbf{x}))$

Softmax Activation Function

For multi-class classification:

we need multiple outputs (1 output per class)
we would like to estimate the conditional probability $p(y=c|\mathbf{x})$

Softmax activation function at the output:

$\mathbf{o}(\mathbf{a}) = \textrm{softmax}(\mathbf{a}) = \left[\frac{\exp{a_1}}{\sum_c\exp{a_c}},\dots,\frac{\exp{a_C}}{\sum_c\exp{a_c}}\right]^T$

strictly positive
sums to one

Predicted class: one with highest estimated probability

Multi-Layer Neural Network

Could have $L$ hidden layers:

layer pre-activation for $k>0$ ($\mathbf{h}^{(0)}(\mathbf{x})=\mathbf{x}$)

$\mathbf{a}^{(k)}(\mathbf{x}) = \mathbf{b}^{(k)} + \mathbf{W}^{(k)}\mathbf{h}^{(k-1)}(\mathbf{x})$

hidden layer activation ($k$ from 1 to $L$):

$\mathbf{h}^{(k)}(\mathbf{x}) = \mathbf{g}(\mathbf{a}^{(k)}(\mathbf{x}))$

output layer activation ($k=L+1$):

$\mathbf{h}^{(L+1)}(\mathbf{x}) = \mathbf{o}(\mathbf{a}^{(L+1)}(\mathbf{x})) = f(\mathbf{x})$

Capacity of Single Hidden Layer Neural Network

Universal Approximation

Universal approximation theorem (Hornik, 1991):

"a single hidden layer neural network with a linear output unit can approximate any continuous function arbitrarily well, given enough hidden units"

The result applies for sigmoid, tanh and many other hidden layer activation functions.
This is a good result, but it doesn't mean there is a learning algorithm that can find the necessary parameter values.
Many other function classes also known to be universal approximators.