DEEP LEARNING

 DEEP Neural Networks

 Deep Matrix Vector Products

SVM

DT

PGM

KNN

DNN

Text

The data are Unstructured and DL models work well for unstructured data (poorly for structured data)

One learning approach (Gradient based) for all these data types 

Paves the way to build a single model that learns from a two (or more) different modalities (Example: Image Captioning)

Model's performance scales up with increase in data size

Data type : Image 

Dataset: CIFAR-10 (Natural images)

Models:  A single neuron, FCNN, CNN

Data Collection

Transformations

Training 

model

Testing model

It is alright if some words are Alien to you 

Matrix (Tensor) of dim \( m \times n \)

Image of size \(height \times width\)

R

G

B

Scalar

(0 dim Tensor)

Vector

(1 dim Tensor)

Matrix

(2 dim Tensor)

Stacked Matrix

(3 dim Tensor)

Provide efficient ways to manipulate them (Accessing elements, mathematical operations, ..)

flatten()

reshape(1,9)

transpose(,0,1)

cat(,dim=1)

sum()

sum(,dim=0)

sum(,dim=1)

Scalar (dim=0)

Vector (dim=1)

Vector (dim=1)

sigmoid()

softmax(,dim=0)

softmax(,dim=1)

across rows

across columns

Of course, not all of them are relevant for a particular problem in most of the situation.

Recognize whether an image contains a single vertical line or not

The output is True for a very small subset of images and False for the rest

How many patterns are possible with a colour image of size \(5 \times 5\)?

This is a challenge for Generative modelling 

Our focus is on discriminative modelling

Two configurations out of million possibilities

The Parameter \(\mathbf{W}\) is randomly initialized

Compute the gradient of  loss w.r.t \(\mathbf{w}\) , \(\nabla \mathbf{w}\) 

Update rule:

The non-linear function \(f\) in neurons is called activation function

Check the performance with a set of criteria. Iteratively update the parameters for improving the performance

Note: Input elements are real-valued in general.

https://cloud.google.com/products/ai/ml-comic-2

Few more design choices

• Loss functions

• Optimizers : SGD, Adaptive

• Hyper-parameters: Learning rate (schedulers), momentum, ..

Just to ensure that the gradients flow across entire network

The network contains \(\text L-1\) hidden layers (2, in this case) having \(\text n\) neurons each

Finally, there is one output layer containing \(\text k\) neurons (say, corresponding to \(\text k\) classes)

\(a_2\)

\(a_3\)

Each neuron in the hidden layer and output layer can be split into two parts : pre-activation and activation (\(a_i\) and \(h_i\) are vectors)

pre-activation

(\(a_i\) and \(h_i\) are vectors)

The input layer can be called the \(0\)-th layer and the output layer can be called the (\(L\))-th layer

\(W_i \in \R^{n \times n}\) and \(b_i \in \R^n\) are the weight and bias between layers \(i-1\) and \(i  (0 < i < L\))

\(W_L \in \R^{k \times n}\) and \(b_L \in \R^k\) are the weight and bias between the last hidden layer and the output layer   (\(L = 3\)) in this case)

\(a_1\)

\(h_L=\hat {y} = \hat{f}(x)\)

\(h_2\)

\(h_1\)

                                                                                   and activation \(a_i\) and \(h_i\) are vectors)

The activation at layer \(i\) is given by

\(h_i(x) = g(a_i(x))\)

where \(g\) is called the activation function (for example, logistic, tanh, linear, etc.)

The activation at the output layer is given by

\(f(x) = h_L(x)=O(a_L(x))\)

where \(O\) is the output activation function (for example, softmax, linear, etc.)

To simplify notation we will refer to \(a_i(x)\) as \(a_i\) and \(h_i(x)\) as \(h_i\)

\(\hat y_i = \hat{f}(x_i) = O(W_3 g(W_2 g(W_1 x + b_1) + b_2) + b_3)\)

Model:

\(\theta = W_1, ..., W_L, b_1, b_2, ..., b_L (L = 3)\)

Algorithm: Gradient Descent with Back-propagation 

Objective/Loss/Error function: Say,

\(min \cfrac {1}{N} \displaystyle \sum_{i=1}^N \sum_{j=1}^k (\hat y_{ij} - y_{ij})^2\)

\(\text {In general,}\) \(min  \mathscr{L}(\theta)\)

Parameters:

where \(\mathscr{L}(\theta)\) is some function of the parameters

\(b_2\) = [0.01,0.02,0.03]

\(b_3\) = [0.01,0.02]

\(W_1=\)

1.5

2.5

3

\(W_2=\)

\(W_3=\)

0.36

0.37

0.38

0.589

0.591

0.593

0.054

0.064

0.074

0.513

0.516

0.518

1.558

1.568

0.497

0.502

\(b_1\) = [0.01,0.02,0.03]

\([h_1]=sigmoid(a_1)\)

\([h_2]=sigmoid(a_2)\)

\([h_3]=softmax(a_3)\)