Roll your sleeves! Let us do some partial derivatives.

Posted August 14, 2020 by Gowri Shankar  ‐  3 min read

In this post, we shall explore a shallow neural network with a single hidden layer and the math behind back propagation algorithm, gradient descent

from IPython.display import Image


Forward and Backpropagation Algorithm

An associated source code will be published shortly.

In this post, we shall explore a shallow Dense Neural Network(DNN) with

• an Input Layer
• a Single Hidden Layer and
• an Output Layer

The typical architecture of a feed forward network has the above units. Where

• $x$ is a vector that represents the input values,
• $\hat y$ is a vector that represents the predictions and
• $h$ is vector that represents the hidden layer, $h$ is a hyperparameter selected based on the context of the problem

Further to the above 3 parameters, connection weights $[W_1, W_2]$ and connection biases $[b_1, b_2]$ are the other matrices and vectors used in a DNN

Architecture of a Shallow DNN

6 Inputs, 9 Hidden Nodes, 4 Outputs

display(Image("/kaggle/input/sample-images/shallow-dnn.png"))


Following are the processes we shall cover in the quest of understanding a simple Neural Network architecture,

• Forward Propagation
• Cross Entropy Loss (covered in a separate post)
• Activation Functions
• Softmax
• ReLU
• Backpropagation
• Extraction of features of the hidden layer

From the above Diagram…

• There are 6 Inputs of dimension $(I \times 1)$ ie (6 X 1)
• There are 9 nodes in the Hidden Layer of dimension $(N \times 1)$ ie (9 X 1)
• There are 4 nodes in the Output Layer of dimension $(O \times 1)$ ie (4 X 1)

Forward Propagation

From the inputs and weights, biases of the hidden layer, $$z_1 = W_1x + b_1 \tag{1}$$ $$h = ReLU(z_1) \tag{2}$$ From hidden layer ($h$) and weights, biases to the output layer $$z_2 = W_2h + b_2 \tag{3}$$ $$\hat y = softmax(z_2) \tag{4}$$

Dimension Analysis

How to select the weights, let us do dimensionality reduction Let us examine the dimensions of all elements in the equation 1 $$z_1 = W_1h + b_1$$ $$dimensions$$ $$N \times 1 = [?, ?] \times [I, 1] + [?, ?]$$ $$naturally$$ $$N \times 1 = [N, I] \times [I, 1] + [N, 1]$$ $$hence$$ $$9 \times 1 = [9, 6] \times [6, 1] + [9, 1]$$ Hence $N \times I$ is the dimension of the weight matrix $W_1$

Let us examine the dimensions of all elements in the equation 2 $$z_2 = W_2h + b_2$$ $$dimensions$$ $$O \times 1 = [?, ?] \times [N, 1] + [?, ?]$$ $$then$$ $$O \times 1 = [O, N] \times [N, 1] + [O, 1]$$ $$hence$$ $$4 \times 1 = [4, 9] \times [9, 1] + [4, 1]$$ Hence $O \times N$ is the dimension of the weight matrix $W_2$

Cross Entropy Loss

Our goal is to minimize the loss $J$ $$J = - \sum\limits_{k=1}^V y_k \log\hat y_k \tag{5}$$ Cost(Loss) Function for Binary Classification - A Deep Dive

Backpropagation - Pull your sleeves and do some partial derivatives

We calculated the $\hat y$ during forward propagation and now our goal is to optimize the weights by minimizing the loss. This is done by calculating the partial derivatives wrt $[W_2, b_2]$ and then $[W_1, b_1]$ $$\frac{\partial J}{\partial W_1} = ReLU\left( W^T_2(\hat y - y)\right)x^T \tag{6}$$ $$\frac{\partial J}{\partial W_2} = (\hat y - y)h^T \tag{7}$$ $$\frac{\partial J}{\partial b_1} = ReLU\left(W^T_2(\hat y - y)\right) \tag{8}$$ $$\frac{\partial J}{\partial b_2} = \hat y - y \tag{9}$$

Gradient descent is the process during which the weights and biases are updated by subtracting $\alpha$ times the learning rate of calculated gradients from the original matrices and biases
$$W_1 := W_1 - \alpha\frac{\partial J}{\partial {W_{1}}} \tag{10}$$ $$W_2 := W_2 - \alpha\frac{\partial J}{\partial W_2} \tag{11}$$ $$b_1 := b_1 - \alpha\frac{\partial J}{\partial b_1} \tag{12}$$ $$b_2 := b_2 - \alpha\frac{\partial J}{\partial b_2} \tag{13}$$