How to Update Bias and Bias’s Weight Using Backpropagation Algorithm?

Answer: In backpropagation, biases are updated by applying the chain rule to the loss function with respect to the bias parameters in each layer during gradient descent.

Let’s explore the details of how biases and their weights are updated using the backpropagation algorithm:

1. Backpropagation Overview:
   • Backpropagation is an algorithm used to train artificial neural networks by efficiently computing the gradients of the loss function with respect to the parameters of the network.
   • It involves two main steps: forward propagation, where the inputs are passed through the network to generate predictions, and backward propagation, where the gradients of the loss function with respect to each parameter are computed recursively using the chain rule.
2. Biases in Neural Networks:
   • Biases are additional parameters in neural network nodes (neurons) that allow the model to capture offsets or shifts in the data.
   • Each neuron typically has its own bias parameter, which is added to the weighted sum of inputs before passing through an activation function.
3. Weight Update Rule:
   • During backpropagation, the gradient of the loss function with respect to each parameter (including biases) is computed.
   • The gradient descent algorithm is then used to update the parameters in the direction that minimizes the loss function.
4. Gradient Calculation for Biases:
   • The gradient of the loss function with respect to a bias parameter in a particular layer is computed using the chain rule.
   • For each bias parameter [Tex]b_i[/Tex] in a layer, the gradient is computed as the partial derivative of the loss function [Tex](L)[/Tex] with respect to the output of that layer [Tex]z_i[/Tex], multiplied by the derivative of the activation function [Tex](\frac{\partial z_i}{\partial b_i})[/Tex]).
   • Mathematically, this can be expressed as:
     [Tex][ \frac{\partial L}{\partial b_i} = \frac{\partial L}{\partial z_i} \times \frac{\partial z_i}{\partial b_i} ][/Tex]
   • The derivative [Tex]\frac{\partial z_i}{\partial b_i}[/Tex] is often simply 1, as biases are directly added to the input of the activation function.

1. Bias Update Rule:
   • Once the gradients of the loss function with respect to the biases are computed, the biases are updated using gradient descent.
   • The update rule for a bias parameter $b_i$ in a particular layer is [Tex][ b_i \leftarrow b_i – \alpha \times \frac{\partial L}{\partial b_i} ][/Tex].
   • Here, [Tex]\alpha[/Tex] is the learning rate, which determines the size of the step taken during gradient descent.
2. Iterative Update:
   • The process of computing gradients and updating biases is repeated iteratively for each mini-batch of data in the training set until convergence or a predefined number of iterations.

By updating biases and their weights using the backpropagation algorithm, neural networks can effectively learn the optimal parameters that minimize the loss function and improve their performance on the given task