How to Update Weights in a Neural Network Without an Activation Function

In a neural network setup without an activation function, the output of each neuron is a linear combination of the inputs. This design simplifies the network's structure but limits its ability to model complex nonlinear relationships. However, the process for updating the weights during training can still be achieved using gradient descent. In this article, we will explore the detailed steps involved in updating the weights in such a neural network.

Weight Update Rule and Forward Pass

The forward pass in a neural network without an activation function involves computing the output of the network. For a single neuron, the output is given by the linear combination of the inputs, taking into account the weight matrix, input vector, and bias term:

[ y_{text{pred}} W cdot X b ]

Where:

y_{text{pred}}: The predicted output of the neuron

W: The weight matrix that defines the relationship between the inputs and the neuron's output

X: The input vector for the neuron

b: The bias term that shifts the linear relationship

This output calculation is similar to linear regression, where the network aims to find the best linear fit for the given input-output relationship.

Loss Calculation

After the forward pass, the next step is to calculate the loss, which measures how well the predicted output matches the actual target values. For regression tasks, the commonly used loss function is the Mean Squared Error (MSE):

[ text{Loss} frac{1}{n} sum (y_{text{true}} - y_{text{pred}})^2 ]

Where:

n: The number of samples in the dataset

y_{text{true}}: The actual target value for the sample

y_{text{pred}}: The predicted output of the sample

The goal is to minimize this loss function over the training process.

Backward Pass and Gradient Calculation

The backward pass involves computing the gradients of the loss with respect to the weights and biases. For the Mean Squared Error, the gradient of the loss with respect to the predicted output is:

[ frac{partial text{Loss}}{partial y_{text{pred}}} -frac{2}{n} (y_{text{true}} - y_{text{pred}}) ]

The gradients with respect to the weights and biases can be calculated as follows:

For weights:

[ frac{partial text{Loss}}{partial W} frac{partial text{Loss}}{partial y_{text{pred}}} cdot frac{partial y_{text{pred}}}{partial W} frac{partial text{Loss}}{partial y_{text{pred}}} cdot X ]

For biases:

[ frac{partial text{Loss}}{partial b} frac{partial text{Loss}}{partial y_{text{pred}}} ]

These gradients indicate the direction and magnitude of the updates needed to minimize the loss function.

Weight Update

Using the calculated gradients, the weights and biases can be updated using the gradient descent approach. The update rule is given by:

[ W leftarrow W - eta cdot frac{partial text{Loss}}{partial W} ]

[ b leftarrow b - eta cdot frac{partial text{Loss}}{partial b} ]

Where:

eta: The learning rate that controls the step size of the updates

By iteratively applying these update rules, the network's parameters can be tuned to minimize the loss function, leading to better predictions.

Summary

In summary, in a neural network without an activation function, the weights are updated using the standard gradient descent approach based on the gradients of the loss function. The absence of activation functions limits the network's capability to model nonlinear relationships; however, it can still be effective for simpler linear problems. The provided steps outline the detailed process of updating the weights and biases to improve the network's performance.

Understanding these concepts is crucial for anyone working with neural networks, whether they are a beginner or an advanced researcher. By mastering the weight update process, you can enhance the performance of your models and achieve better results on various tasks, from simple linear regression to more complex datasets.

Keywords: neural network, activation function, gradient descent, weight update