Weights and gradients are closely related in neural networks and are central to how the model learns during training. Here’s how they’re connected:
1. Weights: The Model’s Knowledge
Weights are the parameters within the model that determine how the input data is transformed at each layer. They decide the strength and direction of influence each neuron has on another. During training, the model adjusts these weights to better fit the data and minimize prediction errors.
Each weight can be thought of as a “knob” that the model tunes to improve its accuracy on a given task.
2. Gradients: Directions for Updating Weights
Gradients are the partial derivatives of the loss (error) with respect to each weight. In other words, a gradient tells us how much and in what direction to adjust each weight to reduce the error.
• Direction: A positive gradient means the weight should decrease to reduce the error, while a negative gradient means it should increase.
• Magnitude: The size of the gradient indicates the step size. A large gradient suggests a bigger change, while a small gradient suggests a smaller change.
Gradients are computed through a process called backpropagation and are used in the optimization step to update weights in a way that minimizes the model’s error.
3. The Role of Backpropagation in Learning
Backpropagation is a key process that relates weights and gradients. Here’s how it works:
1. Forward Pass: The input is passed through the network, and the model makes a prediction.
2. Compute Loss: The prediction is compared to the true label using a loss function, which calculates the error.
3. Backward Pass: Backpropagation calculates the gradient of the loss with respect to each weight. It works by applying the chain rule of calculus layer by layer, from the output layer back to the input layer.
4. Update Weights: An optimizer (like Stochastic Gradient Descent) uses the gradients to adjust the weights. Each weight w is updated based on its gradient ∂L/∂w:
where (the learning rate) controls the size of each weight update step.
4. How Gradients Affect Weight Adjustments
During each update step, weights are adjusted slightly based on their gradients. These adjustments help reduce the model’s loss over time. Ideally, after many updates, the model’s weights reach values that produce low error, meaning the model has effectively learned patterns in the data.
Example: Weight and Gradient Relationship
Suppose a weight has a positive gradient of 0.5. This gradient means that increasing would increase the loss, so the optimizer will decrease in the next step to try to reduce the loss. Similarly, if the gradient was -0.5, the optimizer would increase to reduce the loss.
In summary:
• Weights store the model’s “knowledge.”
• Gradients provide the “instructions” for how to update these weights during training to improve the model’s accuracy.