Stochastic Gradient Descent

$⟨$ $\frac{\partial L}{\partial w} |_{w_{i}}$ $⟩$ $_{D_{i}}$

This expression represents the average gradient of the loss function with respect to the weights, calculated over a batch of training examples. Let's dissect it further:

∂L/∂w: This is the partial derivative of the loss function L with respect to the weights w. It tells us how the loss changes as we adjust the weights.
|wᵢ: This indicates that we're evaluating this derivative at the current weight values wᵢ.
<>Dᵢ: The angle brackets with Dᵢ subscript denote an average over the ith batch of training data.

In practical terms, this means:

For each example in the current batch (Dᵢ), we calculate the gradient of the loss with respect to the weights.
We then average these gradients across all examples in the batch.
This average gradient is used to update the weights, as shown in the update rule.

Why is this important?

Batch processing: Instead of computing the gradient over the entire dataset (which would be computationally expensive), we use a batch of examples. This allows for more frequent weight updates and can lead to faster convergence.
Noise reduction: By averaging over a batch, we reduce the noise in our gradient estimates compared to using single examples.
Stochasticity: Using batches introduces some randomness into the training process, which can help the model escape local minima and potentially find better solutions.

This approach balances computational efficiency with the quality of gradient estimates, making it a key component of effective training in deep learning models like AlexNet.