Mean Squared Error Gradient

Implement a function to compute the gradient of the Mean Squared Error (MSE) loss with respect to predictions, a crucial step in training models using optimization techniques. This task involves understanding the mathematical foundation of MSE and its derivative.

The Mean Squared Error is a measure of the average squared difference between predictions $\hat{y}$ and actual targets $y$, given by the formula $\frac{1}{n}\sum_{i=1}^{n}(\hat{y}_i - y_i)^2$. To minimize this loss, we need to calculate its gradient with respect to each prediction $\hat{y}_i$.

Here are the steps to calculate the gradient:

1. Define the MSE loss function.
2. Apply the chain rule and the sum rule of calculus to differentiate the MSE loss with respect to each $\hat{y}_i$.

This technique is widely used in machine learning for training regression models.