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MSE Loss Gradient

Implement a function to compute the gradient of the Mean Squared Error (MSE) loss with respect to predictions. The MSE loss is a common loss function used in regression problems to measure the difference between predicted and actual values.

The MSE loss is calculated as the average of the squared differences between predicted and actual values, which can be expressed as $MSE = \frac{1}{n}\sum_{i=1}^{n}(\hat{y}_i - y_i)^2$ . To update model parameters using backpropagation, we need to compute the gradient of the MSE loss with respect to each prediction.

Here are the steps to compute the gradient:

Calculate the difference between each predicted value and actual value.
Square each difference and sum them up.
Compute the average of the squared differences.
Calculate the gradient of the MSE loss with respect to each prediction.

\frac{\partial MSE}{\partial \hat{y}_i} = \frac{2}{n}(\hat{y}_i - y_i)

This technique is widely used in deep learning models for regression tasks.

Example:

Input:

mse_grad([2, 4], [1, 3])

Output:

[1.0, 1.0]

Reasoning:

Initialize variables: predictions $\hat{y} = [2, 4]$ , targets $y = [1, 3]$ , and $n = 2$ (number of samples)
Calculate differences: Compute $\hat{y}_i - y_i$ for each element: $[2-1, 4-3] = [1, 1]$
Apply gradient formula: Use $\frac{\partial MSE}{\partial \hat{y}_i} = \frac{2}{n}(\hat{y}_i - y_i)$ for each element: $\frac{2}{2} \cdot [1, 1] = [1.0, 1.0]$ [1][4]
Output result: The gradient is $[1.0, 1.0]$ , indicating how much each prediction should adjust to reduce the loss[1]

Constraints:

Return gradient rounded to 4 decimal places

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You are given the gradient formula already; your task is to understand where it comes from and how to apply it correctly in code.

1. Background Knowledge

Mean Squared Error (MSE) compares predictions $\hat{y}$ to targets $y$ by averaging squared differences:

MSE = \frac{1}{n} \sum_{i=1}^n (\hat{y}_i - y_i)^2 It penalizes larger errors more heavily because of the square. - **Gradients for backpropagation**: during training, we update model parameters by moving them in the direction that *reduces* the loss. This uses the gradient of the loss with respect to: 1. The model outputs (predictions) \hat{y}_i 2. Then, via chain rule, with respect to weights and biases inside the network. - **Chain rule idea**: if a parameter w affects the loss through \hat{y}, then \frac{\partial MSE}{\partial w} = \sum_i \frac{\partial MSE}{\partial \hat{y}_i} \cdot \frac{\partial \hat{y}_i}{\partial w} So the derivative \frac{\partial MSE}{\partial \hat{y}_i} is the “entry point” for backprop. --- ## 2. Algorithm / General Approach For MSE loss: 1. **Define the loss** as average of squared errors: (\hat{y} - y)^2 averaged over all samples (and sometimes over all output dimensions). 2. **Differentiate w.r.t. each prediction** \hat{y}_i using basic calculus (derivative of x^2 is 2x). 3. **Use this gradient** as the “error signal” at the output layer, then propagate it backward through the network using the chain rule. In array/vector form this becomes:

$\nabla$ _{ $\hat{y}$ } $MSE =$ \frac{2}{n} $(\hat{y}$ - y)

--- ## 3. Step-by-Step Strategy To derive and then implement: 1. **Write the loss explicitly**:

$MSE =$ \frac{1}{n}\sum_{i=1} $^n (\hat{y}$ _i - y_i)^2

2. **Focus on a single term** in the sum for index i:

$L_i =$ \frac{1}{n} $(\hat{y}$ _i - y_i)^2

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solution.py

Python 3.13

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