Mean Squared Error Loss Gradienteasy

Mean Squared Error Loss Gradient

easy Neural Networks by Hand 15 min

NumPy allowed · plain Python earns 10 bonus points

Loss moves downhill

The update rule changes the parameters on each step.

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01parameters

02prediction

03loss

04gradient

θ = 4.000loss = 16.000∂L/∂θ = 8.000

θ ← θ − 0.14 · ∂L/∂θ

Compute the analytical gradient of Mean Squared Error loss w.r.t network predictions to initiate backpropagation.

Related Skills

Regression Loss Functions

Task

Implement mse_loss_backward(y_pred, y_true): with MSE = (1 / (N * D)) * sum((y_pred - y_true)^2), compute the derivative with respect to each prediction y_pred[i][j], which is (2 / (N * D)) * (y_pred[i][j] - y_true[i][j]). Here N is the number of rows and D is the number of columns. Round each gradient element to 4 decimal places.

Function Signature

mse_loss_backward(y_pred: list[list[float]], y_true: list[list[float]]) -> list[list[float]]

Examples

Example 1: Single sample 1D output
Input: y_pred = [[3, 5]]
y_true = [[2, 1]]
Output: [[1, 4]]

Example 2: Batch of 2 samples
Input: y_pred = [[1], [4]]
y_true = [[1], [2]]
Output: [[0], [2]]

Example 3: Exact match zero gradient
Input: y_pred = [[2.5, -1]]
y_true = [[2.5, -1]]
Output: [[0, 0]]

Constraints

y_pred and y_true have matching shape (N, D)
1 <= N, D <= 100

Python3Saved ✓

def mse_loss_backward(y_pred: list[list[float]], y_true: list[list[float]]) -> list[list[float]]:

"""Compute gradient of mean squared error with respect to predictions y_pred.

Args:

y_pred: Model predictions of shape (N, D).

y_true: Ground truth targets of shape (N, D).

Returns:

Gradient matrix dL/dy_pred of shape (N, D) rounded to 4 decimals.

"""

pass

Case 1: Single sample 1D output

Input: y_pred = [[3, 5]]
y_true = [[2, 1]]

Expected: [[1, 4]]

Case 2: Batch of 2 samples

Input: y_pred = [[1], [4]]
y_true = [[1], [2]]

Expected: [[0], [2]]

Case 3: Exact match zero gradient

Input: y_pred = [[2.5, -1]]
y_true = [[2.5, -1]]

Expected: [[0, 0]]