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L1 Regularization Gradient

~10 mincode completion

Gradient of L1 Regularization

To include L1 regularization in gradient descent, we need:

where $sign (x) = ⎩ ⎨ ⎧ + 1 0 - 1 x > 0 x = 0 x < 0$

This gradient is added to the data-loss gradient during backpropagation:

Key contrast with L2: The L1 gradient has constant magnitude, it always pushes each weight toward zero by the same step size regardless of weight magnitude. This creates sparsity because small weights get pushed all the way to zero, while L2 only asymptotically approaches zero.

Your task:

Implement l1_gradient(W, lambda_) that returns the gradient of the L1 penalty with respect to each element of W.

Example Tests

Row vector with positive and negative entries, lambda=1

Input: {"W":[[1,-2,3]],"lambda_":1}

Expected: [[1,-1,1]]

Zero maps to zero; lambda=0.5 scales the result

Input: {"W":[[0,5,-5]],"lambda_":0.5}

Expected: [[0,0.5,-0.5]]

2x2 matrix with small lambda

Input: {"W":[[1,-1],[2,-2]],"lambda_":0.1}

Expected: [[0.1,-0.1],[0.1,-0.1]]

Python

import numpy as np

def l1_gradient(W: np.ndarray, lambda_: float) -> np.ndarray:
    """
    Compute the gradient of L1 regularization w.r.t. each weight.

    Args:
        W:       Weight matrix (any shape)
        lambda_: Regularization strength

    Returns:
        Gradient array of same shape as W: lambda_ * sign(W)
    """
    # YOUR CODE HERE
    pass

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