Linear Regression with Gradient Descent

Instead of a formula, train the line y = w × x + b with batch gradient descent on the mean squared error MSE = Σ(w × x + b - y)² / n.

Start with w = 0 and b = 0. In each epoch, using all points:

1. errorᵢ = w × xᵢ + b - yᵢ
2. dw = (2 / n) × Σ errorᵢ × xᵢ and db = (2 / n) × Σ errorᵢ
3. w ← w - learning_rate × dw and b ← b - learning_rate × db (both computed from the same errors)

Implement train_linear_regression(xs, ys, learning_rate, epochs).

Input

xs, ys: The data points (same length).

learning_rate, epochs: The step size and the number of updates.

Output

Return [w, b] after training.

Constraints

1 ≤ number of points ≤ 1,000, 0 ≤ epochs ≤ 10,000

0 < learning_rate ≤ 1

-100 ≤ each value ≤ 100

Examples

Input:
xs = [1.0, 2.0]
ys = [2.0, 4.0]
epochs = 1
learning_rate = 0.1

Expected output:
expected = [1.0, 0.6]

Input:
std::vector<double> xs = {1.0, 2.0};
std::vector<double> ys = {2.0, 4.0};
long long epochs = 1;
double learning_rate = 0.1;

Expected output:
std::vector<double> expected = {1.0, 0.6};

Input:
double[] xs = new double[] {1.0, 2.0};
double[] ys = new double[] {2.0, 4.0};
long epochs = 1;
double learning_rate = 0.1;

Expected output:
double[] expected = new double[] {1.0, 0.6};

Input:
xs = [1.0, 2.0, 3.0]
ys = [3.0, 5.0, 7.0]
epochs = 2000
learning_rate = 0.05

Expected output:
expected = [2.0, 1.0]

Input:
std::vector<double> xs = {1.0, 2.0, 3.0};
std::vector<double> ys = {3.0, 5.0, 7.0};
long long epochs = 2000;
double learning_rate = 0.05;

Expected output:
std::vector<double> expected = {2.0, 1.0};

Input:
double[] xs = new double[] {1.0, 2.0, 3.0};
double[] ys = new double[] {3.0, 5.0, 7.0};
long epochs = 2000;
double learning_rate = 0.05;

Expected output:
double[] expected = new double[] {2.0, 1.0};

References

This challenge is adapted for CodeUtility. Follow the statement and constraints above.

• Stanford CS229 Machine Learning, lecture notes
• scikit-learn, Linear models