from scipy.optimize import minimize

def one_vs_all(X, y, num_labels, learning_rate):

    rows = X.shape[0]

    params = X.shape[1]

    # k X (n + 1) array for the parameters of each of the k classifiers

    all_theta = np.zeros((num_labels, params + 1))

    # insert a column of ones at the beginning for the intercept term

    X = np.insert(X, 0, values=np.ones(rows), axis=1)

    # labels are 1-indexed instead of 0-indexed

    for i in range(1, num_labels + 1):

        theta = np.zeros(params + 1)

        y_i = np.array([1 if label == i else 0 for label in y])

        y_i = np.reshape(y_i, (rows, 1))

        # minimize the objective function

        fmin = minimize(fun=cost, x0=theta, args=(X, y_i, learning_rate), method='TNC', jac=gradient)

        all_theta[i-1,:] = fmin.x

    return all_theta