import numpy as np

def polynomial_regression_with_cem(x: np.ndarray, y: np.ndarray, d: int, num_iters: int = 12, elite_frac: float = 0.2, batch_size: int = 1000): ''' Inputs `x` (np.ndarray): a 1-D numpy array corresponding to the 'x' values of some unknown function f. `y` (np.ndarray): a 1-D numpy array corresponding to the 'y' values of some unknown function f, f(x[i]) = y[i]. Has precisely the same shape as `x`. `d` (integer): a scalar reflecting the maximum degree of the least-squares polynomial whose coefficients we hope to estimate with CEM. `num_iters` (integer): CEM hyperparameter, the total number of CEM iterations to perform. `elite_frac` (float): CEM hyperparameter, the total proportion of "top" samples used to refine CEM parameter estimates at each iteration. `batch_size` (int): CEM hyperparameter, the total number of parameter values sampled from the paramter inference distribution at each iteration. Outputs `dummy_coeffs` (np.ndarray): a 1-D numpy array of shape (d+1,), containing the final estimates for each of the polynomial coefficients regressed with CEM after `num_iters` have been completed. Each value `dummy_coeffs[i]` should correspond to the degree i coefficient of the regressed polynomial.

''' dummy_coeffs = np.zeros((d+1,)) assert isinstance(dummy_coeffs, np.ndarray) and (dummy_coeffs.shape == (d+1,)) return dummy_coeffs

def sample_test(f): x = np.array([[1, 2.5, 3.5, 4, 5, 7, 8.5]]) y = np.array([[0.3, 1.1, 1.5, 2.0, 3.2, 6.6, 8.6]]) d = 1

out = f(x, y, d, num_iters=100, elite_frac=0.01, batch_size = 1000) assert np.allclose(out, [1.03571252, -1.19191605], rtol=1)

sample_test(polynomial_regression_with_cem)

In this problem, your task is to write a complete implementation of Cross-Entropy Method for generalized polynomial regression. More precisely, given - a polynomial degree d:d1 - a dataset of data pairs (xi,yi) represented via two equal-length 1-D numpy arrays; and - CEM hyperparameters num_iters, elite_frac, and batch_size, your code should return the final regressed estimates for the values of each of the (d+1) coefficients of the polynomial minimizing the total sum of squared errors, f(xi^;)=dxid+d1xid1++1xi+0 i(yif(xi^;))2 via the parameter vector [0,,d]. Employ a multi-dimensional Gaussian distribution with diagonal covariance as the distribution underlying parameter inference. The "starting" estimation distribution should be centered on the zero vector and have the identity as its covariance