def grad_max_2deg_penalty(a, f_vals, X, X_grad, ind, n, beta):
"""
Gradient for the smooth maximum
"""
d = X.shape[1]
b = a[:d]
B = a[d:].reshape((d, d))
Y = f_vals[:, ind] + X_grad @ b + qform_q(B + B.T, X_grad,
X) + 2 * np.trace(B)
Y_max = np.max(np.abs(Y))
grad_exp = np.exp(beta * (np.abs(Y) - Y_max)) / np.sum(
np.exp(beta * (np.abs(Y) - Y_max)))
#gradient w.r.t. b
nabla_b = X_grad.T @ (grad_exp * np.sign(Y))
#gradient w.r.t B
nabla_f_B = np.matmul(X_grad.reshape((n, d, 1)), X.reshape((n, 1, d)))
nabla_f_B = nabla_f_B + nabla_f_B.transpose(
(0, 2, 1)) + 2 * np.eye(d).reshape((1, d, d))
nabla_B = np.sum(nabla_f_B * (grad_exp * np.sign(Y)).reshape((n, 1, 1)),
axis=0)
#stack gradients together
grad = np.zeros((d + 1) * d, dtype=np.float64)
grad[:d] = nabla_b
grad[d:] = nabla_B.ravel()
return grad
def grad_qform_2_ESVM(a, f_vals, X, X_grad, W, ind, n, alpha=0.0):
"""
Arguments:
a - np.array of shape (d+1,d),
a[0,:] - np.array of shape(d) - corresponds to coefficients via linear variables
a[1:,:] - np.array of shape (d,d) - to quadratic terms
"""
d = X_grad.shape[1]
b = a[:d]
B = a[d:].reshape((d, d))
Y = f_vals[:, ind] + X_grad @ b + qform_q(B + B.T, X_grad,
X) + 2 * np.trace(B)
#gradient w.r.t. b
nabla_b = 2. / n * (X_grad * PWP_fast(Y, W).reshape((n, 1))).sum(axis=0)
#gradient w.r.t B
nabla_f_B = np.matmul(X_grad.reshape((n, d, 1)), X.reshape((n, 1, d)))
nabla_f_B = nabla_f_B + nabla_f_B.transpose(
(0, 2, 1)) + 2 * np.eye(d).reshape((1, d, d))
nabla_B = 2. / n * np.sum(nabla_f_B * PWP_fast(Y, W).reshape((n, 1, 1)),
axis=0)
#add ridge
nabla_B += 2 * alpha * B
#stack gradients together
grad = np.zeros((d + 1) * d, dtype=np.float64)
grad[:d] = nabla_b
grad[d:] = nabla_B.ravel()
return grad
def qform_2_ESVM(a, f_vals, X, X_grad, W, ind, n, alpha=0.0):
"""
Arguments:
a - np.array of shape (d+1,d),
a[0,:] - np.array of shape(d) - corresponds to coefficients via linear variables
a[1:,:] - np.array of shape (d,d) - to quadratic terms
"""
d = X_grad.shape[1]
b = a[:d]
B = a[d:].reshape((d, d))
x_cur = f_vals[:, ind] + X_grad @ b + qform_q(B + B.T, X_grad,
X) + 2 * np.trace(B)
return Spectral_var(x_cur, W) + alpha * np.sum(B**2)
def qform_2_LS(a, f_vals, X, X_grad, ind, n):
"""
Least squares evaluation for 2nd order polynomials as control variates;
Arguments:
a - np.array of shape (d+1,d),
a[0,:] - np.array of shape(d) - corresponds to coefficients via linear variables
a[1:,:] - np.array of shape (d,d) - to quadratic terms
Returns:
function value for index ind, scalar variable
"""
d = X.shape[1]
b = a[:d]
B = a[d:].reshape((d, d))
x_cur = f_vals[:, ind] + X_grad @ b + qform_q(B + B.T, X_grad,
X) + 2 * np.trace(B)
return np.mean(x_cur**2)
def qform_2_ZV(a, f_vals, X, X_grad, ind, n):
"""
Least squares evaluated for ZV-2 method
Arguments:
a - np.array of shape (d+1,d),
a[0,:] - np.array of shape(d) - corresponds to coefficients via linear variables
a[1:,:] - np.array of shape (d,d) - to quadratic terms
Returns:
function value for index ind, scalar variable
"""
d = X.shape[1]
b = a[:d]
B = a[d:].reshape((d, d))
x_cur = f_vals[:, ind] + X_grad @ b + qform_q(B + B.T, X_grad,
X) + 2 * np.trace(B)
return 1. / (n - 1) * np.dot(x_cur - np.mean(x_cur),
x_cur - np.mean(x_cur))
def max_2deg_penalty(a, f_vals, X, X_grad, ind, n, beta):
"""
Smooth maximum penalization for 2nd order polynomials as control variables;
Arguments:
a - np.array of shape (d+1,d),
a[0,:] - coefficients corresponding to 1st order terms;
a[1:,:] - np.array of shape (d,d) - coefficients for 2nd order terms
beta - smoothness penalization
Returns:
function value for index ind, scalar
"""
d = X.shape[1]
b = a[:d]
B = a[d:].reshape((d, d))
Y = f_vals[:, ind] + X_grad @ b + qform_q(B + B.T, X_grad,
X) + 2 * np.trace(B)
Y_max = np.max(np.abs(Y))
return Y_max + 1. / beta * np.log(
np.sum(np.exp(beta * (np.abs(Y) - Y_max))))
def grad_qform_2_LS(a, f_vals, X, X_grad, ind, n):
"""
Gradient for quadratic form in ZV-2 method
"""
d = X.shape[1]
b = a[:d]
B = a[d:].reshape((d, d))
Y = f_vals[:, ind] + X_grad @ b + qform_q(B + B.T, X_grad,
X) + 2 * np.trace(B)
#gradient w.r.t. b
nabla_b = 2. / n * X_grad.T @ Y
#gradient w.r.t B
nabla_f_B = np.matmul(X_grad.reshape((n, d, 1)), X.reshape((n, 1, d)))
nabla_f_B = nabla_f_B + nabla_f_B.transpose(
(0, 2, 1)) + 2 * np.eye(d).reshape((1, d, d))
nabla_B = 2. / n * np.sum(nabla_f_B * Y.reshape((n, 1, 1)), axis=0)
#stack gradients together
grad = np.zeros((d + 1) * d, dtype=np.float64)
grad[:d] = nabla_b
grad[d:] = nabla_B.ravel()
return grad
