# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public
# License version 3 as published by the Free Software Foundation.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this program; if not, write to the Free Software Foundation,
# Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
# Contributors:
# INITIAL AUTHORS - initial API and implementation and/or initial
# documentation
# :author: Francois Gallard
# OTHER AUTHORS - MACROSCOPIC CHANGES
r"""Approximation of the Hessian matrix from an optimization history.
Notations:
- :math:`f`: the function of interest for which to approximate the Hessian matrix,
- :math:`y`: the output value of :math:`f`,
- :math:`x\in\mathbb{R}^d`: the :math:`d` input variables of :math:`f`,
- :math:`k`: the :math:`k`-th iteration of the optimization history,
- :math:`K`: the iteration of the optimization history
at which to approximate the Hessian matrix,
- :math:`x_k`: the input value at iteration :math:`k`,
- :math:`\Delta x_k=x_{k+1}-x_k`: the variation of :math:`x`
from iteration :math:`k` to iteration :math:`k+1`,
- :math:`y_k`: the output value at iteration :math:`k`,
- :math:`\Delta y_k=y_{k+1}-y_k`: the variation of the function output
from iteration :math:`k` to iteration :math:`k+1`,
- :math:`g_k`: the gradient of :math:`f` at :math:`x_k`,
- :math:`\Delta g_k=g_{k+1}-g_k`: the variation of the gradient
from iteration :math:`k` to iteration :math:`k+1`,
- :math:`B_k`: the approximation of the Hessian of :math:`f` at :math:`x_k`,
- :math:`H_k`: the inverse of :math:`B_k`.
"""
from __future__ import annotations
import logging
from typing import TYPE_CHECKING
from docstring_inheritance import GoogleDocstringInheritanceMeta
from numpy import array
from numpy import atleast_2d
from numpy import concatenate
from numpy import cumsum
from numpy import diag as np_diag
from numpy import eye
from numpy import inf
from numpy import ndarray
from numpy import newaxis
from numpy import sqrt
from numpy import tile
from numpy import trace
from numpy import zeros
from numpy.linalg import LinAlgError
from numpy.linalg import cholesky
from numpy.linalg import inv
from numpy.linalg import norm
from scipy.optimize import leastsq
if TYPE_CHECKING:
from collections.abc import Generator
from gemseo.algos.database import Database
from gemseo.algos.design_space import DesignSpace
LOGGER = logging.getLogger(__name__)
[docs]
class HessianApproximation(metaclass=GoogleDocstringInheritanceMeta):
r"""Approximation of the Hessian matrix from an optimization history."""
history: Database
"""The optimization history containing input values, output values and Jacobian
values."""
x_ref: ndarray | None
"""The value :math:`x_K` of the input variables :math:`x` at the iteration :math:`K`
of the optimization history; this is the point at which the Hessian matrix and its
inverse are approximated."""
fgrad_ref: ndarray | None
"""The value :math:`g_K` of the gradient function :math:`g` of :math:`f` at
:math:`x_K`."""
f_ref: ndarray | None
"""The value :math:`y_K` of the output of :math:`f` at :math:`x_K`."""
b_mat_history: list[ndarray]
r"""The history :math:`B_0,B_1,\ldots,B_K` of the approximations of the Hessian
matrix :math:`B`."""
h_mat_history: list[ndarray]
r"""The history :math:`H_0,H_1,\ldots,H_K` of the approximations of the inverse
Hessian matrix :math:`H`."""
def __init__(
self,
history: Database,
) -> None:
"""
Args:
history: The optimization history
containing input values, output values and Jacobian values.
""" # noqa: D205, D212, D415
self.history = history
self.x_ref = None
self.fgrad_ref = None
self.f_ref = None
self.b_mat_history = []
self.h_mat_history = []
[docs]
def get_x_grad_history(
self,
funcname: str,
first_iter: int = 0,
last_iter: int | None = None,
at_most_niter: int | None = None,
func_index: int | None = None,
normalize_design_space: bool = False,
design_space: DesignSpace | None = None,
) -> tuple[ndarray, ndarray, int, int]:
"""Return the histories of the inputs and gradient.
Args:
funcname: The name of the function for which to get the gradient.
first_iter: The first iteration of the history to be considered.
last_iter: The last iteration of the history to be considered.
If ``None``, consider all the iterations.
at_most_niter: The maximum number of iterations to be considered.
If ``None``, consider all the iterations.
func_index: The index of the output of interest
to be defined if the function has a multidimensional output.
If ``None`` and if the output is multidimensional, an error is raised.
normalize_design_space: Whether to scale the input values between 0 and 1
to work in a normalized input space.
design_space: The input space used to scale the input values
if ``normalize_design_space`` is ``True``.
Returns:
* The history of the input variables.
* The history of the gradient.
* The length of the history.
* The dimension of the input space.
Raises:
ValueError: When either
the gradient history contains a single element,
``func_index`` is ``None`` while the function output is a vector,
``func_index`` is not an output index,
the shape of the history of the input variables
is not consistent with the shape of the history of the gradient
or the optimization history size is insufficient.
"""
grad_hist, x_hist = self.history.get_gradient_history(
funcname, with_x_vect=True
)
if normalize_design_space:
(
x_hist,
grad_hist,
) = self._normalize_x_g(x_hist, grad_hist, design_space)
grad_hist_length = len(grad_hist)
if grad_hist_length < 2:
msg = (
f"Cannot build approximation for function: {funcname} "
f"because its gradient history is too small: {grad_hist_length}."
)
raise ValueError(msg)
if grad_hist.ndim == 1:
grad_hist = grad_hist[:, newaxis]
x_hist_shape = x_hist.shape
grad_hist_shape = grad_hist.shape
grad_hist_shape_from_grad_hist = (grad_hist.shape[0], grad_hist.shape[-1])
if x_hist_shape != grad_hist_shape_from_grad_hist:
msg = (
"The shape of the design variable history "
f"(n_iter,n_x)=({x_hist_shape}) "
"and the shape of the gradient history "
f"(n_iter,[n_y,]n_x)=({grad_hist_shape}) "
"are not consistent."
)
raise ValueError(msg)
# Function is a vector, Jacobian is a 2D matrix
if grad_hist.ndim == 3:
if grad_hist.shape[1] == 1:
func_index = 0
else:
if func_index is None:
msg = (
f"Function {funcname} has a vector output, "
"the function index of the output must be specified."
)
raise ValueError(msg)
output_size = grad_hist.shape[1]
if not 0 <= func_index < output_size:
msg = (
f"Function {funcname} has a vector output "
f"of size {output_size}, "
f"function index {func_index} is out of range."
)
raise ValueError(msg)
grad_hist = grad_hist[:, func_index, :]
if not last_iter:
last_iter = len(x_hist)
x_hist = x_hist[first_iter:last_iter, :]
grad_hist = grad_hist[first_iter:last_iter, :]
n_iterations = len(x_hist)
if at_most_niter and 0 < at_most_niter < n_iterations:
x_hist = x_hist[n_iterations - at_most_niter :, :]
grad_hist = grad_hist[n_iterations - at_most_niter :, :]
n_iterations, input_dimension = x_hist.shape
if n_iterations < 2:
msg = (
f"The number of iterations ({n_iterations}) "
"must greater than or equal to 2."
)
raise ValueError(msg)
self.x_ref = x_hist[-1]
self.fgrad_ref = grad_hist[-1]
self.f_ref = array(self.history.get_function_history(funcname))[:last_iter][-1]
return x_hist, grad_hist, n_iterations, input_dimension
@staticmethod
def _normalize_x_g(
x_hist: ndarray,
x_grad_hist: ndarray,
design_space: DesignSpace,
) -> tuple[ndarray, ndarray]:
"""Scale the design variables between 0 and 1 in the histories.
Args:
x_hist: The history of the input variables.
x_grad_hist: The history of the gradient.
design_space: The input space used to scale the input variables.
Returns:
* The history of the scaled input variables.
* The history of the gradient.
Raises:
ValueError: When the input space is ``None``.
"""
if design_space is None:
msg = (
"Design space must be provided "
"when using a normalize_design_space option."
)
raise ValueError(msg)
scaled_x_hist, scaled_grad_hist = [], []
for x_value, grad_value in zip(x_hist, x_grad_hist):
scaled_x_hist.append(design_space.normalize_vect(x_value))
scaled_grad_hist.append(design_space.normalize_grad(grad_value))
return array(scaled_x_hist), array(scaled_grad_hist)
[docs]
@staticmethod
def get_s_k_y_k(
x_hist: ndarray,
x_grad_hist: ndarray,
iteration: int,
) -> tuple[ndarray, ndarray]:
r"""Compute the variation of the input variables and gradient from an iteration.
The variations from the iteration :math:`k` are defined by:
- :math:`\Delta x_k = x_{k+1}-x_k` for the input variables,
- :math:`\Delta g_k = g_{k+1} - g_k` for the gradient.
Args:
x_hist: The history of the input variables.
x_grad_hist: The history of the gradient.
iteration: The optimization iteration at which to compute the variations.
Returns:
* The difference between the input variables at iteration ``iteration+1``
and the input variables at iteration ``iteration``.
* The difference between the gradient at iteration ``iteration+1``
and the gradient at iteration ``iteration``.
Raises:
ValueError: When the iteration is not stored in the database.
"""
n_iterations = x_grad_hist.shape[0]
if iteration >= n_iterations:
msg = (
f"Iteration {iteration} is higher than the number of gradients "
f"in the database: {n_iterations}."
)
raise ValueError(msg)
input_diff = atleast_2d(x_hist[iteration + 1] - x_hist[iteration]).T
grad_diff = atleast_2d(x_grad_hist[iteration + 1] - x_grad_hist[iteration]).T
return input_diff, grad_diff
[docs]
@staticmethod
def iterate_s_k_y_k(
x_hist: ndarray,
x_grad_hist: ndarray,
) -> Generator[tuple[ndarray, ndarray]]:
r"""Compute the variations of the input variables and gradient.
The variations from the iteration :math:`k` are defined by:
- :math:`\Delta x_k = x_{k+1}-x_k` for the input variables,
- :math:`\Delta g_k = g_{k+1} - g_k` for the gradient.
Args:
x_hist: The history of the input variables.
x_grad_hist: The history of the gradient.
Returns:
* The difference between the input variables at iteration ``iteration``
and the input variables at iteration ``iteration+1``.
* The difference between the gradient at iteration ``iteration``
and the gradient at iteration ``iteration+1``.
"""
for iteration in range(len(x_hist) - 1):
input_diff, grad_diff = HessianApproximation.get_s_k_y_k(
x_hist, x_grad_hist, iteration
)
yield input_diff, grad_diff
[docs]
def build_approximation(
self,
funcname: str,
save_diag: bool = False,
first_iter: int = 0,
last_iter: int | None = -1,
b_mat0: ndarray | None = None,
at_most_niter: int | None = None,
return_x_grad: bool = False,
func_index: int | None = None,
save_matrix: bool = False,
scaling: bool = False,
normalize_design_space: bool = False,
design_space: DesignSpace | None = None,
) -> tuple[ndarray, ndarray, ndarray | None, ndarray | None]:
# pylint: disable=W0221
"""Compute :math:`B`, the approximation of the Hessian matrix.
Args:
funcname: The name of the function
for which to approximate the Hessian matrix.
save_diag: Whether to return the approximations of the Hessian's diagonal.
first_iter: The first iteration of the history to be considered.
last_iter: The last iteration of the history to be considered.
If ``None``, consider all the iterations.
b_mat0: The initial approximation of the Hessian matrix.
at_most_niter: The maximum number of iterations to be considered.
If ``None``, consider all the iterations.
return_x_grad: Whether to return the input variables and gradient
at the last iteration.
func_index: The index of the output of interest
to be defined if the function has a multidimensional output.
If ``None`` and if the output is multidimensional, an error is raised.
save_matrix: Whether to store the approximations of the Hessian
in :attr:`.HessianApproximation.b_mat_history`.
scaling: do scaling step
normalize_design_space: Whether to scale the input values between 0 and 1
to work in a normalized input space.
design_space: The input space used to scale the input values
if ``normalize_design_space`` is ``True``.
Returns:
* :math:`B`, the approximation of the Hessian matrix.
* The diagonal of :math:`B`.
* The history of the input variables if ``return_x_grad`` is ``True``.
* The history of the gradient if ``return_x_grad`` is ``True``.
"""
x_hist, grad_hist, _, _ = self.get_x_grad_history(
funcname,
first_iter,
last_iter,
at_most_niter,
func_index,
normalize_design_space,
design_space,
)
if b_mat0 is None:
last_n_grad = grad_hist.shape[0] - 2
input_diff, grad_diff = self.get_s_k_y_k(x_hist, grad_hist, last_n_grad)
alpha = (grad_diff.T @ input_diff) / (grad_diff.T @ grad_diff)
hessian = (1.0 / alpha) * eye(grad_hist.shape[1])
elif b_mat0.size == 0:
hessian = zeros((x_hist.shape[1],) * 2)
else:
hessian = b_mat0
hessian_diagonal = []
for input_diff, grad_diff in self.iterate_s_k_y_k(x_hist, grad_hist):
self.iterate_approximation(hessian, input_diff, grad_diff, scaling=scaling)
if save_diag:
hessian_diagonal.append(np_diag(hessian).copy())
if save_matrix:
self.b_mat_history.append(hessian.copy())
if return_x_grad:
return hessian, hessian_diagonal, x_hist[-1, :], grad_hist[-1, :]
return hessian, hessian_diagonal, None, None
[docs]
@staticmethod
def compute_scaling(
hessk: ndarray,
hessk_dsk: ndarray,
dskt_hessk_dsk: ndarray,
dyk: ndarray,
dyt_dsk: ndarray,
) -> tuple[float, float]:
r"""Compute the scaling coefficients :math:`c_1` and :math:`c_2`.
- :math:`c_1=\frac{d-1}{\mathrm{Tr}(B_k)-\frac{\|B_k\Delta x_k\|_2^2}
{\Delta x_k^T B_k\Delta x_k}}`,
- :math:`c_2=\frac{\Delta g_k^T\Delta x_k}{\|\Delta g_k\|_2^2}`.
Args:
hessk: The approximation :math:`B_k` of the Hessian matrix
at iteration :math:`k`.
hessk_dsk: The product :math:`B_k\Delta x_k`.
dskt_hessk_dsk: The product :math:`\Delta x_k^T B_k\Delta x_k`.
dyk: The variation of the gradient :math:`\Delta g_k`.
dyt_dsk: The product
:math:`\Delta g_k^T\Delta x_k`.
Returns:
* coeff1: TODO
* coeff2: TODO
"""
coeff1 = (len(hessk_dsk) - 1) / (
trace(hessk) - norm(hessk_dsk) ** 2 / dskt_hessk_dsk
)
coeff2 = dyt_dsk / norm(dyk) ** 2
return coeff1, coeff2
[docs]
@staticmethod
def iterate_approximation(
hessk: ndarray,
dsk: ndarray,
dyk: ndarray,
scaling: bool = False,
) -> None:
r"""Update :math:`B` from iteration :math:`k` to iteration :math:`k+1`.
Based on an iteration of the BFGS algorithm:
:math:`B_{k+1} =
B_k
- c_1\frac{B_k\Delta x_k\Delta x_k^TB_k}{\Delta x_k^TB_k\Delta x_k}
+ c_2\frac{\Delta g_k\Delta g_k^T}{\Delta g_k^T\Delta x_k}`
where :math:`c_1=c_2=1` if ``scaling`` is ``False``, otherwise:
- :math:`c_1=\frac{d-1}{\mathrm{Tr}(B_k)-\frac{\|B_k\Delta x_k\|_2^2}
{\Delta x_k^T B_k\Delta x_k}}`,
- :math:`c_2=\frac{\Delta g_k^T\Delta x_k}{\|\Delta g_k\|_2^2}`.
.. note::
``hessk`` represents :math:`B_k` initially
before to be overwritten by :math:`B_{k+1}` when passed to this method.
.. seealso::
`BFGS algorithm.
<https://en.wikipedia.org/wiki/Broyden-Fletcher-Goldfarb-Shanno_algorithm>`_
Args:
hessk: The approximation :math:`B_k` of the Hessian matrix
at iteration :math:`k`.
dsk: The variation :math:`\Delta x_k` of the input variables.
dyk: The variation :math:`\Delta g_k` of the gradient.
scaling: Whether to use a scaling stage.
"""
dyt_dsk = dyk.T @ dsk
hessk_dsk = hessk @ dsk
dskt_hessk_dsk = dsk.T @ hessk_dsk
# Build the next approximation:
b_first_term = hessk - (hessk_dsk / dskt_hessk_dsk) @ hessk_dsk.T
b_second_term = (dyk @ dyk.T) / dyt_dsk
if not scaling:
hessk[:, :] = b_first_term + b_second_term
else:
c_1, c_2 = HessianApproximation.compute_scaling(
hessk, hessk_dsk, dskt_hessk_dsk, dyk, dyt_dsk
)
hessk[:, :] = c_1 * b_first_term + c_2 * b_second_term
[docs]
def build_inverse_approximation(
self,
funcname: str,
save_diag: int = False,
first_iter: int = 0,
last_iter: int | None = -1,
h_mat0: ndarray | None = None,
at_most_niter: int | None = None,
return_x_grad: bool = False,
func_index: int | None = None,
save_matrix: bool = False,
factorize: bool = False,
scaling: bool = False,
angle_tol: float = 1e-5,
step_tol: float = 1e10,
normalize_design_space: bool = False,
design_space: DesignSpace | None = None,
) -> tuple[ndarray, ndarray, ndarray | None, ndarray | None]:
r"""Compute :math:`H`, the approximation of the inverse of the Hessian matrix.
Args:
funcname: The name of the function
for which to approximate the inverse of the Hessian matrix.
save_diag: Whether to return the list of diagonal approximations.
first_iter: The first iteration of the history to be considered.
last_iter: The last iteration of the history to be considered.
If ``None``, consider all the iterations.
h_mat0: The initial approximation of the inverse of the Hessian matrix.
If ``None``,
use :math:`H_0=\frac{\Delta g_k^T\Delta x_k}
{\Delta g_k^T\Delta g_k}I_d`.
at_most_niter: The maximum number of iterations to be considered.
If ``None``, consider all the iterations.
return_x_grad: Whether to return the input variables and gradient
at the last iteration.
func_index: The output index of the function
to be provided if the function output is a vector.
save_matrix: Whether to store the approximations of the inverse Hessian
in :attr:`.HessianApproximation.h_mat_history`.
factorize: Whether to factorize the approximations of the Hessian matrix
and its inverse, as :math:`A=A_{1/2}A_{1/2}^T` for a matrix :math:`A`.
scaling: do scaling step
angle_tol: The significativity level for
:math:`\Delta g_k^T\Delta x_k`.
step_tol: The significativity level for
:math:`\|\Delta g_k\|_{\infty}`.
normalize_design_space: Whether to scale the input values between 0 and 1
to work in a normalized input space.
design_space: The input space used to scale the input values
if ``normalize_design_space`` is ``True``.
Returns:
* :math:`H`, the approximation of the inverse of the Hessian matrix.
* The diagonal of :math:`H`.
* The history of the input variables if ``return_x_grad`` is ``True``.
* The history of the gradient if ``return_x_grad`` is ``True``.
* The matrix :math:`H_{1/2}` such that :math:`H=H_{1/2}H_{1/2}^T`
if ``factorize`` is ``True``.
* :math:`B`, the approximation of the Hessian matrix.
* A matrix :math:`B_{1/2}` such that :math:`B=B_{1/2}B_{1/2}^T`
if ``factorize`` is ``True``.
Raises:
LinAlgError: When either
the inversion of :math:`H` fails
or the Cholesky decomposition of :math:`H` or :math:`B` fails.
"""
x_hist, grad_hist, _, _ = self.get_x_grad_history(
funcname,
first_iter,
last_iter,
at_most_niter,
func_index,
normalize_design_space,
design_space,
)
h_factor = None # to become a matrix G such that H = G*G', optionally
b_factor = None # to become the inverse of the matrix G
if h_mat0 is None:
last_n_grad = grad_hist.shape[0] - 2
s_k, y_k = self.get_s_k_y_k(x_hist, grad_hist, last_n_grad)
alpha = (y_k.T @ s_k) / (y_k.T @ y_k)
n_x = len(grad_hist[0])
h_mat = alpha * eye(n_x)
b_mat = 1.0 / alpha * eye(n_x)
if factorize:
h_factor = sqrt(alpha) * eye(n_x)
b_factor = eye(n_x) / sqrt(alpha)
elif len(h_mat0) == 0:
n_x = len(x_hist[0])
h_mat = zeros((n_x, n_x))
b_mat = zeros((n_x, n_x))
if factorize:
h_factor = zeros((n_x, n_x))
b_factor = zeros((n_x, n_x))
else:
h_mat = h_mat0
try:
b_mat = inv(h_mat)
except LinAlgError as error:
msg = "The inversion of h_mat failed."
raise LinAlgError(msg) from error
if factorize or scaling:
try:
h_factor = cholesky(h_mat)
b_factor = cholesky(b_mat).T
except LinAlgError as error:
msg = "The Cholesky decomposition of h_factor or b_factor failed."
raise LinAlgError(msg) from error
diag = []
count = 0
for s_k, y_k in self.iterate_s_k_y_k(x_hist, grad_hist):
if (s_k.T @ y_k) > angle_tol and norm(y_k, inf) < step_tol:
count += 1
self.iterate_inverse_approximation(
h_mat,
s_k,
y_k,
h_factor,
b_mat,
b_factor,
factorize=factorize,
scaling=scaling,
)
if save_diag:
diag.append(np_diag(h_mat).copy())
if save_matrix:
self.h_mat_history.append(h_mat.copy())
if return_x_grad:
return h_mat, diag, x_hist[-1, :], grad_hist[-1, :], None, None, None
return h_mat, diag, None, None, h_factor, b_mat, b_factor
[docs]
@staticmethod
def compute_corrections(
x_hist: ndarray,
x_grad_hist: ndarray,
) -> tuple[ndarray, ndarray]:
"""Compute the successive variations of both input variables and gradient.
These variations are called *corrections*.
Args:
x_hist: The history of the input variables.
x_grad_hist: The history of the gradient.
Returns:
* The successive variations of the input variables.
* The successive variations of the gradient.
"""
n_iter = len(x_hist)
x_corr = x_hist[1:n_iter].T - x_hist[: n_iter - 1].T
grad_corr = x_grad_hist[1:n_iter].T - x_grad_hist[: n_iter - 1].T
return x_corr, grad_corr
[docs]
@staticmethod
def rebuild_history(
x_corr: ndarray,
x_0: ndarray,
grad_corr: ndarray,
g_0: ndarray,
) -> tuple[ndarray, ndarray]:
"""Compute the history from the corrections of input variables and gradient.
A *correction* is the variation of a quantity between two successive iterations.
Args:
x_corr: The corrections of the input variables.
x_0: The initial values of the input variables.
grad_corr: The corrections of the gradient.
g_0: The initial value of the gradient.
Returns:
* The history of the input variables.
* The history of the gradient.
"""
# Rebuild the argument history:
x_hist = tile(x_0, (x_corr.shape[1], 1)) + cumsum(x_corr.T, axis=0)
x_hist = concatenate((atleast_2d(x_0), x_hist), axis=0)
# Rebuild the gradient history
x_grad_hist = tile(g_0, (grad_corr.shape[1], 1)) + cumsum(grad_corr.T, axis=0)
x_grad_hist = concatenate((atleast_2d(g_0), x_grad_hist), axis=0)
return x_hist, x_grad_hist
[docs]
@staticmethod
def iterate_inverse_approximation(
h_mat: ndarray,
s_k: ndarray,
y_k: ndarray,
h_factor: ndarray | None = None,
b_mat: ndarray | None = None,
b_factor: ndarray | None = None,
factorize: bool = False,
scaling: bool = False,
) -> None:
r"""Update :math:`H` and :math:`B` from step :math:`k` to step :math:`k+1`.
Use an iteration of the BFGS algorithm:
:math:`B_{k+1} =
B_k
- c_1\frac{B_k\Delta x_k\Delta x_k^TB_k}{\Delta x_k^TB_k\Delta x_k}
+ c_2\frac{\Delta g_k\Delta g_k^T}{\Delta g_k^T\Delta x_k}`
and
:math:`H_{k+1}=c_1^{-1}\Pi_{k+1}H_k\Pi_{k+1}^T
+c_2^{-1}\frac{\Delta x_k\Delta x_k^T}{\Delta g_k^T\Delta x_k}`
where:
:math:`\Pi_{k+1}=I_d-\frac{\Delta x_k\Delta g_k^T}
{\Delta g_k^T\Delta x_k}`
and where :math:`c_1=c_2=1` if ``scaling`` is ``False``, otherwise:
- :math:`c_1=\frac{d-1}{\mathrm{Tr}(B_k)-\frac{\|B_k\Delta x_k\|_2^2}
{\Delta x_k^T B_k\Delta x_k}}`,
- :math:`c_2=\frac{\Delta g_k^T\Delta x_k}{\|\Delta g_k\|_2^2}`.
.. note::
``h_mat`` and ``b_mat`` represent :math:`H_k` and :math:`B_k` initially
before to be overwritten by :math:`H_{k+1}` and :math:`B_{k+1}`
when passed to this method.
.. seealso::
`BFGS algorithm.
<https://en.wikipedia.org/wiki/Broyden-Fletcher-Goldfarb-Shanno_algorithm>`_
Args:
h_mat: The approximation :math:`H_k` of the inverse of the Hessian matrix
at iteration :math:`k`.
s_k: The variation :math:`\Delta x_k` of the input variables.
y_k: The variation :math:`\Delta g_k` of the gradient.
h_factor: The square root of the :math:`H_k` at iteration :math:`k`.
b_mat: The approximation :math:`B_k` of the Hessian matrix
at iteration :math:`k` if ``factorize`` is ``True``.
b_factor: The square root of the :math:`B_k` at iteration :math:`k`
if ``factorize`` is ``True``.
factorize: Whether to update the approximations of the Hessian matrix
and its inverse, as :math:`A=A_{1/2}A_{1/2}^T` for a matrix :math:`A`.
scaling: do scaling step
"""
# Compute the two terms of the non-scaled updated matrix:
yts = y_k.T @ s_k
proj = eye(len(s_k)) - (s_k / yts) @ y_k.T
h_first_term = proj @ (h_mat @ proj.T)
h_second_term = (s_k / yts) @ s_k.T
b_s = b_mat @ s_k
st_b_s = s_k.T @ b_s
# Compute the scaling coefficients:
if scaling:
coeff1, coeff2 = HessianApproximation.compute_scaling(
b_mat, b_s, st_b_s, y_k, yts
)
else:
coeff1, coeff2 = 1.0, 1.0
# Update the inverse approximation H and, optionally, the factor G:
h_mat[:, :] = h_first_term / coeff1 + h_second_term / coeff2
if factorize:
sst_b = s_k @ b_s.T
left = proj / sqrt(coeff1) + sst_b / sqrt(coeff2 * yts * st_b_s)
h_factor[:, :] = left @ h_factor
# b_factor[:, :] = dot(eye(len(s_k)) - sstB.T / stBs / sqrt(coeff1)
# + dot(y_k, s_k.T)
# / sqrt(coeff2 * stBs * yts),
# b_factor)
right = sqrt(coeff1) * (eye(len(s_k)) - sst_b / st_b_s)
right += sqrt(coeff2) * (s_k @ y_k.T) / sqrt(st_b_s * yts)
b_factor[:, :] = b_factor @ right
# Update the Hessian approximation:
b_first_term = b_mat - (b_s / st_b_s) @ b_s.T
b_second_term = (y_k @ y_k.T) / yts
b_mat[:, :] = coeff1 * b_first_term + coeff2 * b_second_term
# b_mat[:, :] = multi_dot((proj.T, b_mat, proj)) \
# + dot(y_k, y_k.T) / yts
[docs]
class BFGSApprox(HessianApproximation):
"""Hessian matrix approximation with the BFGS algorithm."""
[docs]
@staticmethod
def iterate_s_k_y_k( # noqa:D102
x_hist: ndarray,
x_grad_hist: ndarray,
) -> Generator[tuple[ndarray, ndarray]]:
for iteration in range(len(x_hist) - 1):
input_diff, grad_diff = BFGSApprox.get_s_k_y_k(
x_hist, x_grad_hist, iteration
)
# All pairs curvatures shall be positive
# if dot(s_k.T, y_k) > 0.:
if (input_diff.T @ grad_diff) > 1e-16 * (grad_diff.T @ grad_diff):
yield input_diff, grad_diff
[docs]
class SR1Approx(HessianApproximation):
r"""Hessian matrix approximation with the Symmetric Rank One (SR1) algorithm.
The approximation at iteration :math:`k+1` is:
.. math::
B_{k+1}=B_k +
\frac{(\Delta g_k-B_k\Delta x_k)(\Delta g_k-B_k\Delta x_k)^T}
{(\Delta g_k-B_k\Delta x_k)^T\Delta x_k}
This update from iteration :math:`k` to iteration :math:`k+1` is applied only if
:math:`|(\Delta g_k-B_k\Delta x_k)^T\Delta x_k|
\geq \varepsilon\|\Delta x_k\|\|\Delta g_k\|`
where :math:`\varepsilon` is a small number, e.g. :math:`10^{-8}`.
.. seealso::
`SR1 algorithm. <https://en.wikipedia.org/wiki/Symmetric_rank-one>`_
"""
EPSILON = 1e-8
[docs]
@staticmethod
def iterate_approximation( # noqa:D102
b_mat: ndarray,
s_k: ndarray,
y_k: ndarray,
scaling: bool = False,
) -> None:
residuals = y_k - b_mat @ s_k
denominator = residuals.T @ s_k
if abs(denominator) > SR1Approx.EPSILON * norm(s_k) * norm(residuals):
b_mat[:, :] = b_mat + (residuals / denominator) @ residuals.T
else:
LOGGER.debug(
"Denominator of SR1 update is too small, update skipped %s.",
denominator,
)
[docs]
class LSTSQApprox(HessianApproximation):
"""Least squares approximation of a Hessian matrix from an optimization history."""
[docs]
def build_approximation( # noqa:D102
self,
funcname: str,
save_diag: bool = False,
first_iter: int = 0,
last_iter: int | None = -1,
b_mat0: ndarray | None = None,
at_most_niter: int | None = None,
return_x_grad: bool = False,
scaling: bool = False,
func_index: int = -1,
normalize_design_space: bool = False,
design_space: DesignSpace | None = None,
) -> tuple[ndarray, ndarray, ndarray | None, ndarray | None]:
x_hist, grad_hist, n_iterations, input_dimension = self.get_x_grad_history(
funcname,
first_iter,
last_iter,
at_most_niter,
func_index=func_index,
normalize_design_space=normalize_design_space,
design_space=design_space,
)
assert len(grad_hist) == len(x_hist) # TODO: replace with an if/raise
sec_dim = max(input_dimension, n_iterations)
hessian_diagonal = []
def y_to_b(
y_vars: ndarray,
) -> ndarray:
"""Reshape the approximation from vector to matrix.
Args:
y_vars: The vector approximation.
Returns:
The square matrix version of the passed vector.
"""
y_mat = y_vars.reshape((input_dimension, input_dimension))
return y_mat + y_mat.T
def compute_error(
y_vars: ndarray,
) -> ndarray:
"""Create the least square function.
Args:
y_vars: The current approximation vector.
Returns:
The estimated error vector.
"""
hessian = y_to_b(y_vars)
err = zeros((input_dimension, sec_dim))
for item, x_current in enumerate(x_hist):
err[:, item] = hessian @ (x_current - self.x_ref) - grad_hist[item]
err = err.reshape(-1)
if n_iterations < input_dimension:
err += y_vars
return err
x_0 = zeros(input_dimension * input_dimension)
LOGGER.debug("Start least squares problem..")
x_opt, ier = leastsq(compute_error, x0=x_0) # , cov_x, infodict, mesg, ier
LOGGER.debug("End least squares, msg=%s", str(ier))
hessian = y_to_b(x_opt)
if return_x_grad:
return hessian, hessian_diagonal, x_hist[-1, :], grad_hist[-1, :]
return hessian, hessian_diagonal, None, None
