wehub-resource-sync
232 lines
10 KiB
Python
232 lines
10 KiB
Python
# Copyright (c) 2022 PaddlePaddle Authors. All Rights Reserved.
|
|
#
|
|
# Licensed under the Apache License, Version 2.0 (the "License");
|
|
# you may not use this file except in compliance with the License.
|
|
# You may obtain a copy of the License at
|
|
#
|
|
# http://www.apache.org/licenses/LICENSE-2.0
|
|
#
|
|
# Unless required by applicable law or agreed to in writing, software
|
|
# distributed under the License is distributed on an "AS IS" BASIS,
|
|
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
|
|
# See the License for the specific language governing permissions and
|
|
# limitations under the License.
|
|
|
|
from __future__ import annotations
|
|
|
|
from typing import TYPE_CHECKING, Literal
|
|
|
|
import numpy as np
|
|
|
|
import paddle
|
|
|
|
from .line_search import strong_wolfe
|
|
from .utils import (
|
|
_value_and_gradient,
|
|
check_initial_inverse_hessian_estimate,
|
|
check_input_type,
|
|
)
|
|
|
|
if TYPE_CHECKING:
|
|
from collections.abc import Callable
|
|
|
|
from paddle import Tensor
|
|
|
|
|
|
def minimize_bfgs(
|
|
objective_func: Callable[[Tensor], Tensor],
|
|
initial_position: Tensor,
|
|
max_iters: int = 50,
|
|
tolerance_grad: float = 1e-7,
|
|
tolerance_change: float = 1e-9,
|
|
initial_inverse_hessian_estimate: Tensor | None = None,
|
|
line_search_fn: Literal['strong_wolfe'] = 'strong_wolfe',
|
|
max_line_search_iters: int = 50,
|
|
initial_step_length: float = 1.0,
|
|
dtype: Literal['float32', 'float64'] = 'float32',
|
|
name: str | None = None,
|
|
) -> tuple[bool, int, Tensor, Tensor, Tensor, Tensor]:
|
|
r"""
|
|
Minimizes a differentiable function `func` using the BFGS method.
|
|
The BFGS is a quasi-Newton method for solving an unconstrained optimization problem over a differentiable function.
|
|
Closely related is the Newton method for minimization. Consider the iterate update formula:
|
|
|
|
.. math::
|
|
x_{k+1} = x_{k} + H_k \nabla{f_k}
|
|
|
|
If :math:`H_k` is the inverse Hessian of :math:`f` at :math:`x_k`, then it's the Newton method.
|
|
If :math:`H_k` is symmetric and positive definite, used as an approximation of the inverse Hessian, then
|
|
it's a quasi-Newton. In practice, the approximated Hessians are obtained
|
|
by only using the gradients, over either whole or part of the search
|
|
history, the former is BFGS, the latter is L-BFGS.
|
|
|
|
Reference:
|
|
Jorge Nocedal, Stephen J. Wright, Numerical Optimization, Second Edition, 2006. pp140: Algorithm 6.1 (BFGS Method).
|
|
|
|
Args:
|
|
objective_func: the objective function to minimize. ``objective_func`` accepts a 1D Tensor and returns a scalar.
|
|
initial_position (Tensor): the starting point of the iterates, has the same shape with the input of ``objective_func`` .
|
|
max_iters (int, optional): the maximum number of minimization iterations. Default value: 50.
|
|
tolerance_grad (float, optional): terminates if the gradient norm is smaller than this. Currently gradient norm uses inf norm. Default value: 1e-7.
|
|
tolerance_change (float, optional): terminates if the change of function value/position/parameter between two iterations is smaller than this value. Default value: 1e-9.
|
|
initial_inverse_hessian_estimate (Tensor, optional): the initial inverse hessian approximation at initial_position. It must be symmetric and positive definite. If not given, will use an identity matrix of order N, which is size of ``initial_position`` . Default value: None.
|
|
line_search_fn (str, optional): indicate which line search method to use, only support 'strong wolfe' right now. May support 'Hager Zhang' in the future. Default value: 'strong wolfe'.
|
|
max_line_search_iters (int, optional): the maximum number of line search iterations. Default value: 50.
|
|
initial_step_length (float, optional): step length used in first iteration of line search. different initial_step_length may cause different optimal result. For methods like Newton and quasi-Newton the initial trial step length should always be 1.0. Default value: 1.0.
|
|
dtype ('float32' | 'float64', optional): data type used in the algorithm, the data type of the input parameter must be consistent with the dtype. Default value: 'float32'.
|
|
name (str, optional): Name for the operation. For more information, please refer to :ref:`api_guide_Name`. Default value: None.
|
|
|
|
Returns:
|
|
output(tuple):
|
|
|
|
- is_converge (bool): Indicates whether found the minimum within tolerance.
|
|
- num_func_calls (int): number of objective function called.
|
|
- position (Tensor): the position of the last iteration. If the search converged, this value is the argmin of the objective function regarding to the initial position.
|
|
- objective_value (Tensor): objective function value at the `position`.
|
|
- objective_gradient (Tensor): objective function gradient at the `position`.
|
|
- inverse_hessian_estimate (Tensor): the estimate of inverse hessian at the `position`.
|
|
|
|
Examples:
|
|
.. code-block:: pycon
|
|
:name: code-example1
|
|
|
|
>>> # Example1: 1D Grid Parameters
|
|
>>> import paddle
|
|
>>> # Randomly simulate a batch of input data
|
|
>>> inputs = paddle.normal(shape=(100, 1))
|
|
>>> labels = inputs * 2.0
|
|
>>> # define the loss function
|
|
>>> def loss(w):
|
|
... y = w * inputs
|
|
... return paddle.nn.functional.square_error_cost(y, labels).mean()
|
|
>>> # Initialize weight parameters
|
|
>>> w = paddle.normal(shape=(1,))
|
|
>>> # Call the bfgs method to solve the weight that makes the loss the smallest, and update the parameters
|
|
>>> for epoch in range(0, 10):
|
|
... # Call the bfgs method to optimize the loss, note that the third parameter returned represents the weight
|
|
... w_update = paddle.incubate.optimizer.functional.minimize_bfgs(loss, w)[2]
|
|
... # Use paddle.assign to update parameters in place
|
|
... paddle.assign(w_update, w)
|
|
|
|
.. code-block:: pycon
|
|
:name: code-example2
|
|
|
|
>>> # Example2: Multidimensional Grid Parameters
|
|
>>> import paddle
|
|
>>> def flatten(x):
|
|
... return x.flatten()
|
|
>>> def unflatten(x):
|
|
... return x.reshape((2, 2))
|
|
>>> # Assume the network parameters are more than one dimension
|
|
>>> def net(x):
|
|
... assert len(x.shape) > 1
|
|
... return x.square().mean()
|
|
>>> # function to be optimized
|
|
>>> def bfgs_f(flatten_x):
|
|
... return net(unflatten(flatten_x))
|
|
>>> x = paddle.rand([2, 2])
|
|
>>> for i in range(0, 10):
|
|
... # Flatten x before using minimize_bfgs
|
|
... x_update = paddle.incubate.optimizer.functional.minimize_bfgs(bfgs_f, flatten(x))[2]
|
|
... # unflatten x_update, then update parameters
|
|
... paddle.assign(unflatten(x_update), x)
|
|
"""
|
|
|
|
if dtype not in ['float32', 'float64']:
|
|
raise ValueError(
|
|
f"The dtype must be 'float32' or 'float64', but the specified is {dtype}."
|
|
)
|
|
|
|
op_name = 'minimize_bfgs'
|
|
check_input_type(initial_position, 'initial_position', op_name)
|
|
|
|
I = paddle.eye(initial_position.shape[0], dtype=dtype)
|
|
if initial_inverse_hessian_estimate is None:
|
|
initial_inverse_hessian_estimate = I
|
|
else:
|
|
check_input_type(
|
|
initial_inverse_hessian_estimate,
|
|
'initial_inverse_hessian_estimate',
|
|
op_name,
|
|
)
|
|
check_initial_inverse_hessian_estimate(initial_inverse_hessian_estimate)
|
|
|
|
Hk = paddle.assign(initial_inverse_hessian_estimate)
|
|
# use detach and assign to create new tensor rather than =, or xk will share memory and grad with initial_position
|
|
xk = paddle.assign(initial_position.detach())
|
|
|
|
value, g1 = _value_and_gradient(objective_func, xk)
|
|
num_func_calls = paddle.full(shape=[1], fill_value=1, dtype='int64')
|
|
|
|
# when the dim of x is 1000, it needs more than 30 iters to get all element converge to minimum.
|
|
k = paddle.full(shape=[1], fill_value=0, dtype='int64')
|
|
done = paddle.full(shape=[1], fill_value=False, dtype='bool')
|
|
is_converge = paddle.full(shape=[1], fill_value=False, dtype='bool')
|
|
|
|
def cond(k, done, is_converge, num_func_calls, xk, value, g1, Hk):
|
|
return (k < max_iters) & ~done
|
|
|
|
def body(k, done, is_converge, num_func_calls, xk, value, g1, Hk):
|
|
# -------------- compute pk -------------- #
|
|
pk = -paddle.matmul(Hk, g1)
|
|
|
|
# -------------- compute alpha by line search -------------- #
|
|
if line_search_fn == 'strong_wolfe':
|
|
alpha, value, g2, ls_func_calls = strong_wolfe(
|
|
f=objective_func,
|
|
xk=xk,
|
|
pk=pk,
|
|
max_iters=max_line_search_iters,
|
|
initial_step_length=initial_step_length,
|
|
dtype=dtype,
|
|
)
|
|
else:
|
|
raise NotImplementedError(
|
|
f"Currently only support line_search_fn = 'strong_wolfe', but the specified is '{line_search_fn}'"
|
|
)
|
|
num_func_calls += ls_func_calls
|
|
|
|
# -------------- update Hk -------------- #
|
|
sk = alpha * pk
|
|
yk = g2 - g1
|
|
|
|
xk = xk + sk
|
|
g1 = g2
|
|
|
|
sk = paddle.unsqueeze(sk, 0)
|
|
yk = paddle.unsqueeze(yk, 0)
|
|
|
|
rhok_inv = paddle.dot(yk, sk)
|
|
rhok = paddle.static.nn.cond(
|
|
rhok_inv == 0.0,
|
|
lambda: paddle.full(shape=[1], fill_value=1000.0, dtype=dtype),
|
|
lambda: 1.0 / rhok_inv,
|
|
)
|
|
|
|
Vk_transpose = I - rhok * sk * yk.t()
|
|
Vk = I - rhok * yk * sk.t()
|
|
Hk = (
|
|
paddle.matmul(paddle.matmul(Vk_transpose, Hk), Vk)
|
|
+ rhok * sk * sk.t()
|
|
)
|
|
|
|
k += 1
|
|
|
|
# -------------- check convergence -------------- #
|
|
gnorm = paddle.linalg.norm(g1, p=np.inf)
|
|
pk_norm = paddle.linalg.norm(pk, p=np.inf)
|
|
paddle.assign(
|
|
done | (gnorm < tolerance_grad) | (pk_norm < tolerance_change), done
|
|
)
|
|
paddle.assign(done, is_converge)
|
|
# when alpha=0, there is no chance to get xk change.
|
|
paddle.assign(done | (alpha == 0.0), done)
|
|
return [k, done, is_converge, num_func_calls, xk, value, g1, Hk]
|
|
|
|
paddle.static.nn.while_loop(
|
|
cond=cond,
|
|
body=body,
|
|
loop_vars=[k, done, is_converge, num_func_calls, xk, value, g1, Hk],
|
|
)
|
|
return is_converge, num_func_calls, xk, value, g1, Hk
|