# Copyright 2021 Huawei Technologies Co., Ltd
#
# 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
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# ============================================================================
"""line search"""
from typing import NamedTuple
from mindspore import ops
from ... import nn
from ... import numpy as mnp
from ...common import dtype as mstype
from ...common import Tensor
from ..utils import _to_scalar, _to_tensor, grad
class _LineSearchResults(NamedTuple):
"""Results of line search results.
Args:
failed (bool): `True`` if the strong Wolfe criteria were satisfied
nit (int): number of iterations
nfev (int): number of functions evaluations
ngev (int): number of gradients evaluations
k (int): number of iterations
a_k (float): step size
f_k (float): final function value
g_k (Tensor): final gradient value
status (int): end status
"""
failed: bool
nit: int
nfev: int
ngev: int
k: int
a_k: float
f_k: float
g_k: Tensor
status: int
def _cubicmin(a, fa, fpa, b, fb, c, fc):
"""Finds the minimizer for a cubic polynomial that goes through the
points (a,fa), (b,fb), and (c,fc) with derivative at a of fpa.
"""
db = b - a
dc = c - a
denom = (db * dc) ** 2 * (db - dc)
d1 = mnp.zeros((2, 2))
d1[0, 0] = dc ** 2
d1[0, 1] = -db ** 2
d1[1, 0] = -dc ** 3
d1[1, 1] = db ** 3
d2 = mnp.zeros((2,))
d2[0] = fb - fa - fpa * db
d2[1] = fc - fa - fpa * dc
a2, b2 = mnp.dot(d1, d2) / denom
radical = b2 * b2 - 3. * a2 * fpa
xmin = a + (-b2 + mnp.sqrt(radical)) / (3. * a2)
return xmin
def _quadmin(a, fa, fpa, b, fb):
"""Finds the minimizer for a quadratic polynomial that goes through
the points (a,fa), (b,fb) with derivative at a of fpa.
"""
db = b - a
b2 = (fb - fa - fpa * db) / (db ** 2)
xmin = a - fpa / (2. * b2)
return xmin
def _zoom(fn, a_low, phi_low, dphi_low, a_high, phi_high, dphi_high, phi_0, g_0, dphi_0, c1, c2, is_run):
"""Implementation of zoom algorithm.
Algorithm 3.6 from Wright and Nocedal, 'Numerical Optimization', 1999, pg. 59-61.
Tries cubic, quadratic, and bisection methods of zooming.
"""
# Constant tensors which avoid loop unrolling
const_float_one = _to_tensor(1., dtype=a_low.dtype)
const_bool_false = _to_tensor(False)
const_int_zero = _to_tensor(0)
state = {
"done": const_bool_false,
"failed": const_bool_false,
"j": const_int_zero,
"a_low": a_low,
"phi_low": phi_low,
"dphi_low": dphi_low,
"a_high": a_high,
"phi_high": phi_high,
"dphi_high": dphi_high,
"a_rec": (a_low + a_high) / 2.,
"phi_rec": (phi_low + phi_high) / 2.,
"a_star": const_float_one,
"phi_star": phi_low,
"dphi_star": dphi_low,
"g_star": g_0,
"nfev": const_int_zero,
"ngev": const_int_zero,
}
if mnp.logical_not(is_run):
return state
delta1 = 0.2
delta2 = 0.1
maxiter = 10 # scipy: 10 jax: 30
while mnp.logical_not(state["done"]) and state["j"] < maxiter:
dalpha = state["a_high"] - state["a_low"]
a = mnp.minimum(state["a_low"], state["a_high"])
b = mnp.maximum(state["a_low"], state["a_high"])
cchk = delta1 * dalpha
qchk = delta2 * dalpha
a_j_cubic = _cubicmin(state["a_low"], state["phi_low"], state["dphi_low"], state["a_high"],
state["phi_high"], state["a_rec"], state["phi_rec"])
use_cubic = state["j"] > 0 and mnp.isfinite(a_j_cubic) and \
mnp.logical_and(a_j_cubic > a + cchk, a_j_cubic < b - cchk)
a_j_quad = _quadmin(state["a_low"], state["phi_low"], state["dphi_low"], state["a_high"],
state["phi_high"])
use_quad = mnp.logical_not(use_cubic) and mnp.isfinite(a_j_quad) and \
mnp.logical_and(a_j_quad > a + qchk, a_j_quad < b - qchk)
a_j_bisection = (state["a_low"] + state["a_high"]) / 2.0
use_bisection = mnp.logical_not(use_cubic) and mnp.logical_not(use_quad)
a_j = mnp.where(use_cubic, a_j_cubic, state["a_rec"])
a_j = mnp.where(use_quad, a_j_quad, a_j)
a_j = mnp.where(use_bisection, a_j_bisection, a_j)
phi_j, g_j, dphi_j = fn(a_j)
state["nfev"] += 1
state["ngev"] += 1
j_to_high = (phi_j > phi_0 + c1 * a_j * dphi_0) or (phi_j >= state["phi_low"])
state["a_rec"] = mnp.where(j_to_high, state["a_high"], state["a_rec"])
state["phi_rec"] = mnp.where(j_to_high, state["phi_high"], state["phi_rec"])
state["a_high"] = mnp.where(j_to_high, a_j, state["a_high"])
state["phi_high"] = mnp.where(j_to_high, phi_j, state["phi_high"])
state["dphi_high"] = mnp.where(j_to_high, dphi_j, state["dphi_high"])
j_to_star = mnp.logical_not(j_to_high) and mnp.abs(dphi_j) <= ops.negative(ops.add(c2, Tensor(0))) * dphi_0
state["done"] = j_to_star
state["a_star"] = mnp.where(j_to_star, a_j, state["a_star"])
state["phi_star"] = mnp.where(j_to_star, phi_j, state["phi_star"])
state["g_star"] = mnp.where(j_to_star, g_j, state["g_star"])
state["dphi_star"] = mnp.where(j_to_star, dphi_j, state["dphi_star"])
low_to_high = mnp.logical_not(j_to_high) and mnp.logical_not(j_to_star) and \
dphi_j * (state["a_high"] - state["a_low"]) >= 0.
state["a_rec"] = mnp.where(low_to_high, state["a_high"], state["a_rec"])
state["phi_rec"] = mnp.where(low_to_high, state["phi_high"], state["phi_rec"])
state["a_high"] = mnp.where(low_to_high, a_low, state["a_high"])
state["phi_high"] = mnp.where(low_to_high, phi_low, state["phi_high"])
state["dphi_high"] = mnp.where(low_to_high, dphi_low, state["dphi_high"])
j_to_low = mnp.logical_not(j_to_high) and mnp.logical_not(j_to_star)
state["a_rec"] = mnp.where(j_to_low, state["a_low"], state["a_rec"])
state["phi_rec"] = mnp.where(j_to_low, state["phi_low"], state["phi_rec"])
state["a_low"] = mnp.where(j_to_low, a_j, state["a_low"])
state["phi_low"] = mnp.where(j_to_low, phi_j, state["phi_low"])
state["dphi_low"] = mnp.where(j_to_low, dphi_j, state["dphi_low"])
state["j"] += 1
state["failed"] = state["j"] == maxiter
return state
class LineSearch(nn.Cell):
"""Line Search that satisfies strong Wolfe conditions."""
def __init__(self, func, jac):
"""Initialize LineSearch."""
super(LineSearch, self).__init__()
self.func = func
self.jac = jac
def construct(self, xk, pk, old_fval=None, old_old_fval=None, gfk=None, c1=1e-4, c2=0.9, maxiter=20):
def fval_and_grad(alpha):
xkk = xk + alpha * pk
fkk = self.func(xkk)
gkk = self.jac(xkk)
return fkk, gkk, mnp.dot(gkk, pk)
# Constant tensors which avoid loop unrolling
const_float_zero = _to_tensor(0., dtype=xk.dtype)
const_float_one = _to_tensor(1., dtype=xk.dtype)
const_bool_false = _to_tensor(False)
const_int_zero = _to_tensor(0)
const_int_one = _to_tensor(1)
if old_fval is None or gfk is None:
nfev, ngev = const_int_one, const_int_one
phi_0, g_0, dphi_0 = fval_and_grad(const_float_zero)
else:
nfev, ngev = const_int_zero, const_int_zero
phi_0, g_0 = old_fval, gfk
dphi_0 = mnp.dot(g_0, pk)
if old_old_fval is None:
start_value = const_float_one
else:
old_phi0 = old_old_fval
candidate_start_value = 1.01 * 2 * (phi_0 - old_phi0) / dphi_0
start_value = mnp.where(
mnp.isfinite(candidate_start_value),
mnp.minimum(candidate_start_value, const_float_one),
const_float_one
)
state = {
"done": const_bool_false,
"failed": const_bool_false,
"i": const_int_one,
"a_i": const_float_zero,
"phi_i": phi_0,
"dphi_i": dphi_0,
"nfev": nfev,
"ngev": ngev,
"a_star": const_float_zero,
"phi_star": phi_0,
"dphi_star": dphi_0,
"g_star": g_0,
}
while mnp.logical_not(state["done"]) and state["i"] <= maxiter:
a_i = mnp.where(state["i"] > 1, state["a_i"] * 2.0, start_value)
phi_i, g_i, dphi_i = fval_and_grad(a_i)
state["nfev"] += 1
state["ngev"] += 1
# Armijo condition
cond1 = (phi_i > phi_0 + c1 * a_i * dphi_0) or \
(phi_i >= state["phi_i"] and state["i"] > 1)
zoom1 = _zoom(fval_and_grad, state["a_i"], state["phi_i"], state["dphi_i"],
a_i, phi_i, dphi_i, phi_0, g_0, dphi_0, c1, c2, cond1)
state["nfev"] += zoom1["nfev"]
state["ngev"] += zoom1["ngev"]
state["done"] = cond1
state["failed"] = cond1 and zoom1["failed"]
state["a_star"] = mnp.where(cond1, zoom1["a_star"], state["a_star"])
state["phi_star"] = mnp.where(cond1, zoom1["phi_star"], state["phi_star"])
state["g_star"] = mnp.where(cond1, zoom1["g_star"], state["g_star"])
state["dphi_star"] = mnp.where(cond1, zoom1["dphi_star"], state["dphi_star"])
# Curvature condition
cond2 = mnp.logical_not(cond1) and mnp.abs(dphi_i) <= -c2 * dphi_0
state["done"] = state["done"] or cond2
state["a_star"] = mnp.where(cond2, a_i, state["a_star"])
state["phi_star"] = mnp.where(cond2, phi_i, state["phi_star"])
state["g_star"] = mnp.where(cond2, g_i, state["g_star"])
state["dphi_star"] = mnp.where(cond2, dphi_i, state["dphi_star"])
# Satisfying the strong wolf conditions
cond3 = mnp.logical_not(cond1) and mnp.logical_not(cond2) and dphi_i >= 0.
zoom2 = _zoom(fval_and_grad, a_i, phi_i, dphi_i, state["a_i"], state["phi_i"],
state["dphi_i"], phi_0, g_0, dphi_0, c1, c2, cond3)
state["nfev"] += zoom2["nfev"]
state["ngev"] += zoom2["ngev"]
state["done"] = state["done"] or cond3
state["failed"] = state["failed"] or (cond3 and zoom2["failed"])
state["a_star"] = mnp.where(cond3, zoom2["a_star"], state["a_star"])
state["phi_star"] = mnp.where(cond3, zoom2["phi_star"], state["phi_star"])
state["g_star"] = mnp.where(cond3, zoom2["g_star"], state["g_star"])
state["dphi_star"] = mnp.where(cond3, zoom2["dphi_star"], state["dphi_star"])
state["i"] += 1
state["a_i"] = a_i
state["phi_i"] = phi_i
state["dphi_i"] = dphi_i
state["status"] = mnp.where(
state["failed"],
1, # zoom failed
mnp.where(
state["i"] > maxiter,
3, # maxiter reached
0, # passed (should be)
),
)
state["a_star"] = mnp.where(
_to_tensor(state["a_star"].dtype != mstype.float64)
and (mnp.abs(state["a_star"]) < 1e-8),
mnp.sign(state["a_star"]) * 1e-8,
state["a_star"],
)
return state
[文档]def line_search(f, xk, pk, jac=None, gfk=None, old_fval=None, old_old_fval=None, c1=1e-4, c2=0.9, maxiter=20):
"""Inexact line search that satisfies strong Wolfe conditions.
Algorithm 3.5 from Wright and Nocedal, 'Numerical Optimization', 1999, pg. 59-61
Note:
`line_search` is not supported on Windows platform yet.
Args:
f (function): function of the form f(x) where x is a flat Tensor and returns a real
scalar. The function should be composed of operations with vjp defined.
xk (Tensor): initial guess.
pk (Tensor): direction to search in. Assumes the direction is a descent direction.
jac (function): the gradient function at x where x is a flat Tensor and returns a Tensor.
The function can be None if you want to use automatic credits.
gfk (Tensor): initial value of value_and_gradient as position. Default: ``None`` .
old_fval (Tensor): The same as `gfk`. Default: ``None`` .
old_old_fval (Tensor): unused argument, only for scipy API compliance. Default: ``None`` .
c1 (float): Wolfe criteria constant, see ref. Default: ``1e-4`` .
c2 (float): The same as `c1`. Default: ``0.9`` .
maxiter (int): maximum number of iterations to search. Default: ``20`` .
Returns:
LineSearchResults, results of line search results.
Supported Platforms:
``GPU`` ``CPU``
Examples:
>>> import numpy as onp
>>> from mindspore.scipy.optimize import line_search
>>> from mindspore import Tensor
>>> x0 = Tensor(onp.ones(2).astype(onp.float32))
>>> p0 = Tensor(onp.array([-1, -1]).astype(onp.float32))
>>> def func(x):
... return x[0] ** 2 - x[1] ** 3
>>> res = line_search(func, x0, p0)
>>> print(res.a_k)
1.0
"""
if jac is None:
jac = grad(f)
state = LineSearch(f, jac)(xk, pk, old_fval, old_old_fval, gfk, c1, c2, maxiter)
# If running in graph mode, the state is a tuple.
if isinstance(state, tuple):
state = _LineSearchResults(failed=_to_scalar(state[1] or not state[0]),
nit=_to_scalar(state[2] - 1),
nfev=_to_scalar(state[6]),
ngev=_to_scalar(state[7]),
k=_to_scalar(state[2]),
a_k=_to_scalar(state[8]),
f_k=_to_scalar(state[9]),
g_k=state[11],
status=_to_scalar(state[12]))
else:
state = _LineSearchResults(failed=_to_scalar(state["failed"] or not state["done"]),
nit=_to_scalar(state["i"] - 1),
nfev=_to_scalar(state["nfev"]),
ngev=_to_scalar(state["ngev"]),
k=_to_scalar(state["i"]),
a_k=_to_scalar(state["a_star"]),
f_k=_to_scalar(state["phi_star"]),
g_k=state["g_star"],
status=_to_scalar(state["status"]))
return state