258 lines
7.4 KiB
Python
258 lines
7.4 KiB
Python
"""
|
|
Our own implementation of the Newton algorithm
|
|
|
|
Unlike the scipy.optimize version, this version of the Newton conjugate
|
|
gradient solver uses only one function call to retrieve the
|
|
func value, the gradient value and a callable for the Hessian matvec
|
|
product. If the function call is very expensive (e.g. for logistic
|
|
regression with large design matrix), this approach gives very
|
|
significant speedups.
|
|
"""
|
|
# This is a modified file from scipy.optimize
|
|
# Original authors: Travis Oliphant, Eric Jones
|
|
# Modifications by Gael Varoquaux, Mathieu Blondel and Tom Dupre la Tour
|
|
# License: BSD
|
|
|
|
import numpy as np
|
|
import warnings
|
|
from scipy.optimize.linesearch import line_search_wolfe2, line_search_wolfe1
|
|
|
|
from ..exceptions import ConvergenceWarning
|
|
|
|
|
|
class _LineSearchError(RuntimeError):
|
|
pass
|
|
|
|
|
|
def _line_search_wolfe12(f, fprime, xk, pk, gfk, old_fval, old_old_fval,
|
|
**kwargs):
|
|
"""
|
|
Same as line_search_wolfe1, but fall back to line_search_wolfe2 if
|
|
suitable step length is not found, and raise an exception if a
|
|
suitable step length is not found.
|
|
|
|
Raises
|
|
------
|
|
_LineSearchError
|
|
If no suitable step size is found.
|
|
|
|
"""
|
|
ret = line_search_wolfe1(f, fprime, xk, pk, gfk,
|
|
old_fval, old_old_fval,
|
|
**kwargs)
|
|
|
|
if ret[0] is None:
|
|
# line search failed: try different one.
|
|
ret = line_search_wolfe2(f, fprime, xk, pk, gfk,
|
|
old_fval, old_old_fval, **kwargs)
|
|
|
|
if ret[0] is None:
|
|
raise _LineSearchError()
|
|
|
|
return ret
|
|
|
|
|
|
def _cg(fhess_p, fgrad, maxiter, tol):
|
|
"""
|
|
Solve iteratively the linear system 'fhess_p . xsupi = fgrad'
|
|
with a conjugate gradient descent.
|
|
|
|
Parameters
|
|
----------
|
|
fhess_p : callable
|
|
Function that takes the gradient as a parameter and returns the
|
|
matrix product of the Hessian and gradient.
|
|
|
|
fgrad : ndarray of shape (n_features,) or (n_features + 1,)
|
|
Gradient vector.
|
|
|
|
maxiter : int
|
|
Number of CG iterations.
|
|
|
|
tol : float
|
|
Stopping criterion.
|
|
|
|
Returns
|
|
-------
|
|
xsupi : ndarray of shape (n_features,) or (n_features + 1,)
|
|
Estimated solution.
|
|
"""
|
|
xsupi = np.zeros(len(fgrad), dtype=fgrad.dtype)
|
|
ri = fgrad
|
|
psupi = -ri
|
|
i = 0
|
|
dri0 = np.dot(ri, ri)
|
|
|
|
while i <= maxiter:
|
|
if np.sum(np.abs(ri)) <= tol:
|
|
break
|
|
|
|
Ap = fhess_p(psupi)
|
|
# check curvature
|
|
curv = np.dot(psupi, Ap)
|
|
if 0 <= curv <= 3 * np.finfo(np.float64).eps:
|
|
break
|
|
elif curv < 0:
|
|
if i > 0:
|
|
break
|
|
else:
|
|
# fall back to steepest descent direction
|
|
xsupi += dri0 / curv * psupi
|
|
break
|
|
alphai = dri0 / curv
|
|
xsupi += alphai * psupi
|
|
ri = ri + alphai * Ap
|
|
dri1 = np.dot(ri, ri)
|
|
betai = dri1 / dri0
|
|
psupi = -ri + betai * psupi
|
|
i = i + 1
|
|
dri0 = dri1 # update np.dot(ri,ri) for next time.
|
|
|
|
return xsupi
|
|
|
|
|
|
def _newton_cg(grad_hess, func, grad, x0, args=(), tol=1e-4,
|
|
maxiter=100, maxinner=200, line_search=True, warn=True):
|
|
"""
|
|
Minimization of scalar function of one or more variables using the
|
|
Newton-CG algorithm.
|
|
|
|
Parameters
|
|
----------
|
|
grad_hess : callable
|
|
Should return the gradient and a callable returning the matvec product
|
|
of the Hessian.
|
|
|
|
func : callable
|
|
Should return the value of the function.
|
|
|
|
grad : callable
|
|
Should return the function value and the gradient. This is used
|
|
by the linesearch functions.
|
|
|
|
x0 : array of float
|
|
Initial guess.
|
|
|
|
args : tuple, default=()
|
|
Arguments passed to func_grad_hess, func and grad.
|
|
|
|
tol : float, default=1e-4
|
|
Stopping criterion. The iteration will stop when
|
|
``max{|g_i | i = 1, ..., n} <= tol``
|
|
where ``g_i`` is the i-th component of the gradient.
|
|
|
|
maxiter : int, default=100
|
|
Number of Newton iterations.
|
|
|
|
maxinner : int, default=200
|
|
Number of CG iterations.
|
|
|
|
line_search : bool, default=True
|
|
Whether to use a line search or not.
|
|
|
|
warn : bool, default=True
|
|
Whether to warn when didn't converge.
|
|
|
|
Returns
|
|
-------
|
|
xk : ndarray of float
|
|
Estimated minimum.
|
|
"""
|
|
x0 = np.asarray(x0).flatten()
|
|
xk = x0
|
|
k = 0
|
|
|
|
if line_search:
|
|
old_fval = func(x0, *args)
|
|
old_old_fval = None
|
|
|
|
# Outer loop: our Newton iteration
|
|
while k < maxiter:
|
|
# Compute a search direction pk by applying the CG method to
|
|
# del2 f(xk) p = - fgrad f(xk) starting from 0.
|
|
fgrad, fhess_p = grad_hess(xk, *args)
|
|
|
|
absgrad = np.abs(fgrad)
|
|
if np.max(absgrad) <= tol:
|
|
break
|
|
|
|
maggrad = np.sum(absgrad)
|
|
eta = min([0.5, np.sqrt(maggrad)])
|
|
termcond = eta * maggrad
|
|
|
|
# Inner loop: solve the Newton update by conjugate gradient, to
|
|
# avoid inverting the Hessian
|
|
xsupi = _cg(fhess_p, fgrad, maxiter=maxinner, tol=termcond)
|
|
|
|
alphak = 1.0
|
|
|
|
if line_search:
|
|
try:
|
|
alphak, fc, gc, old_fval, old_old_fval, gfkp1 = \
|
|
_line_search_wolfe12(func, grad, xk, xsupi, fgrad,
|
|
old_fval, old_old_fval, args=args)
|
|
except _LineSearchError:
|
|
warnings.warn('Line Search failed')
|
|
break
|
|
|
|
xk = xk + alphak * xsupi # upcast if necessary
|
|
k += 1
|
|
|
|
if warn and k >= maxiter:
|
|
warnings.warn("newton-cg failed to converge. Increase the "
|
|
"number of iterations.", ConvergenceWarning)
|
|
return xk, k
|
|
|
|
|
|
def _check_optimize_result(solver, result, max_iter=None,
|
|
extra_warning_msg=None):
|
|
"""Check the OptimizeResult for successful convergence
|
|
|
|
Parameters
|
|
----------
|
|
solver : str
|
|
Solver name. Currently only `lbfgs` is supported.
|
|
|
|
result : OptimizeResult
|
|
Result of the scipy.optimize.minimize function.
|
|
|
|
max_iter : int, default=None
|
|
Expected maximum number of iterations.
|
|
|
|
extra_warning_msg : str, default=None
|
|
Extra warning message.
|
|
|
|
Returns
|
|
-------
|
|
n_iter : int
|
|
Number of iterations.
|
|
"""
|
|
# handle both scipy and scikit-learn solver names
|
|
if solver == "lbfgs":
|
|
if result.status != 0:
|
|
try:
|
|
# The message is already decoded in scipy>=1.6.0
|
|
result_message = result.message.decode("latin1")
|
|
except AttributeError:
|
|
result_message = result.message
|
|
warning_msg = (
|
|
"{} failed to converge (status={}):\n{}.\n\n"
|
|
"Increase the number of iterations (max_iter) "
|
|
"or scale the data as shown in:\n"
|
|
" https://scikit-learn.org/stable/modules/"
|
|
"preprocessing.html"
|
|
).format(solver, result.status, result_message)
|
|
if extra_warning_msg is not None:
|
|
warning_msg += "\n" + extra_warning_msg
|
|
warnings.warn(warning_msg, ConvergenceWarning, stacklevel=2)
|
|
if max_iter is not None:
|
|
# In scipy <= 1.0.0, nit may exceed maxiter for lbfgs.
|
|
# See https://github.com/scipy/scipy/issues/7854
|
|
n_iter_i = min(result.nit, max_iter)
|
|
else:
|
|
n_iter_i = result.nit
|
|
else:
|
|
raise NotImplementedError
|
|
|
|
return n_iter_i
|