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497 lines
18 KiB
Python
497 lines
18 KiB
Python
"""
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Functions
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---------
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.. autosummary::
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:toctree: generated/
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fmin_l_bfgs_b
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"""
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## License for the Python wrapper
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## ==============================
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## Copyright (c) 2004 David M. Cooke <cookedm@physics.mcmaster.ca>
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## Permission is hereby granted, free of charge, to any person obtaining a
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## copy of this software and associated documentation files (the "Software"),
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## to deal in the Software without restriction, including without limitation
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## the rights to use, copy, modify, merge, publish, distribute, sublicense,
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## and/or sell copies of the Software, and to permit persons to whom the
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## Software is furnished to do so, subject to the following conditions:
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## The above copyright notice and this permission notice shall be included in
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## all copies or substantial portions of the Software.
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## THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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## IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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## FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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## AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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## LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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## FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
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## DEALINGS IN THE SOFTWARE.
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## Modifications by Travis Oliphant and Enthought, Inc. for inclusion in SciPy
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import numpy as np
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from numpy import array, asarray, float64, zeros
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from . import _lbfgsb
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from .optimize import (MemoizeJac, OptimizeResult,
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_check_unknown_options, _prepare_scalar_function)
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from ._constraints import old_bound_to_new
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from scipy.sparse.linalg import LinearOperator
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__all__ = ['fmin_l_bfgs_b', 'LbfgsInvHessProduct']
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def fmin_l_bfgs_b(func, x0, fprime=None, args=(),
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approx_grad=0,
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bounds=None, m=10, factr=1e7, pgtol=1e-5,
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epsilon=1e-8,
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iprint=-1, maxfun=15000, maxiter=15000, disp=None,
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callback=None, maxls=20):
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"""
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Minimize a function func using the L-BFGS-B algorithm.
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Parameters
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----------
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func : callable f(x,*args)
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Function to minimize.
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x0 : ndarray
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Initial guess.
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fprime : callable fprime(x,*args), optional
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The gradient of `func`. If None, then `func` returns the function
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value and the gradient (``f, g = func(x, *args)``), unless
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`approx_grad` is True in which case `func` returns only ``f``.
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args : sequence, optional
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Arguments to pass to `func` and `fprime`.
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approx_grad : bool, optional
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Whether to approximate the gradient numerically (in which case
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`func` returns only the function value).
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bounds : list, optional
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``(min, max)`` pairs for each element in ``x``, defining
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the bounds on that parameter. Use None or +-inf for one of ``min`` or
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``max`` when there is no bound in that direction.
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m : int, optional
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The maximum number of variable metric corrections
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used to define the limited memory matrix. (The limited memory BFGS
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method does not store the full hessian but uses this many terms in an
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approximation to it.)
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factr : float, optional
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The iteration stops when
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``(f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr * eps``,
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where ``eps`` is the machine precision, which is automatically
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generated by the code. Typical values for `factr` are: 1e12 for
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low accuracy; 1e7 for moderate accuracy; 10.0 for extremely
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high accuracy. See Notes for relationship to `ftol`, which is exposed
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(instead of `factr`) by the `scipy.optimize.minimize` interface to
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L-BFGS-B.
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pgtol : float, optional
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The iteration will stop when
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``max{|proj g_i | i = 1, ..., n} <= pgtol``
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where ``pg_i`` is the i-th component of the projected gradient.
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epsilon : float, optional
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Step size used when `approx_grad` is True, for numerically
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calculating the gradient
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iprint : int, optional
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Controls the frequency of output. ``iprint < 0`` means no output;
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``iprint = 0`` print only one line at the last iteration;
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``0 < iprint < 99`` print also f and ``|proj g|`` every iprint iterations;
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``iprint = 99`` print details of every iteration except n-vectors;
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``iprint = 100`` print also the changes of active set and final x;
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``iprint > 100`` print details of every iteration including x and g.
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disp : int, optional
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If zero, then no output. If a positive number, then this over-rides
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`iprint` (i.e., `iprint` gets the value of `disp`).
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maxfun : int, optional
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Maximum number of function evaluations.
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maxiter : int, optional
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Maximum number of iterations.
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callback : callable, optional
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Called after each iteration, as ``callback(xk)``, where ``xk`` is the
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current parameter vector.
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maxls : int, optional
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Maximum number of line search steps (per iteration). Default is 20.
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Returns
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-------
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x : array_like
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Estimated position of the minimum.
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f : float
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Value of `func` at the minimum.
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d : dict
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Information dictionary.
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* d['warnflag'] is
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- 0 if converged,
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- 1 if too many function evaluations or too many iterations,
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- 2 if stopped for another reason, given in d['task']
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* d['grad'] is the gradient at the minimum (should be 0 ish)
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* d['funcalls'] is the number of function calls made.
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* d['nit'] is the number of iterations.
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See also
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--------
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minimize: Interface to minimization algorithms for multivariate
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functions. See the 'L-BFGS-B' `method` in particular. Note that the
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`ftol` option is made available via that interface, while `factr` is
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provided via this interface, where `factr` is the factor multiplying
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the default machine floating-point precision to arrive at `ftol`:
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``ftol = factr * numpy.finfo(float).eps``.
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Notes
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-----
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License of L-BFGS-B (FORTRAN code):
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The version included here (in fortran code) is 3.0
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(released April 25, 2011). It was written by Ciyou Zhu, Richard Byrd,
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and Jorge Nocedal <nocedal@ece.nwu.edu>. It carries the following
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condition for use:
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This software is freely available, but we expect that all publications
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describing work using this software, or all commercial products using it,
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quote at least one of the references given below. This software is released
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under the BSD License.
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References
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----------
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* R. H. Byrd, P. Lu and J. Nocedal. A Limited Memory Algorithm for Bound
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Constrained Optimization, (1995), SIAM Journal on Scientific and
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Statistical Computing, 16, 5, pp. 1190-1208.
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* C. Zhu, R. H. Byrd and J. Nocedal. L-BFGS-B: Algorithm 778: L-BFGS-B,
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FORTRAN routines for large scale bound constrained optimization (1997),
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ACM Transactions on Mathematical Software, 23, 4, pp. 550 - 560.
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* J.L. Morales and J. Nocedal. L-BFGS-B: Remark on Algorithm 778: L-BFGS-B,
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FORTRAN routines for large scale bound constrained optimization (2011),
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ACM Transactions on Mathematical Software, 38, 1.
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"""
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# handle fprime/approx_grad
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if approx_grad:
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fun = func
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jac = None
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elif fprime is None:
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fun = MemoizeJac(func)
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jac = fun.derivative
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else:
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fun = func
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jac = fprime
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# build options
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if disp is None:
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disp = iprint
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opts = {'disp': disp,
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'iprint': iprint,
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'maxcor': m,
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'ftol': factr * np.finfo(float).eps,
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'gtol': pgtol,
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'eps': epsilon,
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'maxfun': maxfun,
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'maxiter': maxiter,
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'callback': callback,
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'maxls': maxls}
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res = _minimize_lbfgsb(fun, x0, args=args, jac=jac, bounds=bounds,
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**opts)
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d = {'grad': res['jac'],
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'task': res['message'],
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'funcalls': res['nfev'],
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'nit': res['nit'],
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'warnflag': res['status']}
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f = res['fun']
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x = res['x']
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return x, f, d
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def _minimize_lbfgsb(fun, x0, args=(), jac=None, bounds=None,
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disp=None, maxcor=10, ftol=2.2204460492503131e-09,
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gtol=1e-5, eps=1e-8, maxfun=15000, maxiter=15000,
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iprint=-1, callback=None, maxls=20,
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finite_diff_rel_step=None, **unknown_options):
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"""
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Minimize a scalar function of one or more variables using the L-BFGS-B
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algorithm.
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Options
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-------
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disp : None or int
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If `disp is None` (the default), then the supplied version of `iprint`
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is used. If `disp is not None`, then it overrides the supplied version
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of `iprint` with the behaviour you outlined.
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maxcor : int
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The maximum number of variable metric corrections used to
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define the limited memory matrix. (The limited memory BFGS
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method does not store the full hessian but uses this many terms
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in an approximation to it.)
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ftol : float
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The iteration stops when ``(f^k -
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f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= ftol``.
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gtol : float
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The iteration will stop when ``max{|proj g_i | i = 1, ..., n}
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<= gtol`` where ``pg_i`` is the i-th component of the
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projected gradient.
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eps : float or ndarray
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If `jac is None` the absolute step size used for numerical
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approximation of the jacobian via forward differences.
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maxfun : int
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Maximum number of function evaluations.
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maxiter : int
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Maximum number of iterations.
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iprint : int, optional
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Controls the frequency of output. ``iprint < 0`` means no output;
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``iprint = 0`` print only one line at the last iteration;
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``0 < iprint < 99`` print also f and ``|proj g|`` every iprint iterations;
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``iprint = 99`` print details of every iteration except n-vectors;
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``iprint = 100`` print also the changes of active set and final x;
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``iprint > 100`` print details of every iteration including x and g.
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callback : callable, optional
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Called after each iteration, as ``callback(xk)``, where ``xk`` is the
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current parameter vector.
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maxls : int, optional
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Maximum number of line search steps (per iteration). Default is 20.
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finite_diff_rel_step : None or array_like, optional
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If `jac in ['2-point', '3-point', 'cs']` the relative step size to
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use for numerical approximation of the jacobian. The absolute step
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size is computed as ``h = rel_step * sign(x0) * max(1, abs(x0))``,
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possibly adjusted to fit into the bounds. For ``method='3-point'``
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the sign of `h` is ignored. If None (default) then step is selected
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automatically.
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Notes
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-----
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The option `ftol` is exposed via the `scipy.optimize.minimize` interface,
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but calling `scipy.optimize.fmin_l_bfgs_b` directly exposes `factr`. The
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relationship between the two is ``ftol = factr * numpy.finfo(float).eps``.
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I.e., `factr` multiplies the default machine floating-point precision to
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arrive at `ftol`.
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"""
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_check_unknown_options(unknown_options)
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m = maxcor
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pgtol = gtol
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factr = ftol / np.finfo(float).eps
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x0 = asarray(x0).ravel()
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n, = x0.shape
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if bounds is None:
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bounds = [(None, None)] * n
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if len(bounds) != n:
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raise ValueError('length of x0 != length of bounds')
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# unbounded variables must use None, not +-inf, for optimizer to work properly
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bounds = [(None if l == -np.inf else l, None if u == np.inf else u) for l, u in bounds]
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# LBFGSB is sent 'old-style' bounds, 'new-style' bounds are required by
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# approx_derivative and ScalarFunction
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new_bounds = old_bound_to_new(bounds)
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# check bounds
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if (new_bounds[0] > new_bounds[1]).any():
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raise ValueError("LBFGSB - one of the lower bounds is greater than an upper bound.")
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# initial vector must lie within the bounds. Otherwise ScalarFunction and
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# approx_derivative will cause problems
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x0 = np.clip(x0, new_bounds[0], new_bounds[1])
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if disp is not None:
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if disp == 0:
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iprint = -1
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else:
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iprint = disp
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sf = _prepare_scalar_function(fun, x0, jac=jac, args=args, epsilon=eps,
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bounds=new_bounds,
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finite_diff_rel_step=finite_diff_rel_step)
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func_and_grad = sf.fun_and_grad
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fortran_int = _lbfgsb.types.intvar.dtype
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nbd = zeros(n, fortran_int)
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low_bnd = zeros(n, float64)
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upper_bnd = zeros(n, float64)
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bounds_map = {(None, None): 0,
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(1, None): 1,
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(1, 1): 2,
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(None, 1): 3}
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for i in range(0, n):
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l, u = bounds[i]
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if l is not None:
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low_bnd[i] = l
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l = 1
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if u is not None:
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upper_bnd[i] = u
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u = 1
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nbd[i] = bounds_map[l, u]
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if not maxls > 0:
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raise ValueError('maxls must be positive.')
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x = array(x0, float64)
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f = array(0.0, float64)
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g = zeros((n,), float64)
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wa = zeros(2*m*n + 5*n + 11*m*m + 8*m, float64)
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iwa = zeros(3*n, fortran_int)
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task = zeros(1, 'S60')
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csave = zeros(1, 'S60')
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lsave = zeros(4, fortran_int)
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isave = zeros(44, fortran_int)
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dsave = zeros(29, float64)
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task[:] = 'START'
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n_iterations = 0
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while 1:
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# x, f, g, wa, iwa, task, csave, lsave, isave, dsave = \
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_lbfgsb.setulb(m, x, low_bnd, upper_bnd, nbd, f, g, factr,
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pgtol, wa, iwa, task, iprint, csave, lsave,
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isave, dsave, maxls)
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task_str = task.tobytes()
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if task_str.startswith(b'FG'):
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# The minimization routine wants f and g at the current x.
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# Note that interruptions due to maxfun are postponed
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# until the completion of the current minimization iteration.
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# Overwrite f and g:
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f, g = func_and_grad(x)
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elif task_str.startswith(b'NEW_X'):
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# new iteration
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n_iterations += 1
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if callback is not None:
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callback(np.copy(x))
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if n_iterations >= maxiter:
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task[:] = 'STOP: TOTAL NO. of ITERATIONS REACHED LIMIT'
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elif sf.nfev > maxfun:
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task[:] = ('STOP: TOTAL NO. of f AND g EVALUATIONS '
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'EXCEEDS LIMIT')
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else:
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break
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task_str = task.tobytes().strip(b'\x00').strip()
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if task_str.startswith(b'CONV'):
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warnflag = 0
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elif sf.nfev > maxfun or n_iterations >= maxiter:
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warnflag = 1
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else:
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warnflag = 2
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# These two portions of the workspace are described in the mainlb
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# subroutine in lbfgsb.f. See line 363.
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s = wa[0: m*n].reshape(m, n)
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y = wa[m*n: 2*m*n].reshape(m, n)
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# See lbfgsb.f line 160 for this portion of the workspace.
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# isave(31) = the total number of BFGS updates prior the current iteration;
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n_bfgs_updates = isave[30]
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n_corrs = min(n_bfgs_updates, maxcor)
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hess_inv = LbfgsInvHessProduct(s[:n_corrs], y[:n_corrs])
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task_str = task_str.decode()
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return OptimizeResult(fun=f, jac=g, nfev=sf.nfev,
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njev=sf.ngev,
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nit=n_iterations, status=warnflag, message=task_str,
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x=x, success=(warnflag == 0), hess_inv=hess_inv)
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class LbfgsInvHessProduct(LinearOperator):
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"""Linear operator for the L-BFGS approximate inverse Hessian.
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This operator computes the product of a vector with the approximate inverse
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of the Hessian of the objective function, using the L-BFGS limited
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memory approximation to the inverse Hessian, accumulated during the
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optimization.
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Objects of this class implement the ``scipy.sparse.linalg.LinearOperator``
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interface.
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Parameters
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----------
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sk : array_like, shape=(n_corr, n)
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Array of `n_corr` most recent updates to the solution vector.
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(See [1]).
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yk : array_like, shape=(n_corr, n)
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Array of `n_corr` most recent updates to the gradient. (See [1]).
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References
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----------
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.. [1] Nocedal, Jorge. "Updating quasi-Newton matrices with limited
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storage." Mathematics of computation 35.151 (1980): 773-782.
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"""
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def __init__(self, sk, yk):
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"""Construct the operator."""
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if sk.shape != yk.shape or sk.ndim != 2:
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raise ValueError('sk and yk must have matching shape, (n_corrs, n)')
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n_corrs, n = sk.shape
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super(LbfgsInvHessProduct, self).__init__(
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dtype=np.float64, shape=(n, n))
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self.sk = sk
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self.yk = yk
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self.n_corrs = n_corrs
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self.rho = 1 / np.einsum('ij,ij->i', sk, yk)
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def _matvec(self, x):
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"""Efficient matrix-vector multiply with the BFGS matrices.
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This calculation is described in Section (4) of [1].
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Parameters
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----------
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x : ndarray
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An array with shape (n,) or (n,1).
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Returns
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-------
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y : ndarray
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The matrix-vector product
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"""
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s, y, n_corrs, rho = self.sk, self.yk, self.n_corrs, self.rho
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q = np.array(x, dtype=self.dtype, copy=True)
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if q.ndim == 2 and q.shape[1] == 1:
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q = q.reshape(-1)
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alpha = np.empty(n_corrs)
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for i in range(n_corrs-1, -1, -1):
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alpha[i] = rho[i] * np.dot(s[i], q)
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q = q - alpha[i]*y[i]
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r = q
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for i in range(n_corrs):
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beta = rho[i] * np.dot(y[i], r)
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r = r + s[i] * (alpha[i] - beta)
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return r
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def todense(self):
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"""Return a dense array representation of this operator.
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|
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Returns
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-------
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|
arr : ndarray, shape=(n, n)
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|
An array with the same shape and containing
|
|
the same data represented by this `LinearOperator`.
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|
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"""
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|
s, y, n_corrs, rho = self.sk, self.yk, self.n_corrs, self.rho
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I = np.eye(*self.shape, dtype=self.dtype)
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|
Hk = I
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|
|
|
for i in range(n_corrs):
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|
A1 = I - s[i][:, np.newaxis] * y[i][np.newaxis, :] * rho[i]
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|
A2 = I - y[i][:, np.newaxis] * s[i][np.newaxis, :] * rho[i]
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|
|
|
Hk = np.dot(A1, np.dot(Hk, A2)) + (rho[i] * s[i][:, np.newaxis] *
|
|
s[i][np.newaxis, :])
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|
return Hk
|