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""" Functions --------- .. autosummary:: :toctree: generated/
fmin_l_bfgs_b
"""
## License for the Python wrapper ## ==============================
## Copyright (c) 2004 David M. Cooke <cookedm@physics.mcmaster.ca>
## Permission is hereby granted, free of charge, to any person obtaining a ## copy of this software and associated documentation files (the "Software"), ## to deal in the Software without restriction, including without limitation ## the rights to use, copy, modify, merge, publish, distribute, sublicense, ## and/or sell copies of the Software, and to permit persons to whom the ## Software is furnished to do so, subject to the following conditions:
## The above copyright notice and this permission notice shall be included in ## all copies or substantial portions of the Software.
## THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR ## IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, ## FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE ## AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER ## LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING ## FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER ## DEALINGS IN THE SOFTWARE.
## Modifications by Travis Oliphant and Enthought, Inc. for inclusion in SciPy
_check_unknown_options, wrap_function, _approx_fprime_helper)
approx_grad=0, bounds=None, m=10, factr=1e7, pgtol=1e-5, epsilon=1e-8, iprint=-1, maxfun=15000, maxiter=15000, disp=None, callback=None, maxls=20): """ Minimize a function func using the L-BFGS-B algorithm.
Parameters ---------- func : callable f(x,*args) Function to minimise. x0 : ndarray Initial guess. fprime : callable fprime(x,*args), optional The gradient of `func`. If None, then `func` returns the function value and the gradient (``f, g = func(x, *args)``), unless `approx_grad` is True in which case `func` returns only ``f``. args : sequence, optional Arguments to pass to `func` and `fprime`. approx_grad : bool, optional Whether to approximate the gradient numerically (in which case `func` returns only the function value). bounds : list, optional ``(min, max)`` pairs for each element in ``x``, defining the bounds on that parameter. Use None or +-inf for one of ``min`` or ``max`` when there is no bound in that direction. m : int, optional The maximum number of variable metric corrections used to define the limited memory matrix. (The limited memory BFGS method does not store the full hessian but uses this many terms in an approximation to it.) factr : float, optional The iteration stops when ``(f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr * eps``, where ``eps`` is the machine precision, which is automatically generated by the code. Typical values for `factr` are: 1e12 for low accuracy; 1e7 for moderate accuracy; 10.0 for extremely high accuracy. See Notes for relationship to `ftol`, which is exposed (instead of `factr`) by the `scipy.optimize.minimize` interface to L-BFGS-B. pgtol : float, optional The iteration will stop when ``max{|proj g_i | i = 1, ..., n} <= pgtol`` where ``pg_i`` is the i-th component of the projected gradient. epsilon : float, optional Step size used when `approx_grad` is True, for numerically calculating the gradient iprint : int, optional Controls the frequency of output. ``iprint < 0`` means no output; ``iprint = 0`` print only one line at the last iteration; ``0 < iprint < 99`` print also f and ``|proj g|`` every iprint iterations; ``iprint = 99`` print details of every iteration except n-vectors; ``iprint = 100`` print also the changes of active set and final x; ``iprint > 100`` print details of every iteration including x and g. disp : int, optional If zero, then no output. If a positive number, then this over-rides `iprint` (i.e., `iprint` gets the value of `disp`). maxfun : int, optional Maximum number of function evaluations. maxiter : int, optional Maximum number of iterations. callback : callable, optional Called after each iteration, as ``callback(xk)``, where ``xk`` is the current parameter vector. maxls : int, optional Maximum number of line search steps (per iteration). Default is 20.
Returns ------- x : array_like Estimated position of the minimum. f : float Value of `func` at the minimum. d : dict Information dictionary.
* d['warnflag'] is
- 0 if converged, - 1 if too many function evaluations or too many iterations, - 2 if stopped for another reason, given in d['task']
* d['grad'] is the gradient at the minimum (should be 0 ish) * d['funcalls'] is the number of function calls made. * d['nit'] is the number of iterations.
See also -------- minimize: Interface to minimization algorithms for multivariate functions. See the 'L-BFGS-B' `method` in particular. Note that the `ftol` option is made available via that interface, while `factr` is provided via this interface, where `factr` is the factor multiplying the default machine floating-point precision to arrive at `ftol`: ``ftol = factr * numpy.finfo(float).eps``.
Notes ----- License of L-BFGS-B (FORTRAN code):
The version included here (in fortran code) is 3.0 (released April 25, 2011). It was written by Ciyou Zhu, Richard Byrd, and Jorge Nocedal <nocedal@ece.nwu.edu>. It carries the following condition for use:
This software is freely available, but we expect that all publications describing work using this software, or all commercial products using it, quote at least one of the references given below. This software is released under the BSD License.
References ---------- * R. H. Byrd, P. Lu and J. Nocedal. A Limited Memory Algorithm for Bound Constrained Optimization, (1995), SIAM Journal on Scientific and Statistical Computing, 16, 5, pp. 1190-1208. * C. Zhu, R. H. Byrd and J. Nocedal. L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization (1997), ACM Transactions on Mathematical Software, 23, 4, pp. 550 - 560. * J.L. Morales and J. Nocedal. L-BFGS-B: Remark on Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization (2011), ACM Transactions on Mathematical Software, 38, 1.
""" # handle fprime/approx_grad if approx_grad: fun = func jac = None elif fprime is None: fun = MemoizeJac(func) jac = fun.derivative else: fun = func jac = fprime
# build options if disp is None: disp = iprint opts = {'disp': disp, 'iprint': iprint, 'maxcor': m, 'ftol': factr * np.finfo(float).eps, 'gtol': pgtol, 'eps': epsilon, 'maxfun': maxfun, 'maxiter': maxiter, 'callback': callback, 'maxls': maxls}
res = _minimize_lbfgsb(fun, x0, args=args, jac=jac, bounds=bounds, **opts) d = {'grad': res['jac'], 'task': res['message'], 'funcalls': res['nfev'], 'nit': res['nit'], 'warnflag': res['status']} f = res['fun'] x = res['x']
return x, f, d
disp=None, maxcor=10, ftol=2.2204460492503131e-09, gtol=1e-5, eps=1e-8, maxfun=15000, maxiter=15000, iprint=-1, callback=None, maxls=20, **unknown_options): """ Minimize a scalar function of one or more variables using the L-BFGS-B algorithm.
Options ------- disp : bool Set to True to print convergence messages. maxcor : int The maximum number of variable metric corrections used to define the limited memory matrix. (The limited memory BFGS method does not store the full hessian but uses this many terms in an approximation to it.) ftol : float The iteration stops when ``(f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= ftol``. gtol : float The iteration will stop when ``max{|proj g_i | i = 1, ..., n} <= gtol`` where ``pg_i`` is the i-th component of the projected gradient. eps : float Step size used for numerical approximation of the jacobian. disp : int Set to True to print convergence messages. maxfun : int Maximum number of function evaluations. maxiter : int Maximum number of iterations. maxls : int, optional Maximum number of line search steps (per iteration). Default is 20.
Notes ----- The option `ftol` is exposed via the `scipy.optimize.minimize` interface, but calling `scipy.optimize.fmin_l_bfgs_b` directly exposes `factr`. The relationship between the two is ``ftol = factr * numpy.finfo(float).eps``. I.e., `factr` multiplies the default machine floating-point precision to arrive at `ftol`.
""" _check_unknown_options(unknown_options) m = maxcor epsilon = eps pgtol = gtol factr = ftol / np.finfo(float).eps
x0 = asarray(x0).ravel() n, = x0.shape
if bounds is None: bounds = [(None, None)] * n if len(bounds) != n: raise ValueError('length of x0 != length of bounds') # unbounded variables must use None, not +-inf, for optimizer to work properly bounds = [(None if l == -np.inf else l, None if u == np.inf else u) for l, u in bounds]
if disp is not None: if disp == 0: iprint = -1 else: iprint = disp
n_function_evals, fun = wrap_function(fun, ()) if jac is None: def func_and_grad(x): f = fun(x, *args) g = _approx_fprime_helper(x, fun, epsilon, args=args, f0=f) return f, g else: def func_and_grad(x): f = fun(x, *args) g = jac(x, *args) return f, g
nbd = zeros(n, int32) low_bnd = zeros(n, float64) upper_bnd = zeros(n, float64) bounds_map = {(None, None): 0, (1, None): 1, (1, 1): 2, (None, 1): 3} for i in range(0, n): l, u = bounds[i] if l is not None: low_bnd[i] = l l = 1 if u is not None: upper_bnd[i] = u u = 1 nbd[i] = bounds_map[l, u]
if not maxls > 0: raise ValueError('maxls must be positive.')
x = array(x0, float64) f = array(0.0, float64) g = zeros((n,), float64) wa = zeros(2*m*n + 5*n + 11*m*m + 8*m, float64) iwa = zeros(3*n, int32) task = zeros(1, 'S60') csave = zeros(1, 'S60') lsave = zeros(4, int32) isave = zeros(44, int32) dsave = zeros(29, float64)
task[:] = 'START'
n_iterations = 0
while 1: # x, f, g, wa, iwa, task, csave, lsave, isave, dsave = \ _lbfgsb.setulb(m, x, low_bnd, upper_bnd, nbd, f, g, factr, pgtol, wa, iwa, task, iprint, csave, lsave, isave, dsave, maxls) task_str = task.tostring() if task_str.startswith(b'FG'): # The minimization routine wants f and g at the current x. # Note that interruptions due to maxfun are postponed # until the completion of the current minimization iteration. # Overwrite f and g: f, g = func_and_grad(x) elif task_str.startswith(b'NEW_X'): # new iteration n_iterations += 1 if callback is not None: callback(np.copy(x))
if n_iterations >= maxiter: task[:] = 'STOP: TOTAL NO. of ITERATIONS REACHED LIMIT' elif n_function_evals[0] > maxfun: task[:] = ('STOP: TOTAL NO. of f AND g EVALUATIONS ' 'EXCEEDS LIMIT') else: break
task_str = task.tostring().strip(b'\x00').strip() if task_str.startswith(b'CONV'): warnflag = 0 elif n_function_evals[0] > maxfun or n_iterations >= maxiter: warnflag = 1 else: warnflag = 2
# These two portions of the workspace are described in the mainlb # subroutine in lbfgsb.f. See line 363. s = wa[0: m*n].reshape(m, n) y = wa[m*n: 2*m*n].reshape(m, n)
# See lbfgsb.f line 160 for this portion of the workspace. # isave(31) = the total number of BFGS updates prior the current iteration; n_bfgs_updates = isave[30]
n_corrs = min(n_bfgs_updates, maxcor) hess_inv = LbfgsInvHessProduct(s[:n_corrs], y[:n_corrs])
return OptimizeResult(fun=f, jac=g, nfev=n_function_evals[0], nit=n_iterations, status=warnflag, message=task_str, x=x, success=(warnflag == 0), hess_inv=hess_inv)
"""Linear operator for the L-BFGS approximate inverse Hessian.
This operator computes the product of a vector with the approximate inverse of the Hessian of the objective function, using the L-BFGS limited memory approximation to the inverse Hessian, accumulated during the optimization.
Objects of this class implement the ``scipy.sparse.linalg.LinearOperator`` interface.
Parameters ---------- sk : array_like, shape=(n_corr, n) Array of `n_corr` most recent updates to the solution vector. (See [1]). yk : array_like, shape=(n_corr, n) Array of `n_corr` most recent updates to the gradient. (See [1]).
References ---------- .. [1] Nocedal, Jorge. "Updating quasi-Newton matrices with limited storage." Mathematics of computation 35.151 (1980): 773-782.
""" """Construct the operator.""" if sk.shape != yk.shape or sk.ndim != 2: raise ValueError('sk and yk must have matching shape, (n_corrs, n)') n_corrs, n = sk.shape
super(LbfgsInvHessProduct, self).__init__( dtype=np.float64, shape=(n, n))
self.sk = sk self.yk = yk self.n_corrs = n_corrs self.rho = 1 / np.einsum('ij,ij->i', sk, yk)
"""Efficient matrix-vector multiply with the BFGS matrices.
This calculation is described in Section (4) of [1].
Parameters ---------- x : ndarray An array with shape (n,) or (n,1).
Returns ------- y : ndarray The matrix-vector product
""" s, y, n_corrs, rho = self.sk, self.yk, self.n_corrs, self.rho q = np.array(x, dtype=self.dtype, copy=True) if q.ndim == 2 and q.shape[1] == 1: q = q.reshape(-1)
alpha = np.zeros(n_corrs)
for i in range(n_corrs-1, -1, -1): alpha[i] = rho[i] * np.dot(s[i], q) q = q - alpha[i]*y[i]
r = q for i in range(n_corrs): beta = rho[i] * np.dot(y[i], r) r = r + s[i] * (alpha[i] - beta)
return r
"""Return a dense array representation of this operator.
Returns ------- arr : ndarray, shape=(n, n) An array with the same shape and containing the same data represented by this `LinearOperator`.
""" s, y, n_corrs, rho = self.sk, self.yk, self.n_corrs, self.rho I = np.eye(*self.shape, dtype=self.dtype) Hk = I
for i in range(n_corrs): A1 = I - s[i][:, np.newaxis] * y[i][np.newaxis, :] * rho[i] A2 = I - y[i][:, np.newaxis] * s[i][np.newaxis, :] * rho[i]
Hk = np.dot(A1, np.dot(Hk, A2)) + (rho[i] * s[i][:, np.newaxis] * s[i][np.newaxis, :]) return Hk |