""" Functions --------- .. autosummary:: :toctree: generated/ fmin_l_bfgs_b """ ## License for the Python wrapper ## ============================== ## Copyright (c) 2004 David M. Cooke ## 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 import numpy as np from numpy import array, asarray, float64, zeros from . import _lbfgsb from ._optimize import (MemoizeJac, OptimizeResult, _call_callback_maybe_halt, _wrap_callback, _check_unknown_options, _prepare_scalar_function) from ._constraints import old_bound_to_new from scipy.sparse.linalg import LinearOperator __all__ = ['fmin_l_bfgs_b', 'LbfgsInvHessProduct'] status_messages = { 0 : "START", 1 : "NEW_X", 2 : "RESTART", 3 : "FG", 4 : "CONVERGENCE", 5 : "STOP", 6 : "WARNING", 7 : "ERROR", 8 : "ABNORMAL" } task_messages = { 0 : "", 301 : "", 302 : "", 401 : "NORM OF PROJECTED GRADIENT <= PGTOL", 402 : "RELATIVE REDUCTION OF F <= FACTR*EPSMCH", 501 : "CPU EXCEEDING THE TIME LIMIT", 502 : "TOTAL NO. OF F,G EVALUATIONS EXCEEDS LIMIT", 503 : "PROJECTED GRADIENT IS SUFFICIENTLY SMALL", 504 : "TOTAL NO. OF ITERATIONS REACHED LIMIT", 505 : "CALLBACK REQUESTED HALT", 601 : "ROUNDING ERRORS PREVENT PROGRESS", 602 : "STP = STPMAX", 603 : "STP = STPMIN", 604 : "XTOL TEST SATISFIED", 701 : "NO FEASIBLE SOLUTION", 702 : "FACTR < 0", 703 : "FTOL < 0", 704 : "GTOL < 0", 705 : "XTOL < 0", 706 : "STP < STPMIN", 707 : "STP > STPMAX", 708 : "STPMIN < 0", 709 : "STPMAX < STPMIN", 710 : "INITIAL G >= 0", 711 : "M <= 0", 712 : "N <= 0", 713 : "INVALID NBD", } def fmin_l_bfgs_b(func, x0, fprime=None, args=(), 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 minimize. 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 ``proj g_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 Deprecated option that previously controlled the text printed on the screen during the problem solution. Now the code does not emit any output and this keyword has no function. .. deprecated:: 1.15.0 This keyword is deprecated and will be removed from SciPy 1.17.0. disp : int, optional Deprecated option that previously controlled the text printed on the screen during the problem solution. Now the code does not emit any output and this keyword has no function. .. deprecated:: 1.15.0 This keyword is deprecated and will be removed from SciPy 1.17.0. maxfun : int, optional Maximum number of function evaluations. Note that this function may violate the limit because of evaluating gradients by numerical differentiation. 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 ----- SciPy uses a C-translated and modified version of the Fortran code, L-BFGS-B v3.0 (released April 25, 2011, BSD-3 licensed). Original Fortran version was written by Ciyou Zhu, Richard Byrd, Jorge Nocedal and, Jose Luis Morales. 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. Examples -------- Solve a linear regression problem via `fmin_l_bfgs_b`. To do this, first we define an objective function ``f(m, b) = (y - y_model)**2``, where `y` describes the observations and `y_model` the prediction of the linear model as ``y_model = m*x + b``. The bounds for the parameters, ``m`` and ``b``, are arbitrarily chosen as ``(0,5)`` and ``(5,10)`` for this example. >>> import numpy as np >>> from scipy.optimize import fmin_l_bfgs_b >>> X = np.arange(0, 10, 1) >>> M = 2 >>> B = 3 >>> Y = M * X + B >>> def func(parameters, *args): ... x = args[0] ... y = args[1] ... m, b = parameters ... y_model = m*x + b ... error = sum(np.power((y - y_model), 2)) ... return error >>> initial_values = np.array([0.0, 1.0]) >>> x_opt, f_opt, info = fmin_l_bfgs_b(func, x0=initial_values, args=(X, Y), ... approx_grad=True) >>> x_opt, f_opt array([1.99999999, 3.00000006]), 1.7746231151323805e-14 # may vary The optimized parameters in ``x_opt`` agree with the ground truth parameters ``m`` and ``b``. Next, let us perform a bound constrained optimization using the `bounds` parameter. >>> bounds = [(0, 5), (5, 10)] >>> x_opt, f_op, info = fmin_l_bfgs_b(func, x0=initial_values, args=(X, Y), ... approx_grad=True, bounds=bounds) >>> x_opt, f_opt array([1.65990508, 5.31649385]), 15.721334516453945 # may vary """ # 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 callback = _wrap_callback(callback) opts = {'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 def _minimize_lbfgsb(fun, x0, args=(), jac=None, bounds=None, disp=None, maxcor=10, ftol=2.2204460492503131e-09, gtol=1e-5, eps=1e-8, maxfun=15000, maxiter=15000, iprint=-1, callback=None, maxls=20, finite_diff_rel_step=None, **unknown_options): """ Minimize a scalar function of one or more variables using the L-BFGS-B algorithm. Options ------- disp : None or int Deprecated option that previously controlled the text printed on the screen during the problem solution. Now the code does not emit any output and this keyword has no function. .. deprecated:: 1.15.0 This keyword is deprecated and will be removed from SciPy 1.17.0. 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 ``proj g_i`` is the i-th component of the projected gradient. eps : float or ndarray If `jac is None` the absolute step size used for numerical approximation of the jacobian via forward differences. maxfun : int Maximum number of function evaluations. Note that this function may violate the limit because of evaluating gradients by numerical differentiation. maxiter : int Maximum number of iterations. iprint : int, optional Deprecated option that previously controlled the text printed on the screen during the problem solution. Now the code does not emit any output and this keyword has no function. .. deprecated:: 1.15.0 This keyword is deprecated and will be removed from SciPy 1.17.0. maxls : int, optional Maximum number of line search steps (per iteration). Default is 20. finite_diff_rel_step : None or array_like, optional If ``jac in ['2-point', '3-point', 'cs']`` the relative step size to use for numerical approximation of the jacobian. The absolute step size is computed as ``h = rel_step * sign(x) * max(1, abs(x))``, possibly adjusted to fit into the bounds. For ``method='3-point'`` the sign of `h` is ignored. If None (default) then step is selected automatically. 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 pgtol = gtol factr = ftol / np.finfo(float).eps x0 = asarray(x0).ravel() n, = x0.shape # historically old-style bounds were/are expected by lbfgsb. # That's still the case but we'll deal with new-style from here on, # it's easier if bounds is None: pass elif len(bounds) != n: raise ValueError('length of x0 != length of bounds') else: bounds = np.array(old_bound_to_new(bounds)) # check bounds if (bounds[0] > bounds[1]).any(): raise ValueError( "LBFGSB - one of the lower bounds is greater than an upper bound." ) # initial vector must lie within the bounds. Otherwise ScalarFunction and # approx_derivative will cause problems x0 = np.clip(x0, bounds[0], bounds[1]) # _prepare_scalar_function can use bounds=None to represent no bounds sf = _prepare_scalar_function(fun, x0, jac=jac, args=args, epsilon=eps, bounds=bounds, finite_diff_rel_step=finite_diff_rel_step) func_and_grad = sf.fun_and_grad nbd = zeros(n, np.int32) low_bnd = zeros(n, float64) upper_bnd = zeros(n, float64) bounds_map = {(-np.inf, np.inf): 0, (1, np.inf): 1, (1, 1): 2, (-np.inf, 1): 3} if bounds is not None: for i in range(0, n): L, U = bounds[0, i], bounds[1, i] if not np.isinf(L): low_bnd[i] = L L = 1 if not np.isinf(U): 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, dtype=np.float64) f = array(0.0, dtype=np.int32) g = zeros((n,), dtype=np.int32) wa = zeros(2*m*n + 5*n + 11*m*m + 8*m, float64) iwa = zeros(3*n, dtype=np.int32) task = zeros(2, dtype=np.int32) ln_task = zeros(2, dtype=np.int32) lsave = zeros(4, dtype=np.int32) isave = zeros(44, dtype=np.int32) dsave = zeros(29, dtype=float64) n_iterations = 0 while True: # g may become float32 if a user provides a function that calculates # the Jacobian in float32 (see gh-18730). The underlying code expects # float64, so upcast it g = g.astype(np.float64) # 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, lsave, isave, dsave, maxls, ln_task) if task[0] == 3: # 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[0] == 1: # new iteration n_iterations += 1 intermediate_result = OptimizeResult(x=x, fun=f) if _call_callback_maybe_halt(callback, intermediate_result): task[0] = 5 task[1] = 505 if n_iterations >= maxiter: task[0] = 5 task[1] = 504 elif sf.nfev > maxfun: task[0] = 5 task[1] = 502 else: break if task[0] == 4: warnflag = 0 elif sf.nfev > maxfun or n_iterations >= maxiter: warnflag = 1 else: warnflag = 2 # These two portions of the workspace are described in the mainlb # function docstring in "__lbfgsb.c", ws and wy arguments. s = wa[0: m*n].reshape(m, n) y = wa[m*n: 2*m*n].reshape(m, n) # 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]) msg = status_messages[task[0]] + ": " + task_messages[task[1]] return OptimizeResult(fun=f, jac=g, nfev=sf.nfev, njev=sf.ngev, nit=n_iterations, status=warnflag, message=msg, x=x, success=(warnflag == 0), hess_inv=hess_inv) class LbfgsInvHessProduct(LinearOperator): """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. """ def __init__(self, sk, yk): """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().__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) def _matvec(self, x): """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.empty(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 def todense(self): """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_arr = np.eye(*self.shape, dtype=self.dtype) Hk = I_arr for i in range(n_corrs): A1 = I_arr - s[i][:, np.newaxis] * y[i][np.newaxis, :] * rho[i] A2 = I_arr - 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