932 lines
37 KiB
Python
932 lines
37 KiB
Python
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"""Generic interface for least-square minimization."""
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from __future__ import division, print_function, absolute_import
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from warnings import warn
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import numpy as np
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from numpy.linalg import norm
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from scipy.sparse import issparse, csr_matrix
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from scipy.sparse.linalg import LinearOperator
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from scipy.optimize import _minpack, OptimizeResult
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from scipy.optimize._numdiff import approx_derivative, group_columns
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from scipy._lib.six import string_types
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from .trf import trf
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from .dogbox import dogbox
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from .common import EPS, in_bounds, make_strictly_feasible
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TERMINATION_MESSAGES = {
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-1: "Improper input parameters status returned from `leastsq`",
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0: "The maximum number of function evaluations is exceeded.",
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1: "`gtol` termination condition is satisfied.",
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2: "`ftol` termination condition is satisfied.",
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3: "`xtol` termination condition is satisfied.",
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4: "Both `ftol` and `xtol` termination conditions are satisfied."
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}
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FROM_MINPACK_TO_COMMON = {
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0: -1, # Improper input parameters from MINPACK.
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1: 2,
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2: 3,
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3: 4,
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4: 1,
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5: 0
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# There are 6, 7, 8 for too small tolerance parameters,
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# but we guard against it by checking ftol, xtol, gtol beforehand.
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}
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def call_minpack(fun, x0, jac, ftol, xtol, gtol, max_nfev, x_scale, diff_step):
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n = x0.size
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if diff_step is None:
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epsfcn = EPS
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else:
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epsfcn = diff_step**2
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# Compute MINPACK's `diag`, which is inverse of our `x_scale` and
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# ``x_scale='jac'`` corresponds to ``diag=None``.
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if isinstance(x_scale, string_types) and x_scale == 'jac':
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diag = None
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else:
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diag = 1 / x_scale
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full_output = True
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col_deriv = False
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factor = 100.0
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if jac is None:
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if max_nfev is None:
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# n squared to account for Jacobian evaluations.
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max_nfev = 100 * n * (n + 1)
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x, info, status = _minpack._lmdif(
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fun, x0, (), full_output, ftol, xtol, gtol,
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max_nfev, epsfcn, factor, diag)
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else:
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if max_nfev is None:
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max_nfev = 100 * n
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x, info, status = _minpack._lmder(
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fun, jac, x0, (), full_output, col_deriv,
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ftol, xtol, gtol, max_nfev, factor, diag)
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f = info['fvec']
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if callable(jac):
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J = jac(x)
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else:
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J = np.atleast_2d(approx_derivative(fun, x))
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cost = 0.5 * np.dot(f, f)
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g = J.T.dot(f)
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g_norm = norm(g, ord=np.inf)
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nfev = info['nfev']
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njev = info.get('njev', None)
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status = FROM_MINPACK_TO_COMMON[status]
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active_mask = np.zeros_like(x0, dtype=int)
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return OptimizeResult(
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x=x, cost=cost, fun=f, jac=J, grad=g, optimality=g_norm,
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active_mask=active_mask, nfev=nfev, njev=njev, status=status)
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def prepare_bounds(bounds, n):
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lb, ub = [np.asarray(b, dtype=float) for b in bounds]
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if lb.ndim == 0:
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lb = np.resize(lb, n)
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if ub.ndim == 0:
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ub = np.resize(ub, n)
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return lb, ub
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def check_tolerance(ftol, xtol, gtol):
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message = "{} is too low, setting to machine epsilon {}."
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if ftol < EPS:
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warn(message.format("`ftol`", EPS))
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ftol = EPS
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if xtol < EPS:
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warn(message.format("`xtol`", EPS))
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xtol = EPS
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if gtol < EPS:
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warn(message.format("`gtol`", EPS))
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gtol = EPS
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return ftol, xtol, gtol
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def check_x_scale(x_scale, x0):
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if isinstance(x_scale, string_types) and x_scale == 'jac':
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return x_scale
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try:
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x_scale = np.asarray(x_scale, dtype=float)
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valid = np.all(np.isfinite(x_scale)) and np.all(x_scale > 0)
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except (ValueError, TypeError):
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valid = False
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if not valid:
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raise ValueError("`x_scale` must be 'jac' or array_like with "
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"positive numbers.")
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if x_scale.ndim == 0:
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x_scale = np.resize(x_scale, x0.shape)
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if x_scale.shape != x0.shape:
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raise ValueError("Inconsistent shapes between `x_scale` and `x0`.")
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return x_scale
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def check_jac_sparsity(jac_sparsity, m, n):
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if jac_sparsity is None:
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return None
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if not issparse(jac_sparsity):
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jac_sparsity = np.atleast_2d(jac_sparsity)
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if jac_sparsity.shape != (m, n):
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raise ValueError("`jac_sparsity` has wrong shape.")
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return jac_sparsity, group_columns(jac_sparsity)
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# Loss functions.
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def huber(z, rho, cost_only):
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mask = z <= 1
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rho[0, mask] = z[mask]
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rho[0, ~mask] = 2 * z[~mask]**0.5 - 1
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if cost_only:
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return
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rho[1, mask] = 1
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rho[1, ~mask] = z[~mask]**-0.5
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rho[2, mask] = 0
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rho[2, ~mask] = -0.5 * z[~mask]**-1.5
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def soft_l1(z, rho, cost_only):
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t = 1 + z
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rho[0] = 2 * (t**0.5 - 1)
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if cost_only:
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return
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rho[1] = t**-0.5
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rho[2] = -0.5 * t**-1.5
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def cauchy(z, rho, cost_only):
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rho[0] = np.log1p(z)
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if cost_only:
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return
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t = 1 + z
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rho[1] = 1 / t
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rho[2] = -1 / t**2
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def arctan(z, rho, cost_only):
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rho[0] = np.arctan(z)
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if cost_only:
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return
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t = 1 + z**2
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rho[1] = 1 / t
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rho[2] = -2 * z / t**2
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IMPLEMENTED_LOSSES = dict(linear=None, huber=huber, soft_l1=soft_l1,
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cauchy=cauchy, arctan=arctan)
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def construct_loss_function(m, loss, f_scale):
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if loss == 'linear':
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return None
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if not callable(loss):
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loss = IMPLEMENTED_LOSSES[loss]
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rho = np.empty((3, m))
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def loss_function(f, cost_only=False):
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z = (f / f_scale) ** 2
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loss(z, rho, cost_only=cost_only)
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if cost_only:
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return 0.5 * f_scale ** 2 * np.sum(rho[0])
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rho[0] *= f_scale ** 2
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rho[2] /= f_scale ** 2
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return rho
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else:
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def loss_function(f, cost_only=False):
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z = (f / f_scale) ** 2
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rho = loss(z)
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if cost_only:
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return 0.5 * f_scale ** 2 * np.sum(rho[0])
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rho[0] *= f_scale ** 2
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rho[2] /= f_scale ** 2
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return rho
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return loss_function
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def least_squares(
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fun, x0, jac='2-point', bounds=(-np.inf, np.inf), method='trf',
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ftol=1e-8, xtol=1e-8, gtol=1e-8, x_scale=1.0, loss='linear',
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f_scale=1.0, diff_step=None, tr_solver=None, tr_options={},
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jac_sparsity=None, max_nfev=None, verbose=0, args=(), kwargs={}):
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"""Solve a nonlinear least-squares problem with bounds on the variables.
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Given the residuals f(x) (an m-dimensional real function of n real
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variables) and the loss function rho(s) (a scalar function), `least_squares`
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finds a local minimum of the cost function F(x)::
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minimize F(x) = 0.5 * sum(rho(f_i(x)**2), i = 0, ..., m - 1)
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subject to lb <= x <= ub
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The purpose of the loss function rho(s) is to reduce the influence of
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outliers on the solution.
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Parameters
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----------
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fun : callable
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Function which computes the vector of residuals, with the signature
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``fun(x, *args, **kwargs)``, i.e., the minimization proceeds with
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respect to its first argument. The argument ``x`` passed to this
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function is an ndarray of shape (n,) (never a scalar, even for n=1).
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It must return a 1-d array_like of shape (m,) or a scalar. If the
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argument ``x`` is complex or the function ``fun`` returns complex
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residuals, it must be wrapped in a real function of real arguments,
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as shown at the end of the Examples section.
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x0 : array_like with shape (n,) or float
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Initial guess on independent variables. If float, it will be treated
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as a 1-d array with one element.
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jac : {'2-point', '3-point', 'cs', callable}, optional
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Method of computing the Jacobian matrix (an m-by-n matrix, where
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element (i, j) is the partial derivative of f[i] with respect to
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x[j]). The keywords select a finite difference scheme for numerical
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estimation. The scheme '3-point' is more accurate, but requires
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twice as much operations compared to '2-point' (default). The
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scheme 'cs' uses complex steps, and while potentially the most
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accurate, it is applicable only when `fun` correctly handles
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complex inputs and can be analytically continued to the complex
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plane. Method 'lm' always uses the '2-point' scheme. If callable,
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it is used as ``jac(x, *args, **kwargs)`` and should return a
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good approximation (or the exact value) for the Jacobian as an
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array_like (np.atleast_2d is applied), a sparse matrix or a
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`scipy.sparse.linalg.LinearOperator`.
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bounds : 2-tuple of array_like, optional
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Lower and upper bounds on independent variables. Defaults to no bounds.
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Each array must match the size of `x0` or be a scalar, in the latter
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case a bound will be the same for all variables. Use ``np.inf`` with
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an appropriate sign to disable bounds on all or some variables.
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method : {'trf', 'dogbox', 'lm'}, optional
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Algorithm to perform minimization.
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* 'trf' : Trust Region Reflective algorithm, particularly suitable
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for large sparse problems with bounds. Generally robust method.
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* 'dogbox' : dogleg algorithm with rectangular trust regions,
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typical use case is small problems with bounds. Not recommended
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for problems with rank-deficient Jacobian.
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* 'lm' : Levenberg-Marquardt algorithm as implemented in MINPACK.
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Doesn't handle bounds and sparse Jacobians. Usually the most
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efficient method for small unconstrained problems.
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Default is 'trf'. See Notes for more information.
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ftol : float, optional
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Tolerance for termination by the change of the cost function. Default
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is 1e-8. The optimization process is stopped when ``dF < ftol * F``,
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and there was an adequate agreement between a local quadratic model and
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the true model in the last step.
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xtol : float, optional
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Tolerance for termination by the change of the independent variables.
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Default is 1e-8. The exact condition depends on the `method` used:
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* For 'trf' and 'dogbox' : ``norm(dx) < xtol * (xtol + norm(x))``
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* For 'lm' : ``Delta < xtol * norm(xs)``, where ``Delta`` is
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a trust-region radius and ``xs`` is the value of ``x``
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scaled according to `x_scale` parameter (see below).
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gtol : float, optional
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Tolerance for termination by the norm of the gradient. Default is 1e-8.
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The exact condition depends on a `method` used:
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* For 'trf' : ``norm(g_scaled, ord=np.inf) < gtol``, where
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``g_scaled`` is the value of the gradient scaled to account for
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the presence of the bounds [STIR]_.
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* For 'dogbox' : ``norm(g_free, ord=np.inf) < gtol``, where
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``g_free`` is the gradient with respect to the variables which
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are not in the optimal state on the boundary.
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* For 'lm' : the maximum absolute value of the cosine of angles
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between columns of the Jacobian and the residual vector is less
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than `gtol`, or the residual vector is zero.
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x_scale : array_like or 'jac', optional
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Characteristic scale of each variable. Setting `x_scale` is equivalent
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to reformulating the problem in scaled variables ``xs = x / x_scale``.
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An alternative view is that the size of a trust region along j-th
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dimension is proportional to ``x_scale[j]``. Improved convergence may
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be achieved by setting `x_scale` such that a step of a given size
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along any of the scaled variables has a similar effect on the cost
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function. If set to 'jac', the scale is iteratively updated using the
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inverse norms of the columns of the Jacobian matrix (as described in
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[JJMore]_).
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loss : str or callable, optional
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Determines the loss function. The following keyword values are allowed:
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* 'linear' (default) : ``rho(z) = z``. Gives a standard
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least-squares problem.
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* 'soft_l1' : ``rho(z) = 2 * ((1 + z)**0.5 - 1)``. The smooth
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approximation of l1 (absolute value) loss. Usually a good
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choice for robust least squares.
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* 'huber' : ``rho(z) = z if z <= 1 else 2*z**0.5 - 1``. Works
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similarly to 'soft_l1'.
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* 'cauchy' : ``rho(z) = ln(1 + z)``. Severely weakens outliers
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influence, but may cause difficulties in optimization process.
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* 'arctan' : ``rho(z) = arctan(z)``. Limits a maximum loss on
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a single residual, has properties similar to 'cauchy'.
|
||
|
|
|
||
|
|
If callable, it must take a 1-d ndarray ``z=f**2`` and return an
|
||
|
|
array_like with shape (3, m) where row 0 contains function values,
|
||
|
|
row 1 contains first derivatives and row 2 contains second
|
||
|
|
derivatives. Method 'lm' supports only 'linear' loss.
|
||
|
|
f_scale : float, optional
|
||
|
|
Value of soft margin between inlier and outlier residuals, default
|
||
|
|
is 1.0. The loss function is evaluated as follows
|
||
|
|
``rho_(f**2) = C**2 * rho(f**2 / C**2)``, where ``C`` is `f_scale`,
|
||
|
|
and ``rho`` is determined by `loss` parameter. This parameter has
|
||
|
|
no effect with ``loss='linear'``, but for other `loss` values it is
|
||
|
|
of crucial importance.
|
||
|
|
max_nfev : None or int, optional
|
||
|
|
Maximum number of function evaluations before the termination.
|
||
|
|
If None (default), the value is chosen automatically:
|
||
|
|
|
||
|
|
* For 'trf' and 'dogbox' : 100 * n.
|
||
|
|
* For 'lm' : 100 * n if `jac` is callable and 100 * n * (n + 1)
|
||
|
|
otherwise (because 'lm' counts function calls in Jacobian
|
||
|
|
estimation).
|
||
|
|
|
||
|
|
diff_step : None or array_like, optional
|
||
|
|
Determines the relative step size for the finite difference
|
||
|
|
approximation of the Jacobian. The actual step is computed as
|
||
|
|
``x * diff_step``. If None (default), then `diff_step` is taken to be
|
||
|
|
a conventional "optimal" power of machine epsilon for the finite
|
||
|
|
difference scheme used [NR]_.
|
||
|
|
tr_solver : {None, 'exact', 'lsmr'}, optional
|
||
|
|
Method for solving trust-region subproblems, relevant only for 'trf'
|
||
|
|
and 'dogbox' methods.
|
||
|
|
|
||
|
|
* 'exact' is suitable for not very large problems with dense
|
||
|
|
Jacobian matrices. The computational complexity per iteration is
|
||
|
|
comparable to a singular value decomposition of the Jacobian
|
||
|
|
matrix.
|
||
|
|
* 'lsmr' is suitable for problems with sparse and large Jacobian
|
||
|
|
matrices. It uses the iterative procedure
|
||
|
|
`scipy.sparse.linalg.lsmr` for finding a solution of a linear
|
||
|
|
least-squares problem and only requires matrix-vector product
|
||
|
|
evaluations.
|
||
|
|
|
||
|
|
If None (default) the solver is chosen based on the type of Jacobian
|
||
|
|
returned on the first iteration.
|
||
|
|
tr_options : dict, optional
|
||
|
|
Keyword options passed to trust-region solver.
|
||
|
|
|
||
|
|
* ``tr_solver='exact'``: `tr_options` are ignored.
|
||
|
|
* ``tr_solver='lsmr'``: options for `scipy.sparse.linalg.lsmr`.
|
||
|
|
Additionally ``method='trf'`` supports 'regularize' option
|
||
|
|
(bool, default is True) which adds a regularization term to the
|
||
|
|
normal equation, which improves convergence if the Jacobian is
|
||
|
|
rank-deficient [Byrd]_ (eq. 3.4).
|
||
|
|
|
||
|
|
jac_sparsity : {None, array_like, sparse matrix}, optional
|
||
|
|
Defines the sparsity structure of the Jacobian matrix for finite
|
||
|
|
difference estimation, its shape must be (m, n). If the Jacobian has
|
||
|
|
only few non-zero elements in *each* row, providing the sparsity
|
||
|
|
structure will greatly speed up the computations [Curtis]_. A zero
|
||
|
|
entry means that a corresponding element in the Jacobian is identically
|
||
|
|
zero. If provided, forces the use of 'lsmr' trust-region solver.
|
||
|
|
If None (default) then dense differencing will be used. Has no effect
|
||
|
|
for 'lm' method.
|
||
|
|
verbose : {0, 1, 2}, optional
|
||
|
|
Level of algorithm's verbosity:
|
||
|
|
|
||
|
|
* 0 (default) : work silently.
|
||
|
|
* 1 : display a termination report.
|
||
|
|
* 2 : display progress during iterations (not supported by 'lm'
|
||
|
|
method).
|
||
|
|
|
||
|
|
args, kwargs : tuple and dict, optional
|
||
|
|
Additional arguments passed to `fun` and `jac`. Both empty by default.
|
||
|
|
The calling signature is ``fun(x, *args, **kwargs)`` and the same for
|
||
|
|
`jac`.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
`OptimizeResult` with the following fields defined:
|
||
|
|
x : ndarray, shape (n,)
|
||
|
|
Solution found.
|
||
|
|
cost : float
|
||
|
|
Value of the cost function at the solution.
|
||
|
|
fun : ndarray, shape (m,)
|
||
|
|
Vector of residuals at the solution.
|
||
|
|
jac : ndarray, sparse matrix or LinearOperator, shape (m, n)
|
||
|
|
Modified Jacobian matrix at the solution, in the sense that J^T J
|
||
|
|
is a Gauss-Newton approximation of the Hessian of the cost function.
|
||
|
|
The type is the same as the one used by the algorithm.
|
||
|
|
grad : ndarray, shape (m,)
|
||
|
|
Gradient of the cost function at the solution.
|
||
|
|
optimality : float
|
||
|
|
First-order optimality measure. In unconstrained problems, it is always
|
||
|
|
the uniform norm of the gradient. In constrained problems, it is the
|
||
|
|
quantity which was compared with `gtol` during iterations.
|
||
|
|
active_mask : ndarray of int, shape (n,)
|
||
|
|
Each component shows whether a corresponding constraint is active
|
||
|
|
(that is, whether a variable is at the bound):
|
||
|
|
|
||
|
|
* 0 : a constraint is not active.
|
||
|
|
* -1 : a lower bound is active.
|
||
|
|
* 1 : an upper bound is active.
|
||
|
|
|
||
|
|
Might be somewhat arbitrary for 'trf' method as it generates a sequence
|
||
|
|
of strictly feasible iterates and `active_mask` is determined within a
|
||
|
|
tolerance threshold.
|
||
|
|
nfev : int
|
||
|
|
Number of function evaluations done. Methods 'trf' and 'dogbox' do not
|
||
|
|
count function calls for numerical Jacobian approximation, as opposed
|
||
|
|
to 'lm' method.
|
||
|
|
njev : int or None
|
||
|
|
Number of Jacobian evaluations done. If numerical Jacobian
|
||
|
|
approximation is used in 'lm' method, it is set to None.
|
||
|
|
status : int
|
||
|
|
The reason for algorithm termination:
|
||
|
|
|
||
|
|
* -1 : improper input parameters status returned from MINPACK.
|
||
|
|
* 0 : the maximum number of function evaluations is exceeded.
|
||
|
|
* 1 : `gtol` termination condition is satisfied.
|
||
|
|
* 2 : `ftol` termination condition is satisfied.
|
||
|
|
* 3 : `xtol` termination condition is satisfied.
|
||
|
|
* 4 : Both `ftol` and `xtol` termination conditions are satisfied.
|
||
|
|
|
||
|
|
message : str
|
||
|
|
Verbal description of the termination reason.
|
||
|
|
success : bool
|
||
|
|
True if one of the convergence criteria is satisfied (`status` > 0).
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
leastsq : A legacy wrapper for the MINPACK implementation of the
|
||
|
|
Levenberg-Marquadt algorithm.
|
||
|
|
curve_fit : Least-squares minimization applied to a curve fitting problem.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Method 'lm' (Levenberg-Marquardt) calls a wrapper over least-squares
|
||
|
|
algorithms implemented in MINPACK (lmder, lmdif). It runs the
|
||
|
|
Levenberg-Marquardt algorithm formulated as a trust-region type algorithm.
|
||
|
|
The implementation is based on paper [JJMore]_, it is very robust and
|
||
|
|
efficient with a lot of smart tricks. It should be your first choice
|
||
|
|
for unconstrained problems. Note that it doesn't support bounds. Also
|
||
|
|
it doesn't work when m < n.
|
||
|
|
|
||
|
|
Method 'trf' (Trust Region Reflective) is motivated by the process of
|
||
|
|
solving a system of equations, which constitute the first-order optimality
|
||
|
|
condition for a bound-constrained minimization problem as formulated in
|
||
|
|
[STIR]_. The algorithm iteratively solves trust-region subproblems
|
||
|
|
augmented by a special diagonal quadratic term and with trust-region shape
|
||
|
|
determined by the distance from the bounds and the direction of the
|
||
|
|
gradient. This enhancements help to avoid making steps directly into bounds
|
||
|
|
and efficiently explore the whole space of variables. To further improve
|
||
|
|
convergence, the algorithm considers search directions reflected from the
|
||
|
|
bounds. To obey theoretical requirements, the algorithm keeps iterates
|
||
|
|
strictly feasible. With dense Jacobians trust-region subproblems are
|
||
|
|
solved by an exact method very similar to the one described in [JJMore]_
|
||
|
|
(and implemented in MINPACK). The difference from the MINPACK
|
||
|
|
implementation is that a singular value decomposition of a Jacobian
|
||
|
|
matrix is done once per iteration, instead of a QR decomposition and series
|
||
|
|
of Givens rotation eliminations. For large sparse Jacobians a 2-d subspace
|
||
|
|
approach of solving trust-region subproblems is used [STIR]_, [Byrd]_.
|
||
|
|
The subspace is spanned by a scaled gradient and an approximate
|
||
|
|
Gauss-Newton solution delivered by `scipy.sparse.linalg.lsmr`. When no
|
||
|
|
constraints are imposed the algorithm is very similar to MINPACK and has
|
||
|
|
generally comparable performance. The algorithm works quite robust in
|
||
|
|
unbounded and bounded problems, thus it is chosen as a default algorithm.
|
||
|
|
|
||
|
|
Method 'dogbox' operates in a trust-region framework, but considers
|
||
|
|
rectangular trust regions as opposed to conventional ellipsoids [Voglis]_.
|
||
|
|
The intersection of a current trust region and initial bounds is again
|
||
|
|
rectangular, so on each iteration a quadratic minimization problem subject
|
||
|
|
to bound constraints is solved approximately by Powell's dogleg method
|
||
|
|
[NumOpt]_. The required Gauss-Newton step can be computed exactly for
|
||
|
|
dense Jacobians or approximately by `scipy.sparse.linalg.lsmr` for large
|
||
|
|
sparse Jacobians. The algorithm is likely to exhibit slow convergence when
|
||
|
|
the rank of Jacobian is less than the number of variables. The algorithm
|
||
|
|
often outperforms 'trf' in bounded problems with a small number of
|
||
|
|
variables.
|
||
|
|
|
||
|
|
Robust loss functions are implemented as described in [BA]_. The idea
|
||
|
|
is to modify a residual vector and a Jacobian matrix on each iteration
|
||
|
|
such that computed gradient and Gauss-Newton Hessian approximation match
|
||
|
|
the true gradient and Hessian approximation of the cost function. Then
|
||
|
|
the algorithm proceeds in a normal way, i.e. robust loss functions are
|
||
|
|
implemented as a simple wrapper over standard least-squares algorithms.
|
||
|
|
|
||
|
|
.. versionadded:: 0.17.0
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [STIR] M. A. Branch, T. F. Coleman, and Y. Li, "A Subspace, Interior,
|
||
|
|
and Conjugate Gradient Method for Large-Scale Bound-Constrained
|
||
|
|
Minimization Problems," SIAM Journal on Scientific Computing,
|
||
|
|
Vol. 21, Number 1, pp 1-23, 1999.
|
||
|
|
.. [NR] William H. Press et. al., "Numerical Recipes. The Art of Scientific
|
||
|
|
Computing. 3rd edition", Sec. 5.7.
|
||
|
|
.. [Byrd] R. H. Byrd, R. B. Schnabel and G. A. Shultz, "Approximate
|
||
|
|
solution of the trust region problem by minimization over
|
||
|
|
two-dimensional subspaces", Math. Programming, 40, pp. 247-263,
|
||
|
|
1988.
|
||
|
|
.. [Curtis] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of
|
||
|
|
sparse Jacobian matrices", Journal of the Institute of
|
||
|
|
Mathematics and its Applications, 13, pp. 117-120, 1974.
|
||
|
|
.. [JJMore] J. J. More, "The Levenberg-Marquardt Algorithm: Implementation
|
||
|
|
and Theory," Numerical Analysis, ed. G. A. Watson, Lecture
|
||
|
|
Notes in Mathematics 630, Springer Verlag, pp. 105-116, 1977.
|
||
|
|
.. [Voglis] C. Voglis and I. E. Lagaris, "A Rectangular Trust Region
|
||
|
|
Dogleg Approach for Unconstrained and Bound Constrained
|
||
|
|
Nonlinear Optimization", WSEAS International Conference on
|
||
|
|
Applied Mathematics, Corfu, Greece, 2004.
|
||
|
|
.. [NumOpt] J. Nocedal and S. J. Wright, "Numerical optimization,
|
||
|
|
2nd edition", Chapter 4.
|
||
|
|
.. [BA] B. Triggs et. al., "Bundle Adjustment - A Modern Synthesis",
|
||
|
|
Proceedings of the International Workshop on Vision Algorithms:
|
||
|
|
Theory and Practice, pp. 298-372, 1999.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
In this example we find a minimum of the Rosenbrock function without bounds
|
||
|
|
on independent variables.
|
||
|
|
|
||
|
|
>>> def fun_rosenbrock(x):
|
||
|
|
... return np.array([10 * (x[1] - x[0]**2), (1 - x[0])])
|
||
|
|
|
||
|
|
Notice that we only provide the vector of the residuals. The algorithm
|
||
|
|
constructs the cost function as a sum of squares of the residuals, which
|
||
|
|
gives the Rosenbrock function. The exact minimum is at ``x = [1.0, 1.0]``.
|
||
|
|
|
||
|
|
>>> from scipy.optimize import least_squares
|
||
|
|
>>> x0_rosenbrock = np.array([2, 2])
|
||
|
|
>>> res_1 = least_squares(fun_rosenbrock, x0_rosenbrock)
|
||
|
|
>>> res_1.x
|
||
|
|
array([ 1., 1.])
|
||
|
|
>>> res_1.cost
|
||
|
|
9.8669242910846867e-30
|
||
|
|
>>> res_1.optimality
|
||
|
|
8.8928864934219529e-14
|
||
|
|
|
||
|
|
We now constrain the variables, in such a way that the previous solution
|
||
|
|
becomes infeasible. Specifically, we require that ``x[1] >= 1.5``, and
|
||
|
|
``x[0]`` left unconstrained. To this end, we specify the `bounds` parameter
|
||
|
|
to `least_squares` in the form ``bounds=([-np.inf, 1.5], np.inf)``.
|
||
|
|
|
||
|
|
We also provide the analytic Jacobian:
|
||
|
|
|
||
|
|
>>> def jac_rosenbrock(x):
|
||
|
|
... return np.array([
|
||
|
|
... [-20 * x[0], 10],
|
||
|
|
... [-1, 0]])
|
||
|
|
|
||
|
|
Putting this all together, we see that the new solution lies on the bound:
|
||
|
|
|
||
|
|
>>> res_2 = least_squares(fun_rosenbrock, x0_rosenbrock, jac_rosenbrock,
|
||
|
|
... bounds=([-np.inf, 1.5], np.inf))
|
||
|
|
>>> res_2.x
|
||
|
|
array([ 1.22437075, 1.5 ])
|
||
|
|
>>> res_2.cost
|
||
|
|
0.025213093946805685
|
||
|
|
>>> res_2.optimality
|
||
|
|
1.5885401433157753e-07
|
||
|
|
|
||
|
|
Now we solve a system of equations (i.e., the cost function should be zero
|
||
|
|
at a minimum) for a Broyden tridiagonal vector-valued function of 100000
|
||
|
|
variables:
|
||
|
|
|
||
|
|
>>> def fun_broyden(x):
|
||
|
|
... f = (3 - x) * x + 1
|
||
|
|
... f[1:] -= x[:-1]
|
||
|
|
... f[:-1] -= 2 * x[1:]
|
||
|
|
... return f
|
||
|
|
|
||
|
|
The corresponding Jacobian matrix is sparse. We tell the algorithm to
|
||
|
|
estimate it by finite differences and provide the sparsity structure of
|
||
|
|
Jacobian to significantly speed up this process.
|
||
|
|
|
||
|
|
>>> from scipy.sparse import lil_matrix
|
||
|
|
>>> def sparsity_broyden(n):
|
||
|
|
... sparsity = lil_matrix((n, n), dtype=int)
|
||
|
|
... i = np.arange(n)
|
||
|
|
... sparsity[i, i] = 1
|
||
|
|
... i = np.arange(1, n)
|
||
|
|
... sparsity[i, i - 1] = 1
|
||
|
|
... i = np.arange(n - 1)
|
||
|
|
... sparsity[i, i + 1] = 1
|
||
|
|
... return sparsity
|
||
|
|
...
|
||
|
|
>>> n = 100000
|
||
|
|
>>> x0_broyden = -np.ones(n)
|
||
|
|
...
|
||
|
|
>>> res_3 = least_squares(fun_broyden, x0_broyden,
|
||
|
|
... jac_sparsity=sparsity_broyden(n))
|
||
|
|
>>> res_3.cost
|
||
|
|
4.5687069299604613e-23
|
||
|
|
>>> res_3.optimality
|
||
|
|
1.1650454296851518e-11
|
||
|
|
|
||
|
|
Let's also solve a curve fitting problem using robust loss function to
|
||
|
|
take care of outliers in the data. Define the model function as
|
||
|
|
``y = a + b * exp(c * t)``, where t is a predictor variable, y is an
|
||
|
|
observation and a, b, c are parameters to estimate.
|
||
|
|
|
||
|
|
First, define the function which generates the data with noise and
|
||
|
|
outliers, define the model parameters, and generate data:
|
||
|
|
|
||
|
|
>>> def gen_data(t, a, b, c, noise=0, n_outliers=0, random_state=0):
|
||
|
|
... y = a + b * np.exp(t * c)
|
||
|
|
...
|
||
|
|
... rnd = np.random.RandomState(random_state)
|
||
|
|
... error = noise * rnd.randn(t.size)
|
||
|
|
... outliers = rnd.randint(0, t.size, n_outliers)
|
||
|
|
... error[outliers] *= 10
|
||
|
|
...
|
||
|
|
... return y + error
|
||
|
|
...
|
||
|
|
>>> a = 0.5
|
||
|
|
>>> b = 2.0
|
||
|
|
>>> c = -1
|
||
|
|
>>> t_min = 0
|
||
|
|
>>> t_max = 10
|
||
|
|
>>> n_points = 15
|
||
|
|
...
|
||
|
|
>>> t_train = np.linspace(t_min, t_max, n_points)
|
||
|
|
>>> y_train = gen_data(t_train, a, b, c, noise=0.1, n_outliers=3)
|
||
|
|
|
||
|
|
Define function for computing residuals and initial estimate of
|
||
|
|
parameters.
|
||
|
|
|
||
|
|
>>> def fun(x, t, y):
|
||
|
|
... return x[0] + x[1] * np.exp(x[2] * t) - y
|
||
|
|
...
|
||
|
|
>>> x0 = np.array([1.0, 1.0, 0.0])
|
||
|
|
|
||
|
|
Compute a standard least-squares solution:
|
||
|
|
|
||
|
|
>>> res_lsq = least_squares(fun, x0, args=(t_train, y_train))
|
||
|
|
|
||
|
|
Now compute two solutions with two different robust loss functions. The
|
||
|
|
parameter `f_scale` is set to 0.1, meaning that inlier residuals should
|
||
|
|
not significantly exceed 0.1 (the noise level used).
|
||
|
|
|
||
|
|
>>> res_soft_l1 = least_squares(fun, x0, loss='soft_l1', f_scale=0.1,
|
||
|
|
... args=(t_train, y_train))
|
||
|
|
>>> res_log = least_squares(fun, x0, loss='cauchy', f_scale=0.1,
|
||
|
|
... args=(t_train, y_train))
|
||
|
|
|
||
|
|
And finally plot all the curves. We see that by selecting an appropriate
|
||
|
|
`loss` we can get estimates close to optimal even in the presence of
|
||
|
|
strong outliers. But keep in mind that generally it is recommended to try
|
||
|
|
'soft_l1' or 'huber' losses first (if at all necessary) as the other two
|
||
|
|
options may cause difficulties in optimization process.
|
||
|
|
|
||
|
|
>>> t_test = np.linspace(t_min, t_max, n_points * 10)
|
||
|
|
>>> y_true = gen_data(t_test, a, b, c)
|
||
|
|
>>> y_lsq = gen_data(t_test, *res_lsq.x)
|
||
|
|
>>> y_soft_l1 = gen_data(t_test, *res_soft_l1.x)
|
||
|
|
>>> y_log = gen_data(t_test, *res_log.x)
|
||
|
|
...
|
||
|
|
>>> import matplotlib.pyplot as plt
|
||
|
|
>>> plt.plot(t_train, y_train, 'o')
|
||
|
|
>>> plt.plot(t_test, y_true, 'k', linewidth=2, label='true')
|
||
|
|
>>> plt.plot(t_test, y_lsq, label='linear loss')
|
||
|
|
>>> plt.plot(t_test, y_soft_l1, label='soft_l1 loss')
|
||
|
|
>>> plt.plot(t_test, y_log, label='cauchy loss')
|
||
|
|
>>> plt.xlabel("t")
|
||
|
|
>>> plt.ylabel("y")
|
||
|
|
>>> plt.legend()
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
In the next example, we show how complex-valued residual functions of
|
||
|
|
complex variables can be optimized with ``least_squares()``. Consider the
|
||
|
|
following function:
|
||
|
|
|
||
|
|
>>> def f(z):
|
||
|
|
... return z - (0.5 + 0.5j)
|
||
|
|
|
||
|
|
We wrap it into a function of real variables that returns real residuals
|
||
|
|
by simply handling the real and imaginary parts as independent variables:
|
||
|
|
|
||
|
|
>>> def f_wrap(x):
|
||
|
|
... fx = f(x[0] + 1j*x[1])
|
||
|
|
... return np.array([fx.real, fx.imag])
|
||
|
|
|
||
|
|
Thus, instead of the original m-dimensional complex function of n complex
|
||
|
|
variables we optimize a 2m-dimensional real function of 2n real variables:
|
||
|
|
|
||
|
|
>>> from scipy.optimize import least_squares
|
||
|
|
>>> res_wrapped = least_squares(f_wrap, (0.1, 0.1), bounds=([0, 0], [1, 1]))
|
||
|
|
>>> z = res_wrapped.x[0] + res_wrapped.x[1]*1j
|
||
|
|
>>> z
|
||
|
|
(0.49999999999925893+0.49999999999925893j)
|
||
|
|
|
||
|
|
"""
|
||
|
|
if method not in ['trf', 'dogbox', 'lm']:
|
||
|
|
raise ValueError("`method` must be 'trf', 'dogbox' or 'lm'.")
|
||
|
|
|
||
|
|
if jac not in ['2-point', '3-point', 'cs'] and not callable(jac):
|
||
|
|
raise ValueError("`jac` must be '2-point', '3-point', 'cs' or "
|
||
|
|
"callable.")
|
||
|
|
|
||
|
|
if tr_solver not in [None, 'exact', 'lsmr']:
|
||
|
|
raise ValueError("`tr_solver` must be None, 'exact' or 'lsmr'.")
|
||
|
|
|
||
|
|
if loss not in IMPLEMENTED_LOSSES and not callable(loss):
|
||
|
|
raise ValueError("`loss` must be one of {0} or a callable."
|
||
|
|
.format(IMPLEMENTED_LOSSES.keys()))
|
||
|
|
|
||
|
|
if method == 'lm' and loss != 'linear':
|
||
|
|
raise ValueError("method='lm' supports only 'linear' loss function.")
|
||
|
|
|
||
|
|
if verbose not in [0, 1, 2]:
|
||
|
|
raise ValueError("`verbose` must be in [0, 1, 2].")
|
||
|
|
|
||
|
|
if len(bounds) != 2:
|
||
|
|
raise ValueError("`bounds` must contain 2 elements.")
|
||
|
|
|
||
|
|
if max_nfev is not None and max_nfev <= 0:
|
||
|
|
raise ValueError("`max_nfev` must be None or positive integer.")
|
||
|
|
|
||
|
|
if np.iscomplexobj(x0):
|
||
|
|
raise ValueError("`x0` must be real.")
|
||
|
|
|
||
|
|
x0 = np.atleast_1d(x0).astype(float)
|
||
|
|
|
||
|
|
if x0.ndim > 1:
|
||
|
|
raise ValueError("`x0` must have at most 1 dimension.")
|
||
|
|
|
||
|
|
lb, ub = prepare_bounds(bounds, x0.shape[0])
|
||
|
|
|
||
|
|
if method == 'lm' and not np.all((lb == -np.inf) & (ub == np.inf)):
|
||
|
|
raise ValueError("Method 'lm' doesn't support bounds.")
|
||
|
|
|
||
|
|
if lb.shape != x0.shape or ub.shape != x0.shape:
|
||
|
|
raise ValueError("Inconsistent shapes between bounds and `x0`.")
|
||
|
|
|
||
|
|
if np.any(lb >= ub):
|
||
|
|
raise ValueError("Each lower bound must be strictly less than each "
|
||
|
|
"upper bound.")
|
||
|
|
|
||
|
|
if not in_bounds(x0, lb, ub):
|
||
|
|
raise ValueError("`x0` is infeasible.")
|
||
|
|
|
||
|
|
x_scale = check_x_scale(x_scale, x0)
|
||
|
|
|
||
|
|
ftol, xtol, gtol = check_tolerance(ftol, xtol, gtol)
|
||
|
|
|
||
|
|
def fun_wrapped(x):
|
||
|
|
return np.atleast_1d(fun(x, *args, **kwargs))
|
||
|
|
|
||
|
|
if method == 'trf':
|
||
|
|
x0 = make_strictly_feasible(x0, lb, ub)
|
||
|
|
|
||
|
|
f0 = fun_wrapped(x0)
|
||
|
|
|
||
|
|
if f0.ndim != 1:
|
||
|
|
raise ValueError("`fun` must return at most 1-d array_like.")
|
||
|
|
|
||
|
|
if not np.all(np.isfinite(f0)):
|
||
|
|
raise ValueError("Residuals are not finite in the initial point.")
|
||
|
|
|
||
|
|
n = x0.size
|
||
|
|
m = f0.size
|
||
|
|
|
||
|
|
if method == 'lm' and m < n:
|
||
|
|
raise ValueError("Method 'lm' doesn't work when the number of "
|
||
|
|
"residuals is less than the number of variables.")
|
||
|
|
|
||
|
|
loss_function = construct_loss_function(m, loss, f_scale)
|
||
|
|
if callable(loss):
|
||
|
|
rho = loss_function(f0)
|
||
|
|
if rho.shape != (3, m):
|
||
|
|
raise ValueError("The return value of `loss` callable has wrong "
|
||
|
|
"shape.")
|
||
|
|
initial_cost = 0.5 * np.sum(rho[0])
|
||
|
|
elif loss_function is not None:
|
||
|
|
initial_cost = loss_function(f0, cost_only=True)
|
||
|
|
else:
|
||
|
|
initial_cost = 0.5 * np.dot(f0, f0)
|
||
|
|
|
||
|
|
if callable(jac):
|
||
|
|
J0 = jac(x0, *args, **kwargs)
|
||
|
|
|
||
|
|
if issparse(J0):
|
||
|
|
J0 = csr_matrix(J0)
|
||
|
|
|
||
|
|
def jac_wrapped(x, _=None):
|
||
|
|
return csr_matrix(jac(x, *args, **kwargs))
|
||
|
|
|
||
|
|
elif isinstance(J0, LinearOperator):
|
||
|
|
def jac_wrapped(x, _=None):
|
||
|
|
return jac(x, *args, **kwargs)
|
||
|
|
|
||
|
|
else:
|
||
|
|
J0 = np.atleast_2d(J0)
|
||
|
|
|
||
|
|
def jac_wrapped(x, _=None):
|
||
|
|
return np.atleast_2d(jac(x, *args, **kwargs))
|
||
|
|
|
||
|
|
else: # Estimate Jacobian by finite differences.
|
||
|
|
if method == 'lm':
|
||
|
|
if jac_sparsity is not None:
|
||
|
|
raise ValueError("method='lm' does not support "
|
||
|
|
"`jac_sparsity`.")
|
||
|
|
|
||
|
|
if jac != '2-point':
|
||
|
|
warn("jac='{0}' works equivalently to '2-point' "
|
||
|
|
"for method='lm'.".format(jac))
|
||
|
|
|
||
|
|
J0 = jac_wrapped = None
|
||
|
|
else:
|
||
|
|
if jac_sparsity is not None and tr_solver == 'exact':
|
||
|
|
raise ValueError("tr_solver='exact' is incompatible "
|
||
|
|
"with `jac_sparsity`.")
|
||
|
|
|
||
|
|
jac_sparsity = check_jac_sparsity(jac_sparsity, m, n)
|
||
|
|
|
||
|
|
def jac_wrapped(x, f):
|
||
|
|
J = approx_derivative(fun, x, rel_step=diff_step, method=jac,
|
||
|
|
f0=f, bounds=bounds, args=args,
|
||
|
|
kwargs=kwargs, sparsity=jac_sparsity)
|
||
|
|
if J.ndim != 2: # J is guaranteed not sparse.
|
||
|
|
J = np.atleast_2d(J)
|
||
|
|
|
||
|
|
return J
|
||
|
|
|
||
|
|
J0 = jac_wrapped(x0, f0)
|
||
|
|
|
||
|
|
if J0 is not None:
|
||
|
|
if J0.shape != (m, n):
|
||
|
|
raise ValueError(
|
||
|
|
"The return value of `jac` has wrong shape: expected {0}, "
|
||
|
|
"actual {1}.".format((m, n), J0.shape))
|
||
|
|
|
||
|
|
if not isinstance(J0, np.ndarray):
|
||
|
|
if method == 'lm':
|
||
|
|
raise ValueError("method='lm' works only with dense "
|
||
|
|
"Jacobian matrices.")
|
||
|
|
|
||
|
|
if tr_solver == 'exact':
|
||
|
|
raise ValueError(
|
||
|
|
"tr_solver='exact' works only with dense "
|
||
|
|
"Jacobian matrices.")
|
||
|
|
|
||
|
|
jac_scale = isinstance(x_scale, string_types) and x_scale == 'jac'
|
||
|
|
if isinstance(J0, LinearOperator) and jac_scale:
|
||
|
|
raise ValueError("x_scale='jac' can't be used when `jac` "
|
||
|
|
"returns LinearOperator.")
|
||
|
|
|
||
|
|
if tr_solver is None:
|
||
|
|
if isinstance(J0, np.ndarray):
|
||
|
|
tr_solver = 'exact'
|
||
|
|
else:
|
||
|
|
tr_solver = 'lsmr'
|
||
|
|
|
||
|
|
if method == 'lm':
|
||
|
|
result = call_minpack(fun_wrapped, x0, jac_wrapped, ftol, xtol, gtol,
|
||
|
|
max_nfev, x_scale, diff_step)
|
||
|
|
|
||
|
|
elif method == 'trf':
|
||
|
|
result = trf(fun_wrapped, jac_wrapped, x0, f0, J0, lb, ub, ftol, xtol,
|
||
|
|
gtol, max_nfev, x_scale, loss_function, tr_solver,
|
||
|
|
tr_options.copy(), verbose)
|
||
|
|
|
||
|
|
elif method == 'dogbox':
|
||
|
|
if tr_solver == 'lsmr' and 'regularize' in tr_options:
|
||
|
|
warn("The keyword 'regularize' in `tr_options` is not relevant "
|
||
|
|
"for 'dogbox' method.")
|
||
|
|
tr_options = tr_options.copy()
|
||
|
|
del tr_options['regularize']
|
||
|
|
|
||
|
|
result = dogbox(fun_wrapped, jac_wrapped, x0, f0, J0, lb, ub, ftol,
|
||
|
|
xtol, gtol, max_nfev, x_scale, loss_function,
|
||
|
|
tr_solver, tr_options, verbose)
|
||
|
|
|
||
|
|
result.message = TERMINATION_MESSAGES[result.status]
|
||
|
|
result.success = result.status > 0
|
||
|
|
|
||
|
|
if verbose >= 1:
|
||
|
|
print(result.message)
|
||
|
|
print("Function evaluations {0}, initial cost {1:.4e}, final cost "
|
||
|
|
"{2:.4e}, first-order optimality {3:.2e}."
|
||
|
|
.format(result.nfev, initial_cost, result.cost,
|
||
|
|
result.optimality))
|
||
|
|
|
||
|
|
return result
|