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782 lines
30 KiB
Python
782 lines
30 KiB
Python
from itertools import product
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import numpy as np
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from numpy.linalg import norm
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from numpy.testing import (assert_, assert_allclose,
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assert_equal, suppress_warnings)
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from pytest import raises as assert_raises
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from scipy.sparse import issparse, lil_matrix
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from scipy.sparse.linalg import aslinearoperator
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from scipy.optimize import least_squares
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from scipy.optimize._lsq.least_squares import IMPLEMENTED_LOSSES
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from scipy.optimize._lsq.common import EPS, make_strictly_feasible
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def fun_trivial(x, a=0):
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return (x - a)**2 + 5.0
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def jac_trivial(x, a=0.0):
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return 2 * (x - a)
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def fun_2d_trivial(x):
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return np.array([x[0], x[1]])
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def jac_2d_trivial(x):
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return np.identity(2)
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def fun_rosenbrock(x):
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return np.array([10 * (x[1] - x[0]**2), (1 - x[0])])
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def jac_rosenbrock(x):
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return np.array([
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[-20 * x[0], 10],
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[-1, 0]
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])
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def jac_rosenbrock_bad_dim(x):
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return np.array([
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[-20 * x[0], 10],
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[-1, 0],
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[0.0, 0.0]
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])
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def fun_rosenbrock_cropped(x):
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return fun_rosenbrock(x)[0]
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def jac_rosenbrock_cropped(x):
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return jac_rosenbrock(x)[0]
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# When x is 1-D array, return is 2-D array.
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def fun_wrong_dimensions(x):
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return np.array([x, x**2, x**3])
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def jac_wrong_dimensions(x, a=0.0):
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return np.atleast_3d(jac_trivial(x, a=a))
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def fun_bvp(x):
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n = int(np.sqrt(x.shape[0]))
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u = np.zeros((n + 2, n + 2))
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x = x.reshape((n, n))
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u[1:-1, 1:-1] = x
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y = u[:-2, 1:-1] + u[2:, 1:-1] + u[1:-1, :-2] + u[1:-1, 2:] - 4 * x + x**3
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return y.ravel()
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class BroydenTridiagonal(object):
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def __init__(self, n=100, mode='sparse'):
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np.random.seed(0)
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self.n = n
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self.x0 = -np.ones(n)
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self.lb = np.linspace(-2, -1.5, n)
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self.ub = np.linspace(-0.8, 0.0, n)
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self.lb += 0.1 * np.random.randn(n)
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self.ub += 0.1 * np.random.randn(n)
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self.x0 += 0.1 * np.random.randn(n)
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self.x0 = make_strictly_feasible(self.x0, self.lb, self.ub)
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if mode == 'sparse':
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self.sparsity = lil_matrix((n, n), dtype=int)
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i = np.arange(n)
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self.sparsity[i, i] = 1
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i = np.arange(1, n)
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self.sparsity[i, i - 1] = 1
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i = np.arange(n - 1)
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self.sparsity[i, i + 1] = 1
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self.jac = self._jac
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elif mode == 'operator':
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self.jac = lambda x: aslinearoperator(self._jac(x))
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elif mode == 'dense':
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self.sparsity = None
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self.jac = lambda x: self._jac(x).toarray()
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else:
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assert_(False)
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def fun(self, x):
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f = (3 - x) * x + 1
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f[1:] -= x[:-1]
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f[:-1] -= 2 * x[1:]
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return f
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def _jac(self, x):
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J = lil_matrix((self.n, self.n))
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i = np.arange(self.n)
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J[i, i] = 3 - 2 * x
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i = np.arange(1, self.n)
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J[i, i - 1] = -1
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i = np.arange(self.n - 1)
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J[i, i + 1] = -2
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return J
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class ExponentialFittingProblem(object):
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"""Provide data and function for exponential fitting in the form
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y = a + exp(b * x) + noise."""
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def __init__(self, a, b, noise, n_outliers=1, x_range=(-1, 1),
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n_points=11, random_seed=None):
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np.random.seed(random_seed)
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self.m = n_points
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self.n = 2
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self.p0 = np.zeros(2)
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self.x = np.linspace(x_range[0], x_range[1], n_points)
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self.y = a + np.exp(b * self.x)
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self.y += noise * np.random.randn(self.m)
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outliers = np.random.randint(0, self.m, n_outliers)
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self.y[outliers] += 50 * noise * np.random.rand(n_outliers)
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self.p_opt = np.array([a, b])
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def fun(self, p):
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return p[0] + np.exp(p[1] * self.x) - self.y
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def jac(self, p):
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J = np.empty((self.m, self.n))
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J[:, 0] = 1
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J[:, 1] = self.x * np.exp(p[1] * self.x)
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return J
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def cubic_soft_l1(z):
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rho = np.empty((3, z.size))
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t = 1 + z
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rho[0] = 3 * (t**(1/3) - 1)
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rho[1] = t ** (-2/3)
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rho[2] = -2/3 * t**(-5/3)
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return rho
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LOSSES = list(IMPLEMENTED_LOSSES.keys()) + [cubic_soft_l1]
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class BaseMixin(object):
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def test_basic(self):
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# Test that the basic calling sequence works.
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res = least_squares(fun_trivial, 2., method=self.method)
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assert_allclose(res.x, 0, atol=1e-4)
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assert_allclose(res.fun, fun_trivial(res.x))
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def test_args_kwargs(self):
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# Test that args and kwargs are passed correctly to the functions.
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a = 3.0
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for jac in ['2-point', '3-point', 'cs', jac_trivial]:
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with suppress_warnings() as sup:
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sup.filter(UserWarning,
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"jac='(3-point|cs)' works equivalently to '2-point' for method='lm'")
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res = least_squares(fun_trivial, 2.0, jac, args=(a,),
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method=self.method)
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res1 = least_squares(fun_trivial, 2.0, jac, kwargs={'a': a},
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method=self.method)
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assert_allclose(res.x, a, rtol=1e-4)
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assert_allclose(res1.x, a, rtol=1e-4)
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assert_raises(TypeError, least_squares, fun_trivial, 2.0,
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args=(3, 4,), method=self.method)
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assert_raises(TypeError, least_squares, fun_trivial, 2.0,
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kwargs={'kaboom': 3}, method=self.method)
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def test_jac_options(self):
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for jac in ['2-point', '3-point', 'cs', jac_trivial]:
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with suppress_warnings() as sup:
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sup.filter(UserWarning,
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"jac='(3-point|cs)' works equivalently to '2-point' for method='lm'")
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res = least_squares(fun_trivial, 2.0, jac, method=self.method)
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assert_allclose(res.x, 0, atol=1e-4)
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assert_raises(ValueError, least_squares, fun_trivial, 2.0, jac='oops',
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method=self.method)
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def test_nfev_options(self):
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for max_nfev in [None, 20]:
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res = least_squares(fun_trivial, 2.0, max_nfev=max_nfev,
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method=self.method)
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assert_allclose(res.x, 0, atol=1e-4)
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def test_x_scale_options(self):
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for x_scale in [1.0, np.array([0.5]), 'jac']:
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res = least_squares(fun_trivial, 2.0, x_scale=x_scale)
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assert_allclose(res.x, 0)
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assert_raises(ValueError, least_squares, fun_trivial,
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2.0, x_scale='auto', method=self.method)
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assert_raises(ValueError, least_squares, fun_trivial,
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2.0, x_scale=-1.0, method=self.method)
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assert_raises(ValueError, least_squares, fun_trivial,
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2.0, x_scale=None, method=self.method)
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assert_raises(ValueError, least_squares, fun_trivial,
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2.0, x_scale=1.0+2.0j, method=self.method)
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def test_diff_step(self):
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# res1 and res2 should be equivalent.
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# res2 and res3 should be different.
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res1 = least_squares(fun_trivial, 2.0, diff_step=1e-1,
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method=self.method)
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res2 = least_squares(fun_trivial, 2.0, diff_step=-1e-1,
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method=self.method)
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res3 = least_squares(fun_trivial, 2.0,
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diff_step=None, method=self.method)
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assert_allclose(res1.x, 0, atol=1e-4)
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assert_allclose(res2.x, 0, atol=1e-4)
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assert_allclose(res3.x, 0, atol=1e-4)
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assert_equal(res1.x, res2.x)
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assert_equal(res1.nfev, res2.nfev)
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assert_(res2.nfev != res3.nfev)
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def test_incorrect_options_usage(self):
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assert_raises(TypeError, least_squares, fun_trivial, 2.0,
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method=self.method, options={'no_such_option': 100})
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assert_raises(TypeError, least_squares, fun_trivial, 2.0,
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method=self.method, options={'max_nfev': 100})
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def test_full_result(self):
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# MINPACK doesn't work very well with factor=100 on this problem,
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# thus using low 'atol'.
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res = least_squares(fun_trivial, 2.0, method=self.method)
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assert_allclose(res.x, 0, atol=1e-4)
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assert_allclose(res.cost, 12.5)
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assert_allclose(res.fun, 5)
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assert_allclose(res.jac, 0, atol=1e-4)
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assert_allclose(res.grad, 0, atol=1e-2)
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assert_allclose(res.optimality, 0, atol=1e-2)
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assert_equal(res.active_mask, 0)
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if self.method == 'lm':
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assert_(res.nfev < 30)
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assert_(res.njev is None)
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else:
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assert_(res.nfev < 10)
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assert_(res.njev < 10)
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assert_(res.status > 0)
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assert_(res.success)
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def test_full_result_single_fev(self):
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# MINPACK checks the number of nfev after the iteration,
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# so it's hard to tell what he is going to compute.
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if self.method == 'lm':
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return
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res = least_squares(fun_trivial, 2.0, method=self.method,
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max_nfev=1)
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assert_equal(res.x, np.array([2]))
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assert_equal(res.cost, 40.5)
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assert_equal(res.fun, np.array([9]))
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assert_equal(res.jac, np.array([[4]]))
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assert_equal(res.grad, np.array([36]))
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assert_equal(res.optimality, 36)
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assert_equal(res.active_mask, np.array([0]))
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assert_equal(res.nfev, 1)
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assert_equal(res.njev, 1)
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assert_equal(res.status, 0)
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assert_equal(res.success, 0)
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def test_rosenbrock(self):
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x0 = [-2, 1]
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x_opt = [1, 1]
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for jac, x_scale, tr_solver in product(
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['2-point', '3-point', 'cs', jac_rosenbrock],
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[1.0, np.array([1.0, 0.2]), 'jac'],
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['exact', 'lsmr']):
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with suppress_warnings() as sup:
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sup.filter(UserWarning,
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"jac='(3-point|cs)' works equivalently to '2-point' for method='lm'")
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res = least_squares(fun_rosenbrock, x0, jac, x_scale=x_scale,
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tr_solver=tr_solver, method=self.method)
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assert_allclose(res.x, x_opt)
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def test_rosenbrock_cropped(self):
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x0 = [-2, 1]
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if self.method == 'lm':
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assert_raises(ValueError, least_squares, fun_rosenbrock_cropped,
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x0, method='lm')
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else:
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for jac, x_scale, tr_solver in product(
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['2-point', '3-point', 'cs', jac_rosenbrock_cropped],
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[1.0, np.array([1.0, 0.2]), 'jac'],
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['exact', 'lsmr']):
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res = least_squares(
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fun_rosenbrock_cropped, x0, jac, x_scale=x_scale,
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tr_solver=tr_solver, method=self.method)
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assert_allclose(res.cost, 0, atol=1e-14)
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def test_fun_wrong_dimensions(self):
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assert_raises(ValueError, least_squares, fun_wrong_dimensions,
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2.0, method=self.method)
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def test_jac_wrong_dimensions(self):
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assert_raises(ValueError, least_squares, fun_trivial,
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2.0, jac_wrong_dimensions, method=self.method)
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def test_fun_and_jac_inconsistent_dimensions(self):
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x0 = [1, 2]
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assert_raises(ValueError, least_squares, fun_rosenbrock, x0,
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jac_rosenbrock_bad_dim, method=self.method)
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def test_x0_multidimensional(self):
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x0 = np.ones(4).reshape(2, 2)
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assert_raises(ValueError, least_squares, fun_trivial, x0,
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method=self.method)
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def test_x0_complex_scalar(self):
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x0 = 2.0 + 0.0*1j
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assert_raises(ValueError, least_squares, fun_trivial, x0,
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method=self.method)
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def test_x0_complex_array(self):
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x0 = [1.0, 2.0 + 0.0*1j]
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assert_raises(ValueError, least_squares, fun_trivial, x0,
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method=self.method)
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def test_bvp(self):
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# This test was introduced with fix #5556. It turned out that
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# dogbox solver had a bug with trust-region radius update, which
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# could block its progress and create an infinite loop. And this
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# discrete boundary value problem is the one which triggers it.
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n = 10
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x0 = np.ones(n**2)
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if self.method == 'lm':
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max_nfev = 5000 # To account for Jacobian estimation.
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else:
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max_nfev = 100
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res = least_squares(fun_bvp, x0, ftol=1e-2, method=self.method,
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max_nfev=max_nfev)
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assert_(res.nfev < max_nfev)
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assert_(res.cost < 0.5)
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def test_error_raised_when_all_tolerances_below_eps(self):
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# Test that all 0 tolerances are not allowed.
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assert_raises(ValueError, least_squares, fun_trivial, 2.0,
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method=self.method, ftol=None, xtol=None, gtol=None)
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def test_convergence_with_only_one_tolerance_enabled(self):
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if self.method == 'lm':
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return # should not do test
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x0 = [-2, 1]
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x_opt = [1, 1]
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for ftol, xtol, gtol in [(1e-8, None, None),
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(None, 1e-8, None),
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(None, None, 1e-8)]:
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res = least_squares(fun_rosenbrock, x0, jac=jac_rosenbrock,
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ftol=ftol, gtol=gtol, xtol=xtol,
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method=self.method)
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assert_allclose(res.x, x_opt)
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class BoundsMixin(object):
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def test_inconsistent(self):
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assert_raises(ValueError, least_squares, fun_trivial, 2.0,
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bounds=(10.0, 0.0), method=self.method)
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def test_infeasible(self):
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assert_raises(ValueError, least_squares, fun_trivial, 2.0,
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bounds=(3., 4), method=self.method)
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def test_wrong_number(self):
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assert_raises(ValueError, least_squares, fun_trivial, 2.,
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bounds=(1., 2, 3), method=self.method)
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def test_inconsistent_shape(self):
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assert_raises(ValueError, least_squares, fun_trivial, 2.0,
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bounds=(1.0, [2.0, 3.0]), method=self.method)
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# 1-D array wont't be broadcasted
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assert_raises(ValueError, least_squares, fun_rosenbrock, [1.0, 2.0],
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bounds=([0.0], [3.0, 4.0]), method=self.method)
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def test_in_bounds(self):
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for jac in ['2-point', '3-point', 'cs', jac_trivial]:
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res = least_squares(fun_trivial, 2.0, jac=jac,
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bounds=(-1.0, 3.0), method=self.method)
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assert_allclose(res.x, 0.0, atol=1e-4)
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assert_equal(res.active_mask, [0])
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assert_(-1 <= res.x <= 3)
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res = least_squares(fun_trivial, 2.0, jac=jac,
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bounds=(0.5, 3.0), method=self.method)
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assert_allclose(res.x, 0.5, atol=1e-4)
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assert_equal(res.active_mask, [-1])
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assert_(0.5 <= res.x <= 3)
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def test_bounds_shape(self):
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for jac in ['2-point', '3-point', 'cs', jac_2d_trivial]:
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x0 = [1.0, 1.0]
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res = least_squares(fun_2d_trivial, x0, jac=jac)
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assert_allclose(res.x, [0.0, 0.0])
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res = least_squares(fun_2d_trivial, x0, jac=jac,
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bounds=(0.5, [2.0, 2.0]), method=self.method)
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assert_allclose(res.x, [0.5, 0.5])
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res = least_squares(fun_2d_trivial, x0, jac=jac,
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bounds=([0.3, 0.2], 3.0), method=self.method)
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assert_allclose(res.x, [0.3, 0.2])
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res = least_squares(
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fun_2d_trivial, x0, jac=jac, bounds=([-1, 0.5], [1.0, 3.0]),
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method=self.method)
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assert_allclose(res.x, [0.0, 0.5], atol=1e-5)
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def test_rosenbrock_bounds(self):
|
|
x0_1 = np.array([-2.0, 1.0])
|
|
x0_2 = np.array([2.0, 2.0])
|
|
x0_3 = np.array([-2.0, 2.0])
|
|
x0_4 = np.array([0.0, 2.0])
|
|
x0_5 = np.array([-1.2, 1.0])
|
|
problems = [
|
|
(x0_1, ([-np.inf, -1.5], np.inf)),
|
|
(x0_2, ([-np.inf, 1.5], np.inf)),
|
|
(x0_3, ([-np.inf, 1.5], np.inf)),
|
|
(x0_4, ([-np.inf, 1.5], [1.0, np.inf])),
|
|
(x0_2, ([1.0, 1.5], [3.0, 3.0])),
|
|
(x0_5, ([-50.0, 0.0], [0.5, 100]))
|
|
]
|
|
for x0, bounds in problems:
|
|
for jac, x_scale, tr_solver in product(
|
|
['2-point', '3-point', 'cs', jac_rosenbrock],
|
|
[1.0, [1.0, 0.5], 'jac'],
|
|
['exact', 'lsmr']):
|
|
res = least_squares(fun_rosenbrock, x0, jac, bounds,
|
|
x_scale=x_scale, tr_solver=tr_solver,
|
|
method=self.method)
|
|
assert_allclose(res.optimality, 0.0, atol=1e-5)
|
|
|
|
|
|
class SparseMixin(object):
|
|
def test_exact_tr_solver(self):
|
|
p = BroydenTridiagonal()
|
|
assert_raises(ValueError, least_squares, p.fun, p.x0, p.jac,
|
|
tr_solver='exact', method=self.method)
|
|
assert_raises(ValueError, least_squares, p.fun, p.x0,
|
|
tr_solver='exact', jac_sparsity=p.sparsity,
|
|
method=self.method)
|
|
|
|
def test_equivalence(self):
|
|
sparse = BroydenTridiagonal(mode='sparse')
|
|
dense = BroydenTridiagonal(mode='dense')
|
|
res_sparse = least_squares(
|
|
sparse.fun, sparse.x0, jac=sparse.jac,
|
|
method=self.method)
|
|
res_dense = least_squares(
|
|
dense.fun, dense.x0, jac=sparse.jac,
|
|
method=self.method)
|
|
assert_equal(res_sparse.nfev, res_dense.nfev)
|
|
assert_allclose(res_sparse.x, res_dense.x, atol=1e-20)
|
|
assert_allclose(res_sparse.cost, 0, atol=1e-20)
|
|
assert_allclose(res_dense.cost, 0, atol=1e-20)
|
|
|
|
def test_tr_options(self):
|
|
p = BroydenTridiagonal()
|
|
res = least_squares(p.fun, p.x0, p.jac, method=self.method,
|
|
tr_options={'btol': 1e-10})
|
|
assert_allclose(res.cost, 0, atol=1e-20)
|
|
|
|
def test_wrong_parameters(self):
|
|
p = BroydenTridiagonal()
|
|
assert_raises(ValueError, least_squares, p.fun, p.x0, p.jac,
|
|
tr_solver='best', method=self.method)
|
|
assert_raises(TypeError, least_squares, p.fun, p.x0, p.jac,
|
|
tr_solver='lsmr', tr_options={'tol': 1e-10})
|
|
|
|
def test_solver_selection(self):
|
|
sparse = BroydenTridiagonal(mode='sparse')
|
|
dense = BroydenTridiagonal(mode='dense')
|
|
res_sparse = least_squares(sparse.fun, sparse.x0, jac=sparse.jac,
|
|
method=self.method)
|
|
res_dense = least_squares(dense.fun, dense.x0, jac=dense.jac,
|
|
method=self.method)
|
|
assert_allclose(res_sparse.cost, 0, atol=1e-20)
|
|
assert_allclose(res_dense.cost, 0, atol=1e-20)
|
|
assert_(issparse(res_sparse.jac))
|
|
assert_(isinstance(res_dense.jac, np.ndarray))
|
|
|
|
def test_numerical_jac(self):
|
|
p = BroydenTridiagonal()
|
|
for jac in ['2-point', '3-point', 'cs']:
|
|
res_dense = least_squares(p.fun, p.x0, jac, method=self.method)
|
|
res_sparse = least_squares(
|
|
p.fun, p.x0, jac,method=self.method,
|
|
jac_sparsity=p.sparsity)
|
|
assert_equal(res_dense.nfev, res_sparse.nfev)
|
|
assert_allclose(res_dense.x, res_sparse.x, atol=1e-20)
|
|
assert_allclose(res_dense.cost, 0, atol=1e-20)
|
|
assert_allclose(res_sparse.cost, 0, atol=1e-20)
|
|
|
|
def test_with_bounds(self):
|
|
p = BroydenTridiagonal()
|
|
for jac, jac_sparsity in product(
|
|
[p.jac, '2-point', '3-point', 'cs'], [None, p.sparsity]):
|
|
res_1 = least_squares(
|
|
p.fun, p.x0, jac, bounds=(p.lb, np.inf),
|
|
method=self.method,jac_sparsity=jac_sparsity)
|
|
res_2 = least_squares(
|
|
p.fun, p.x0, jac, bounds=(-np.inf, p.ub),
|
|
method=self.method, jac_sparsity=jac_sparsity)
|
|
res_3 = least_squares(
|
|
p.fun, p.x0, jac, bounds=(p.lb, p.ub),
|
|
method=self.method, jac_sparsity=jac_sparsity)
|
|
assert_allclose(res_1.optimality, 0, atol=1e-10)
|
|
assert_allclose(res_2.optimality, 0, atol=1e-10)
|
|
assert_allclose(res_3.optimality, 0, atol=1e-10)
|
|
|
|
def test_wrong_jac_sparsity(self):
|
|
p = BroydenTridiagonal()
|
|
sparsity = p.sparsity[:-1]
|
|
assert_raises(ValueError, least_squares, p.fun, p.x0,
|
|
jac_sparsity=sparsity, method=self.method)
|
|
|
|
def test_linear_operator(self):
|
|
p = BroydenTridiagonal(mode='operator')
|
|
res = least_squares(p.fun, p.x0, p.jac, method=self.method)
|
|
assert_allclose(res.cost, 0.0, atol=1e-20)
|
|
assert_raises(ValueError, least_squares, p.fun, p.x0, p.jac,
|
|
method=self.method, tr_solver='exact')
|
|
|
|
def test_x_scale_jac_scale(self):
|
|
p = BroydenTridiagonal()
|
|
res = least_squares(p.fun, p.x0, p.jac, method=self.method,
|
|
x_scale='jac')
|
|
assert_allclose(res.cost, 0.0, atol=1e-20)
|
|
|
|
p = BroydenTridiagonal(mode='operator')
|
|
assert_raises(ValueError, least_squares, p.fun, p.x0, p.jac,
|
|
method=self.method, x_scale='jac')
|
|
|
|
|
|
class LossFunctionMixin(object):
|
|
def test_options(self):
|
|
for loss in LOSSES:
|
|
res = least_squares(fun_trivial, 2.0, loss=loss,
|
|
method=self.method)
|
|
assert_allclose(res.x, 0, atol=1e-15)
|
|
|
|
assert_raises(ValueError, least_squares, fun_trivial, 2.0,
|
|
loss='hinge', method=self.method)
|
|
|
|
def test_fun(self):
|
|
# Test that res.fun is actual residuals, and not modified by loss
|
|
# function stuff.
|
|
for loss in LOSSES:
|
|
res = least_squares(fun_trivial, 2.0, loss=loss,
|
|
method=self.method)
|
|
assert_equal(res.fun, fun_trivial(res.x))
|
|
|
|
def test_grad(self):
|
|
# Test that res.grad is true gradient of loss function at the
|
|
# solution. Use max_nfev = 1, to avoid reaching minimum.
|
|
x = np.array([2.0]) # res.x will be this.
|
|
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss='linear',
|
|
max_nfev=1, method=self.method)
|
|
assert_equal(res.grad, 2 * x * (x**2 + 5))
|
|
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss='huber',
|
|
max_nfev=1, method=self.method)
|
|
assert_equal(res.grad, 2 * x)
|
|
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss='soft_l1',
|
|
max_nfev=1, method=self.method)
|
|
assert_allclose(res.grad,
|
|
2 * x * (x**2 + 5) / (1 + (x**2 + 5)**2)**0.5)
|
|
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss='cauchy',
|
|
max_nfev=1, method=self.method)
|
|
assert_allclose(res.grad, 2 * x * (x**2 + 5) / (1 + (x**2 + 5)**2))
|
|
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss='arctan',
|
|
max_nfev=1, method=self.method)
|
|
assert_allclose(res.grad, 2 * x * (x**2 + 5) / (1 + (x**2 + 5)**4))
|
|
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss=cubic_soft_l1,
|
|
max_nfev=1, method=self.method)
|
|
assert_allclose(res.grad,
|
|
2 * x * (x**2 + 5) / (1 + (x**2 + 5)**2)**(2/3))
|
|
|
|
def test_jac(self):
|
|
# Test that res.jac.T.dot(res.jac) gives Gauss-Newton approximation
|
|
# of Hessian. This approximation is computed by doubly differentiating
|
|
# the cost function and dropping the part containing second derivative
|
|
# of f. For a scalar function it is computed as
|
|
# H = (rho' + 2 * rho'' * f**2) * f'**2, if the expression inside the
|
|
# brackets is less than EPS it is replaced by EPS. Here, we check
|
|
# against the root of H.
|
|
|
|
x = 2.0 # res.x will be this.
|
|
f = x**2 + 5 # res.fun will be this.
|
|
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss='linear',
|
|
max_nfev=1, method=self.method)
|
|
assert_equal(res.jac, 2 * x)
|
|
|
|
# For `huber` loss the Jacobian correction is identically zero
|
|
# in outlier region, in such cases it is modified to be equal EPS**0.5.
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss='huber',
|
|
max_nfev=1, method=self.method)
|
|
assert_equal(res.jac, 2 * x * EPS**0.5)
|
|
|
|
# Now, let's apply `loss_scale` to turn the residual into an inlier.
|
|
# The loss function becomes linear.
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss='huber',
|
|
f_scale=10, max_nfev=1)
|
|
assert_equal(res.jac, 2 * x)
|
|
|
|
# 'soft_l1' always gives a positive scaling.
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss='soft_l1',
|
|
max_nfev=1, method=self.method)
|
|
assert_allclose(res.jac, 2 * x * (1 + f**2)**-0.75)
|
|
|
|
# For 'cauchy' the correction term turns out to be negative, and it
|
|
# replaced by EPS**0.5.
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss='cauchy',
|
|
max_nfev=1, method=self.method)
|
|
assert_allclose(res.jac, 2 * x * EPS**0.5)
|
|
|
|
# Now use scaling to turn the residual to inlier.
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss='cauchy',
|
|
f_scale=10, max_nfev=1, method=self.method)
|
|
fs = f / 10
|
|
assert_allclose(res.jac, 2 * x * (1 - fs**2)**0.5 / (1 + fs**2))
|
|
|
|
# 'arctan' gives an outlier.
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss='arctan',
|
|
max_nfev=1, method=self.method)
|
|
assert_allclose(res.jac, 2 * x * EPS**0.5)
|
|
|
|
# Turn to inlier.
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss='arctan',
|
|
f_scale=20.0, max_nfev=1, method=self.method)
|
|
fs = f / 20
|
|
assert_allclose(res.jac, 2 * x * (1 - 3 * fs**4)**0.5 / (1 + fs**4))
|
|
|
|
# cubic_soft_l1 will give an outlier.
|
|
res = least_squares(fun_trivial, x, jac_trivial, loss=cubic_soft_l1,
|
|
max_nfev=1)
|
|
assert_allclose(res.jac, 2 * x * EPS**0.5)
|
|
|
|
# Turn to inlier.
|
|
res = least_squares(fun_trivial, x, jac_trivial,
|
|
loss=cubic_soft_l1, f_scale=6, max_nfev=1)
|
|
fs = f / 6
|
|
assert_allclose(res.jac,
|
|
2 * x * (1 - fs**2 / 3)**0.5 * (1 + fs**2)**(-5/6))
|
|
|
|
def test_robustness(self):
|
|
for noise in [0.1, 1.0]:
|
|
p = ExponentialFittingProblem(1, 0.1, noise, random_seed=0)
|
|
|
|
for jac in ['2-point', '3-point', 'cs', p.jac]:
|
|
res_lsq = least_squares(p.fun, p.p0, jac=jac,
|
|
method=self.method)
|
|
assert_allclose(res_lsq.optimality, 0, atol=1e-2)
|
|
for loss in LOSSES:
|
|
if loss == 'linear':
|
|
continue
|
|
res_robust = least_squares(
|
|
p.fun, p.p0, jac=jac, loss=loss, f_scale=noise,
|
|
method=self.method)
|
|
assert_allclose(res_robust.optimality, 0, atol=1e-2)
|
|
assert_(norm(res_robust.x - p.p_opt) <
|
|
norm(res_lsq.x - p.p_opt))
|
|
|
|
|
|
class TestDogbox(BaseMixin, BoundsMixin, SparseMixin, LossFunctionMixin):
|
|
method = 'dogbox'
|
|
|
|
|
|
class TestTRF(BaseMixin, BoundsMixin, SparseMixin, LossFunctionMixin):
|
|
method = 'trf'
|
|
|
|
def test_lsmr_regularization(self):
|
|
p = BroydenTridiagonal()
|
|
for regularize in [True, False]:
|
|
res = least_squares(p.fun, p.x0, p.jac, method='trf',
|
|
tr_options={'regularize': regularize})
|
|
assert_allclose(res.cost, 0, atol=1e-20)
|
|
|
|
|
|
class TestLM(BaseMixin):
|
|
method = 'lm'
|
|
|
|
def test_bounds_not_supported(self):
|
|
assert_raises(ValueError, least_squares, fun_trivial,
|
|
2.0, bounds=(-3.0, 3.0), method='lm')
|
|
|
|
def test_m_less_n_not_supported(self):
|
|
x0 = [-2, 1]
|
|
assert_raises(ValueError, least_squares, fun_rosenbrock_cropped, x0,
|
|
method='lm')
|
|
|
|
def test_sparse_not_supported(self):
|
|
p = BroydenTridiagonal()
|
|
assert_raises(ValueError, least_squares, p.fun, p.x0, p.jac,
|
|
method='lm')
|
|
|
|
def test_jac_sparsity_not_supported(self):
|
|
assert_raises(ValueError, least_squares, fun_trivial, 2.0,
|
|
jac_sparsity=[1], method='lm')
|
|
|
|
def test_LinearOperator_not_supported(self):
|
|
p = BroydenTridiagonal(mode="operator")
|
|
assert_raises(ValueError, least_squares, p.fun, p.x0, p.jac,
|
|
method='lm')
|
|
|
|
def test_loss(self):
|
|
res = least_squares(fun_trivial, 2.0, loss='linear', method='lm')
|
|
assert_allclose(res.x, 0.0, atol=1e-4)
|
|
|
|
assert_raises(ValueError, least_squares, fun_trivial, 2.0,
|
|
method='lm', loss='huber')
|
|
|
|
|
|
def test_basic():
|
|
# test that 'method' arg is really optional
|
|
res = least_squares(fun_trivial, 2.0)
|
|
assert_allclose(res.x, 0, atol=1e-10)
|
|
|
|
|
|
def test_small_tolerances_for_lm():
|
|
for ftol, xtol, gtol in [(None, 1e-13, 1e-13),
|
|
(1e-13, None, 1e-13),
|
|
(1e-13, 1e-13, None)]:
|
|
assert_raises(ValueError, least_squares, fun_trivial, 2.0, xtol=xtol,
|
|
ftol=ftol, gtol=gtol, method='lm')
|
|
|
|
|
|
def test_fp32_gh12991():
|
|
# checks that smaller FP sizes can be used in least_squares
|
|
# this is the minimum working example reported for gh12991
|
|
np.random.seed(1)
|
|
|
|
x = np.linspace(0, 1, 100).astype("float32")
|
|
y = np.random.random(100).astype("float32")
|
|
|
|
def func(p, x):
|
|
return p[0] + p[1] * x
|
|
|
|
def err(p, x, y):
|
|
return func(p, x) - y
|
|
|
|
res = least_squares(err, [-1.0, -1.0], args=(x, y))
|
|
# previously the initial jacobian calculated for this would be all 0
|
|
# and the minimize would terminate immediately, with nfev=1, would
|
|
# report a successful minimization (it shouldn't have done), but be
|
|
# unchanged from the initial solution.
|
|
# It was terminating early because the underlying approx_derivative
|
|
# used a step size for FP64 when the working space was FP32.
|
|
assert res.nfev > 3
|
|
assert_allclose(res.x, np.array([0.4082241, 0.15530563]), atol=5e-5)
|