734 lines
20 KiB
Python
734 lines
20 KiB
Python
"""Functions used by least-squares algorithms."""
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from math import copysign
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import numpy as np
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from numpy.linalg import norm
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from scipy.linalg import cho_factor, cho_solve, LinAlgError
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from scipy.sparse import issparse
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from scipy.sparse.linalg import LinearOperator, aslinearoperator
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EPS = np.finfo(float).eps
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# Functions related to a trust-region problem.
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def intersect_trust_region(x, s, Delta):
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"""Find the intersection of a line with the boundary of a trust region.
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This function solves the quadratic equation with respect to t
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||(x + s*t)||**2 = Delta**2.
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Returns
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-------
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t_neg, t_pos : tuple of float
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Negative and positive roots.
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Raises
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------
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ValueError
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If `s` is zero or `x` is not within the trust region.
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"""
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a = np.dot(s, s)
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if a == 0:
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raise ValueError("`s` is zero.")
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b = np.dot(x, s)
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c = np.dot(x, x) - Delta**2
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if c > 0:
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raise ValueError("`x` is not within the trust region.")
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d = np.sqrt(b*b - a*c) # Root from one fourth of the discriminant.
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# Computations below avoid loss of significance, see "Numerical Recipes".
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q = -(b + copysign(d, b))
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t1 = q / a
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t2 = c / q
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if t1 < t2:
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return t1, t2
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else:
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return t2, t1
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def solve_lsq_trust_region(n, m, uf, s, V, Delta, initial_alpha=None,
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rtol=0.01, max_iter=10):
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"""Solve a trust-region problem arising in least-squares minimization.
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This function implements a method described by J. J. More [1]_ and used
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in MINPACK, but it relies on a single SVD of Jacobian instead of series
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of Cholesky decompositions. Before running this function, compute:
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``U, s, VT = svd(J, full_matrices=False)``.
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Parameters
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----------
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n : int
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Number of variables.
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m : int
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Number of residuals.
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uf : ndarray
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Computed as U.T.dot(f).
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s : ndarray
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Singular values of J.
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V : ndarray
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Transpose of VT.
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Delta : float
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Radius of a trust region.
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initial_alpha : float, optional
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Initial guess for alpha, which might be available from a previous
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iteration. If None, determined automatically.
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rtol : float, optional
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Stopping tolerance for the root-finding procedure. Namely, the
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solution ``p`` will satisfy ``abs(norm(p) - Delta) < rtol * Delta``.
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max_iter : int, optional
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Maximum allowed number of iterations for the root-finding procedure.
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Returns
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-------
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p : ndarray, shape (n,)
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Found solution of a trust-region problem.
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alpha : float
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Positive value such that (J.T*J + alpha*I)*p = -J.T*f.
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Sometimes called Levenberg-Marquardt parameter.
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n_iter : int
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Number of iterations made by root-finding procedure. Zero means
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that Gauss-Newton step was selected as the solution.
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References
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----------
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.. [1] More, J. J., "The Levenberg-Marquardt Algorithm: Implementation
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and Theory," Numerical Analysis, ed. G. A. Watson, Lecture Notes
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in Mathematics 630, Springer Verlag, pp. 105-116, 1977.
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"""
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def phi_and_derivative(alpha, suf, s, Delta):
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"""Function of which to find zero.
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It is defined as "norm of regularized (by alpha) least-squares
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solution minus `Delta`". Refer to [1]_.
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"""
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denom = s**2 + alpha
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p_norm = norm(suf / denom)
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phi = p_norm - Delta
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phi_prime = -np.sum(suf ** 2 / denom**3) / p_norm
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return phi, phi_prime
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suf = s * uf
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# Check if J has full rank and try Gauss-Newton step.
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if m >= n:
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threshold = EPS * m * s[0]
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full_rank = s[-1] > threshold
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else:
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full_rank = False
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if full_rank:
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p = -V.dot(uf / s)
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if norm(p) <= Delta:
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return p, 0.0, 0
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alpha_upper = norm(suf) / Delta
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if full_rank:
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phi, phi_prime = phi_and_derivative(0.0, suf, s, Delta)
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alpha_lower = -phi / phi_prime
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else:
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alpha_lower = 0.0
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if initial_alpha is None or not full_rank and initial_alpha == 0:
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alpha = max(0.001 * alpha_upper, (alpha_lower * alpha_upper)**0.5)
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else:
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alpha = initial_alpha
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for it in range(max_iter):
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if alpha < alpha_lower or alpha > alpha_upper:
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alpha = max(0.001 * alpha_upper, (alpha_lower * alpha_upper)**0.5)
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phi, phi_prime = phi_and_derivative(alpha, suf, s, Delta)
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if phi < 0:
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alpha_upper = alpha
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ratio = phi / phi_prime
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alpha_lower = max(alpha_lower, alpha - ratio)
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alpha -= (phi + Delta) * ratio / Delta
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if np.abs(phi) < rtol * Delta:
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break
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p = -V.dot(suf / (s**2 + alpha))
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# Make the norm of p equal to Delta, p is changed only slightly during
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# this. It is done to prevent p lie outside the trust region (which can
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# cause problems later).
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p *= Delta / norm(p)
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return p, alpha, it + 1
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def solve_trust_region_2d(B, g, Delta):
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"""Solve a general trust-region problem in 2 dimensions.
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The problem is reformulated as a 4th order algebraic equation,
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the solution of which is found by numpy.roots.
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Parameters
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----------
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B : ndarray, shape (2, 2)
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Symmetric matrix, defines a quadratic term of the function.
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g : ndarray, shape (2,)
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Defines a linear term of the function.
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Delta : float
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Radius of a trust region.
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Returns
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-------
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p : ndarray, shape (2,)
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Found solution.
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newton_step : bool
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Whether the returned solution is the Newton step which lies within
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the trust region.
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"""
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try:
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R, lower = cho_factor(B)
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p = -cho_solve((R, lower), g)
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if np.dot(p, p) <= Delta**2:
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return p, True
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except LinAlgError:
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pass
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a = B[0, 0] * Delta**2
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b = B[0, 1] * Delta**2
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c = B[1, 1] * Delta**2
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d = g[0] * Delta
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f = g[1] * Delta
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coeffs = np.array(
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[-b + d, 2 * (a - c + f), 6 * b, 2 * (-a + c + f), -b - d])
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t = np.roots(coeffs) # Can handle leading zeros.
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t = np.real(t[np.isreal(t)])
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p = Delta * np.vstack((2 * t / (1 + t**2), (1 - t**2) / (1 + t**2)))
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value = 0.5 * np.sum(p * B.dot(p), axis=0) + np.dot(g, p)
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i = np.argmin(value)
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p = p[:, i]
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return p, False
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def update_tr_radius(Delta, actual_reduction, predicted_reduction,
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step_norm, bound_hit):
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"""Update the radius of a trust region based on the cost reduction.
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Returns
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-------
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Delta : float
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New radius.
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ratio : float
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Ratio between actual and predicted reductions.
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"""
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if predicted_reduction > 0:
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ratio = actual_reduction / predicted_reduction
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elif predicted_reduction == actual_reduction == 0:
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ratio = 1
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else:
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ratio = 0
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if ratio < 0.25:
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Delta = 0.25 * step_norm
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elif ratio > 0.75 and bound_hit:
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Delta *= 2.0
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return Delta, ratio
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# Construction and minimization of quadratic functions.
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def build_quadratic_1d(J, g, s, diag=None, s0=None):
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"""Parameterize a multivariate quadratic function along a line.
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The resulting univariate quadratic function is given as follows::
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f(t) = 0.5 * (s0 + s*t).T * (J.T*J + diag) * (s0 + s*t) +
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g.T * (s0 + s*t)
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Parameters
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----------
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J : ndarray, sparse matrix or LinearOperator shape (m, n)
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Jacobian matrix, affects the quadratic term.
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g : ndarray, shape (n,)
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Gradient, defines the linear term.
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s : ndarray, shape (n,)
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Direction vector of a line.
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diag : None or ndarray with shape (n,), optional
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Addition diagonal part, affects the quadratic term.
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If None, assumed to be 0.
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s0 : None or ndarray with shape (n,), optional
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Initial point. If None, assumed to be 0.
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Returns
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-------
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a : float
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Coefficient for t**2.
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b : float
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Coefficient for t.
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c : float
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Free term. Returned only if `s0` is provided.
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"""
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v = J.dot(s)
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a = np.dot(v, v)
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if diag is not None:
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a += np.dot(s * diag, s)
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a *= 0.5
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b = np.dot(g, s)
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if s0 is not None:
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u = J.dot(s0)
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b += np.dot(u, v)
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c = 0.5 * np.dot(u, u) + np.dot(g, s0)
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if diag is not None:
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b += np.dot(s0 * diag, s)
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c += 0.5 * np.dot(s0 * diag, s0)
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return a, b, c
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else:
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return a, b
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def minimize_quadratic_1d(a, b, lb, ub, c=0):
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"""Minimize a 1-D quadratic function subject to bounds.
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The free term `c` is 0 by default. Bounds must be finite.
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Returns
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-------
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t : float
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Minimum point.
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y : float
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Minimum value.
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"""
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t = [lb, ub]
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if a != 0:
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extremum = -0.5 * b / a
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if lb < extremum < ub:
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t.append(extremum)
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t = np.asarray(t)
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y = t * (a * t + b) + c
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min_index = np.argmin(y)
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return t[min_index], y[min_index]
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def evaluate_quadratic(J, g, s, diag=None):
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"""Compute values of a quadratic function arising in least squares.
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The function is 0.5 * s.T * (J.T * J + diag) * s + g.T * s.
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Parameters
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----------
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J : ndarray, sparse matrix or LinearOperator, shape (m, n)
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Jacobian matrix, affects the quadratic term.
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g : ndarray, shape (n,)
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Gradient, defines the linear term.
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s : ndarray, shape (k, n) or (n,)
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Array containing steps as rows.
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diag : ndarray, shape (n,), optional
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Addition diagonal part, affects the quadratic term.
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If None, assumed to be 0.
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Returns
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-------
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values : ndarray with shape (k,) or float
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Values of the function. If `s` was 2-D, then ndarray is
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returned, otherwise, float is returned.
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"""
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if s.ndim == 1:
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Js = J.dot(s)
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q = np.dot(Js, Js)
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if diag is not None:
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q += np.dot(s * diag, s)
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else:
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Js = J.dot(s.T)
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q = np.sum(Js**2, axis=0)
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if diag is not None:
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q += np.sum(diag * s**2, axis=1)
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l = np.dot(s, g)
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return 0.5 * q + l
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# Utility functions to work with bound constraints.
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def in_bounds(x, lb, ub):
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"""Check if a point lies within bounds."""
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return np.all((x >= lb) & (x <= ub))
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def step_size_to_bound(x, s, lb, ub):
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"""Compute a min_step size required to reach a bound.
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The function computes a positive scalar t, such that x + s * t is on
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the bound.
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Returns
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-------
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step : float
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Computed step. Non-negative value.
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hits : ndarray of int with shape of x
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Each element indicates whether a corresponding variable reaches the
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bound:
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* 0 - the bound was not hit.
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* -1 - the lower bound was hit.
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* 1 - the upper bound was hit.
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"""
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non_zero = np.nonzero(s)
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s_non_zero = s[non_zero]
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steps = np.empty_like(x)
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steps.fill(np.inf)
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with np.errstate(over='ignore'):
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steps[non_zero] = np.maximum((lb - x)[non_zero] / s_non_zero,
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(ub - x)[non_zero] / s_non_zero)
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min_step = np.min(steps)
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return min_step, np.equal(steps, min_step) * np.sign(s).astype(int)
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def find_active_constraints(x, lb, ub, rtol=1e-10):
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"""Determine which constraints are active in a given point.
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The threshold is computed using `rtol` and the absolute value of the
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closest bound.
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Returns
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-------
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active : ndarray of int with shape of x
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Each component shows whether the corresponding constraint is active:
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* 0 - a constraint is not active.
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* -1 - a lower bound is active.
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* 1 - a upper bound is active.
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"""
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active = np.zeros_like(x, dtype=int)
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if rtol == 0:
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active[x <= lb] = -1
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active[x >= ub] = 1
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return active
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lower_dist = x - lb
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upper_dist = ub - x
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lower_threshold = rtol * np.maximum(1, np.abs(lb))
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upper_threshold = rtol * np.maximum(1, np.abs(ub))
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lower_active = (np.isfinite(lb) &
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(lower_dist <= np.minimum(upper_dist, lower_threshold)))
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active[lower_active] = -1
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upper_active = (np.isfinite(ub) &
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(upper_dist <= np.minimum(lower_dist, upper_threshold)))
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active[upper_active] = 1
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return active
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def make_strictly_feasible(x, lb, ub, rstep=1e-10):
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"""Shift a point to the interior of a feasible region.
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Each element of the returned vector is at least at a relative distance
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`rstep` from the closest bound. If ``rstep=0`` then `np.nextafter` is used.
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"""
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x_new = x.copy()
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active = find_active_constraints(x, lb, ub, rstep)
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lower_mask = np.equal(active, -1)
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upper_mask = np.equal(active, 1)
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if rstep == 0:
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x_new[lower_mask] = np.nextafter(lb[lower_mask], ub[lower_mask])
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x_new[upper_mask] = np.nextafter(ub[upper_mask], lb[upper_mask])
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else:
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x_new[lower_mask] = (lb[lower_mask] +
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rstep * np.maximum(1, np.abs(lb[lower_mask])))
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x_new[upper_mask] = (ub[upper_mask] -
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rstep * np.maximum(1, np.abs(ub[upper_mask])))
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tight_bounds = (x_new < lb) | (x_new > ub)
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x_new[tight_bounds] = 0.5 * (lb[tight_bounds] + ub[tight_bounds])
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return x_new
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def CL_scaling_vector(x, g, lb, ub):
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"""Compute Coleman-Li scaling vector and its derivatives.
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Components of a vector v are defined as follows::
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| ub[i] - x[i], if g[i] < 0 and ub[i] < np.inf
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v[i] = | x[i] - lb[i], if g[i] > 0 and lb[i] > -np.inf
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| 1, otherwise
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According to this definition v[i] >= 0 for all i. It differs from the
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definition in paper [1]_ (eq. (2.2)), where the absolute value of v is
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used. Both definitions are equivalent down the line.
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Derivatives of v with respect to x take value 1, -1 or 0 depending on a
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case.
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Returns
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-------
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v : ndarray with shape of x
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Scaling vector.
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dv : ndarray with shape of x
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Derivatives of v[i] with respect to x[i], diagonal elements of v's
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Jacobian.
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References
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----------
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.. [1] M.A. Branch, T.F. Coleman, and Y. Li, "A Subspace, Interior,
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and Conjugate Gradient Method for Large-Scale Bound-Constrained
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Minimization Problems," SIAM Journal on Scientific Computing,
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Vol. 21, Number 1, pp 1-23, 1999.
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"""
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v = np.ones_like(x)
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dv = np.zeros_like(x)
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mask = (g < 0) & np.isfinite(ub)
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v[mask] = ub[mask] - x[mask]
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dv[mask] = -1
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mask = (g > 0) & np.isfinite(lb)
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v[mask] = x[mask] - lb[mask]
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dv[mask] = 1
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return v, dv
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def reflective_transformation(y, lb, ub):
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"""Compute reflective transformation and its gradient."""
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if in_bounds(y, lb, ub):
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return y, np.ones_like(y)
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lb_finite = np.isfinite(lb)
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ub_finite = np.isfinite(ub)
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x = y.copy()
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g_negative = np.zeros_like(y, dtype=bool)
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mask = lb_finite & ~ub_finite
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x[mask] = np.maximum(y[mask], 2 * lb[mask] - y[mask])
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g_negative[mask] = y[mask] < lb[mask]
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mask = ~lb_finite & ub_finite
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x[mask] = np.minimum(y[mask], 2 * ub[mask] - y[mask])
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g_negative[mask] = y[mask] > ub[mask]
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mask = lb_finite & ub_finite
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d = ub - lb
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t = np.remainder(y[mask] - lb[mask], 2 * d[mask])
|
|
x[mask] = lb[mask] + np.minimum(t, 2 * d[mask] - t)
|
|
g_negative[mask] = t > d[mask]
|
|
|
|
g = np.ones_like(y)
|
|
g[g_negative] = -1
|
|
|
|
return x, g
|
|
|
|
|
|
# Functions to display algorithm's progress.
|
|
|
|
|
|
def print_header_nonlinear():
|
|
print("{:^15}{:^15}{:^15}{:^15}{:^15}{:^15}"
|
|
.format("Iteration", "Total nfev", "Cost", "Cost reduction",
|
|
"Step norm", "Optimality"))
|
|
|
|
|
|
def print_iteration_nonlinear(iteration, nfev, cost, cost_reduction,
|
|
step_norm, optimality):
|
|
if cost_reduction is None:
|
|
cost_reduction = " " * 15
|
|
else:
|
|
cost_reduction = f"{cost_reduction:^15.2e}"
|
|
|
|
if step_norm is None:
|
|
step_norm = " " * 15
|
|
else:
|
|
step_norm = f"{step_norm:^15.2e}"
|
|
|
|
print("{:^15}{:^15}{:^15.4e}{}{}{:^15.2e}"
|
|
.format(iteration, nfev, cost, cost_reduction,
|
|
step_norm, optimality))
|
|
|
|
|
|
def print_header_linear():
|
|
print("{:^15}{:^15}{:^15}{:^15}{:^15}"
|
|
.format("Iteration", "Cost", "Cost reduction", "Step norm",
|
|
"Optimality"))
|
|
|
|
|
|
def print_iteration_linear(iteration, cost, cost_reduction, step_norm,
|
|
optimality):
|
|
if cost_reduction is None:
|
|
cost_reduction = " " * 15
|
|
else:
|
|
cost_reduction = f"{cost_reduction:^15.2e}"
|
|
|
|
if step_norm is None:
|
|
step_norm = " " * 15
|
|
else:
|
|
step_norm = f"{step_norm:^15.2e}"
|
|
|
|
print(f"{iteration:^15}{cost:^15.4e}{cost_reduction}{step_norm}{optimality:^15.2e}")
|
|
|
|
|
|
# Simple helper functions.
|
|
|
|
|
|
def compute_grad(J, f):
|
|
"""Compute gradient of the least-squares cost function."""
|
|
if isinstance(J, LinearOperator):
|
|
return J.rmatvec(f)
|
|
else:
|
|
return J.T.dot(f)
|
|
|
|
|
|
def compute_jac_scale(J, scale_inv_old=None):
|
|
"""Compute variables scale based on the Jacobian matrix."""
|
|
if issparse(J):
|
|
scale_inv = np.asarray(J.power(2).sum(axis=0)).ravel()**0.5
|
|
else:
|
|
scale_inv = np.sum(J**2, axis=0)**0.5
|
|
|
|
if scale_inv_old is None:
|
|
scale_inv[scale_inv == 0] = 1
|
|
else:
|
|
scale_inv = np.maximum(scale_inv, scale_inv_old)
|
|
|
|
return 1 / scale_inv, scale_inv
|
|
|
|
|
|
def left_multiplied_operator(J, d):
|
|
"""Return diag(d) J as LinearOperator."""
|
|
J = aslinearoperator(J)
|
|
|
|
def matvec(x):
|
|
return d * J.matvec(x)
|
|
|
|
def matmat(X):
|
|
return d[:, np.newaxis] * J.matmat(X)
|
|
|
|
def rmatvec(x):
|
|
return J.rmatvec(x.ravel() * d)
|
|
|
|
return LinearOperator(J.shape, matvec=matvec, matmat=matmat,
|
|
rmatvec=rmatvec)
|
|
|
|
|
|
def right_multiplied_operator(J, d):
|
|
"""Return J diag(d) as LinearOperator."""
|
|
J = aslinearoperator(J)
|
|
|
|
def matvec(x):
|
|
return J.matvec(np.ravel(x) * d)
|
|
|
|
def matmat(X):
|
|
return J.matmat(X * d[:, np.newaxis])
|
|
|
|
def rmatvec(x):
|
|
return d * J.rmatvec(x)
|
|
|
|
return LinearOperator(J.shape, matvec=matvec, matmat=matmat,
|
|
rmatvec=rmatvec)
|
|
|
|
|
|
def regularized_lsq_operator(J, diag):
|
|
"""Return a matrix arising in regularized least squares as LinearOperator.
|
|
|
|
The matrix is
|
|
[ J ]
|
|
[ D ]
|
|
where D is diagonal matrix with elements from `diag`.
|
|
"""
|
|
J = aslinearoperator(J)
|
|
m, n = J.shape
|
|
|
|
def matvec(x):
|
|
return np.hstack((J.matvec(x), diag * x))
|
|
|
|
def rmatvec(x):
|
|
x1 = x[:m]
|
|
x2 = x[m:]
|
|
return J.rmatvec(x1) + diag * x2
|
|
|
|
return LinearOperator((m + n, n), matvec=matvec, rmatvec=rmatvec)
|
|
|
|
|
|
def right_multiply(J, d, copy=True):
|
|
"""Compute J diag(d).
|
|
|
|
If `copy` is False, `J` is modified in place (unless being LinearOperator).
|
|
"""
|
|
if copy and not isinstance(J, LinearOperator):
|
|
J = J.copy()
|
|
|
|
if issparse(J):
|
|
J.data *= d.take(J.indices, mode='clip') # scikit-learn recipe.
|
|
elif isinstance(J, LinearOperator):
|
|
J = right_multiplied_operator(J, d)
|
|
else:
|
|
J *= d
|
|
|
|
return J
|
|
|
|
|
|
def left_multiply(J, d, copy=True):
|
|
"""Compute diag(d) J.
|
|
|
|
If `copy` is False, `J` is modified in place (unless being LinearOperator).
|
|
"""
|
|
if copy and not isinstance(J, LinearOperator):
|
|
J = J.copy()
|
|
|
|
if issparse(J):
|
|
J.data *= np.repeat(d, np.diff(J.indptr)) # scikit-learn recipe.
|
|
elif isinstance(J, LinearOperator):
|
|
J = left_multiplied_operator(J, d)
|
|
else:
|
|
J *= d[:, np.newaxis]
|
|
|
|
return J
|
|
|
|
|
|
def check_termination(dF, F, dx_norm, x_norm, ratio, ftol, xtol):
|
|
"""Check termination condition for nonlinear least squares."""
|
|
ftol_satisfied = dF < ftol * F and ratio > 0.25
|
|
xtol_satisfied = dx_norm < xtol * (xtol + x_norm)
|
|
|
|
if ftol_satisfied and xtol_satisfied:
|
|
return 4
|
|
elif ftol_satisfied:
|
|
return 2
|
|
elif xtol_satisfied:
|
|
return 3
|
|
else:
|
|
return None
|
|
|
|
|
|
def scale_for_robust_loss_function(J, f, rho):
|
|
"""Scale Jacobian and residuals for a robust loss function.
|
|
|
|
Arrays are modified in place.
|
|
"""
|
|
J_scale = rho[1] + 2 * rho[2] * f**2
|
|
J_scale[J_scale < EPS] = EPS
|
|
J_scale **= 0.5
|
|
|
|
f *= rho[1] / J_scale
|
|
|
|
return left_multiply(J, J_scale, copy=False), f
|