380 lines
12 KiB
Python
380 lines
12 KiB
Python
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"""
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Our own implementation of the Newton algorithm
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Unlike the scipy.optimize version, this version of the Newton conjugate
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gradient solver uses only one function call to retrieve the
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func value, the gradient value and a callable for the Hessian matvec
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product. If the function call is very expensive (e.g. for logistic
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regression with large design matrix), this approach gives very
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significant speedups.
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"""
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# This is a modified file from scipy.optimize
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# Original authors: Travis Oliphant, Eric Jones
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# Modifications by Gael Varoquaux, Mathieu Blondel and Tom Dupre la Tour
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# License: BSD
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import warnings
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import numpy as np
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import scipy
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from ..exceptions import ConvergenceWarning
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from .fixes import line_search_wolfe1, line_search_wolfe2
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class _LineSearchError(RuntimeError):
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pass
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def _line_search_wolfe12(
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f, fprime, xk, pk, gfk, old_fval, old_old_fval, verbose=0, **kwargs
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):
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"""
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Same as line_search_wolfe1, but fall back to line_search_wolfe2 if
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suitable step length is not found, and raise an exception if a
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suitable step length is not found.
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Raises
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------
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_LineSearchError
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If no suitable step size is found.
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"""
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is_verbose = verbose >= 2
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eps = 16 * np.finfo(np.asarray(old_fval).dtype).eps
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if is_verbose:
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print(" Line Search")
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print(f" eps=16 * finfo.eps={eps}")
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print(" try line search wolfe1")
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ret = line_search_wolfe1(f, fprime, xk, pk, gfk, old_fval, old_old_fval, **kwargs)
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if is_verbose:
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_not_ = "not " if ret[0] is None else ""
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print(" wolfe1 line search was " + _not_ + "successful")
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if ret[0] is None:
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# Have a look at the line_search method of our NewtonSolver class. We borrow
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# the logic from there
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# Deal with relative loss differences around machine precision.
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args = kwargs.get("args", tuple())
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fval = f(xk + pk, *args)
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tiny_loss = np.abs(old_fval * eps)
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loss_improvement = fval - old_fval
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check = np.abs(loss_improvement) <= tiny_loss
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if is_verbose:
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print(
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" check loss |improvement| <= eps * |loss_old|:"
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f" {np.abs(loss_improvement)} <= {tiny_loss} {check}"
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)
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if check:
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# 2.1 Check sum of absolute gradients as alternative condition.
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sum_abs_grad_old = scipy.linalg.norm(gfk, ord=1)
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grad = fprime(xk + pk, *args)
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sum_abs_grad = scipy.linalg.norm(grad, ord=1)
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check = sum_abs_grad < sum_abs_grad_old
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if is_verbose:
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print(
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" check sum(|gradient|) < sum(|gradient_old|): "
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f"{sum_abs_grad} < {sum_abs_grad_old} {check}"
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)
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if check:
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ret = (
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1.0, # step size
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ret[1] + 1, # number of function evaluations
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ret[2] + 1, # number of gradient evaluations
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fval,
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old_fval,
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grad,
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)
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if ret[0] is None:
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# line search failed: try different one.
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# TODO: It seems that the new check for the sum of absolute gradients above
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# catches all cases that, earlier, ended up here. In fact, our tests never
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# trigger this "if branch" here and we can consider to remove it.
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if is_verbose:
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print(" last resort: try line search wolfe2")
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ret = line_search_wolfe2(
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f, fprime, xk, pk, gfk, old_fval, old_old_fval, **kwargs
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)
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if is_verbose:
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_not_ = "not " if ret[0] is None else ""
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print(" wolfe2 line search was " + _not_ + "successful")
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if ret[0] is None:
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raise _LineSearchError()
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return ret
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def _cg(fhess_p, fgrad, maxiter, tol, verbose=0):
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"""
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Solve iteratively the linear system 'fhess_p . xsupi = fgrad'
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with a conjugate gradient descent.
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Parameters
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----------
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fhess_p : callable
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Function that takes the gradient as a parameter and returns the
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matrix product of the Hessian and gradient.
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fgrad : ndarray of shape (n_features,) or (n_features + 1,)
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Gradient vector.
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maxiter : int
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Number of CG iterations.
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tol : float
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Stopping criterion.
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Returns
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-------
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xsupi : ndarray of shape (n_features,) or (n_features + 1,)
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Estimated solution.
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"""
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eps = 16 * np.finfo(np.float64).eps
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xsupi = np.zeros(len(fgrad), dtype=fgrad.dtype)
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ri = np.copy(fgrad) # residual = fgrad - fhess_p @ xsupi
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psupi = -ri
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i = 0
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dri0 = np.dot(ri, ri)
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# We also keep track of |p_i|^2.
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psupi_norm2 = dri0
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is_verbose = verbose >= 2
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while i <= maxiter:
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if np.sum(np.abs(ri)) <= tol:
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if is_verbose:
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print(
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f" Inner CG solver iteration {i} stopped with\n"
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f" sum(|residuals|) <= tol: {np.sum(np.abs(ri))} <= {tol}"
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)
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break
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Ap = fhess_p(psupi)
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# check curvature
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curv = np.dot(psupi, Ap)
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if 0 <= curv <= eps * psupi_norm2:
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# See https://arxiv.org/abs/1803.02924, Algo 1 Capped Conjugate Gradient.
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if is_verbose:
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print(
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f" Inner CG solver iteration {i} stopped with\n"
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f" tiny_|p| = eps * ||p||^2, eps = {eps}, "
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f"squred L2 norm ||p||^2 = {psupi_norm2}\n"
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f" curvature <= tiny_|p|: {curv} <= {eps * psupi_norm2}"
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)
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break
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elif curv < 0:
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if i > 0:
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if is_verbose:
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print(
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f" Inner CG solver iteration {i} stopped with negative "
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f"curvature, curvature = {curv}"
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)
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break
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else:
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# fall back to steepest descent direction
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xsupi += dri0 / curv * psupi
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if is_verbose:
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print(" Inner CG solver iteration 0 fell back to steepest descent")
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break
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alphai = dri0 / curv
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xsupi += alphai * psupi
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ri += alphai * Ap
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dri1 = np.dot(ri, ri)
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betai = dri1 / dri0
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psupi = -ri + betai * psupi
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# We use |p_i|^2 = |r_i|^2 + beta_i^2 |p_{i-1}|^2
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psupi_norm2 = dri1 + betai**2 * psupi_norm2
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i = i + 1
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dri0 = dri1 # update np.dot(ri,ri) for next time.
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if is_verbose and i > maxiter:
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print(
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f" Inner CG solver stopped reaching maxiter={i - 1} with "
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f"sum(|residuals|) = {np.sum(np.abs(ri))}"
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)
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return xsupi
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def _newton_cg(
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grad_hess,
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func,
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grad,
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x0,
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args=(),
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tol=1e-4,
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maxiter=100,
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maxinner=200,
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line_search=True,
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warn=True,
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verbose=0,
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):
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"""
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Minimization of scalar function of one or more variables using the
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Newton-CG algorithm.
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Parameters
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----------
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grad_hess : callable
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Should return the gradient and a callable returning the matvec product
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of the Hessian.
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func : callable
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Should return the value of the function.
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grad : callable
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Should return the function value and the gradient. This is used
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by the linesearch functions.
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x0 : array of float
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Initial guess.
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args : tuple, default=()
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Arguments passed to func_grad_hess, func and grad.
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tol : float, default=1e-4
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Stopping criterion. The iteration will stop when
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``max{|g_i | i = 1, ..., n} <= tol``
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where ``g_i`` is the i-th component of the gradient.
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maxiter : int, default=100
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Number of Newton iterations.
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maxinner : int, default=200
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Number of CG iterations.
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line_search : bool, default=True
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Whether to use a line search or not.
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warn : bool, default=True
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Whether to warn when didn't converge.
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Returns
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-------
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xk : ndarray of float
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Estimated minimum.
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"""
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x0 = np.asarray(x0).flatten()
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xk = np.copy(x0)
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k = 0
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if line_search:
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old_fval = func(x0, *args)
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old_old_fval = None
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else:
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old_fval = 0
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is_verbose = verbose > 0
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# Outer loop: our Newton iteration
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while k < maxiter:
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# Compute a search direction pk by applying the CG method to
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# del2 f(xk) p = - fgrad f(xk) starting from 0.
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fgrad, fhess_p = grad_hess(xk, *args)
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absgrad = np.abs(fgrad)
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max_absgrad = np.max(absgrad)
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check = max_absgrad <= tol
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if is_verbose:
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print(f"Newton-CG iter = {k}")
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print(" Check Convergence")
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print(f" max |gradient| <= tol: {max_absgrad} <= {tol} {check}")
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if check:
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break
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maggrad = np.sum(absgrad)
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eta = min([0.5, np.sqrt(maggrad)])
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termcond = eta * maggrad
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# Inner loop: solve the Newton update by conjugate gradient, to
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# avoid inverting the Hessian
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xsupi = _cg(fhess_p, fgrad, maxiter=maxinner, tol=termcond, verbose=verbose)
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alphak = 1.0
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if line_search:
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try:
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alphak, fc, gc, old_fval, old_old_fval, gfkp1 = _line_search_wolfe12(
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func,
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grad,
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xk,
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xsupi,
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fgrad,
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old_fval,
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old_old_fval,
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verbose=verbose,
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args=args,
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)
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except _LineSearchError:
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warnings.warn("Line Search failed")
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break
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xk += alphak * xsupi # upcast if necessary
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k += 1
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if warn and k >= maxiter:
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warnings.warn(
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(
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f"newton-cg failed to converge at loss = {old_fval}. Increase the"
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" number of iterations."
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),
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ConvergenceWarning,
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)
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elif is_verbose:
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print(f" Solver did converge at loss = {old_fval}.")
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return xk, k
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def _check_optimize_result(solver, result, max_iter=None, extra_warning_msg=None):
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"""Check the OptimizeResult for successful convergence
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Parameters
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----------
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solver : str
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Solver name. Currently only `lbfgs` is supported.
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result : OptimizeResult
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Result of the scipy.optimize.minimize function.
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max_iter : int, default=None
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Expected maximum number of iterations.
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extra_warning_msg : str, default=None
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Extra warning message.
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Returns
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-------
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n_iter : int
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Number of iterations.
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"""
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# handle both scipy and scikit-learn solver names
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if solver == "lbfgs":
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if result.status != 0:
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try:
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# The message is already decoded in scipy>=1.6.0
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result_message = result.message.decode("latin1")
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except AttributeError:
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result_message = result.message
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warning_msg = (
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"{} failed to converge (status={}):\n{}.\n\n"
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"Increase the number of iterations (max_iter) "
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"or scale the data as shown in:\n"
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" https://scikit-learn.org/stable/modules/"
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"preprocessing.html"
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).format(solver, result.status, result_message)
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if extra_warning_msg is not None:
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warning_msg += "\n" + extra_warning_msg
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warnings.warn(warning_msg, ConvergenceWarning, stacklevel=2)
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if max_iter is not None:
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# In scipy <= 1.0.0, nit may exceed maxiter for lbfgs.
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# See https://github.com/scipy/scipy/issues/7854
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n_iter_i = min(result.nit, max_iter)
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else:
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n_iter_i = result.nit
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else:
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raise NotImplementedError
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return n_iter_i
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