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"""Functions used by least-squares algorithms."""
# Functions related to a trust-region problem.
"""Find the intersection of a line with the boundary of a trust region.
This function solves the quadratic equation with respect to t ||(x + s*t)||**2 = Delta**2.
Returns ------- t_neg, t_pos : tuple of float Negative and positive roots.
Raises ------ ValueError If `s` is zero or `x` is not within the trust region. """ a = np.dot(s, s) if a == 0: raise ValueError("`s` is zero.")
b = np.dot(x, s)
c = np.dot(x, x) - Delta**2 if c > 0: raise ValueError("`x` is not within the trust region.")
d = np.sqrt(b*b - a*c) # Root from one fourth of the discriminant.
# Computations below avoid loss of significance, see "Numerical Recipes". q = -(b + copysign(d, b)) t1 = q / a t2 = c / q
if t1 < t2: return t1, t2 else: return t2, t1
rtol=0.01, max_iter=10): """Solve a trust-region problem arising in least-squares minimization.
This function implements a method described by J. J. More [1]_ and used in MINPACK, but it relies on a single SVD of Jacobian instead of series of Cholesky decompositions. Before running this function, compute: ``U, s, VT = svd(J, full_matrices=False)``.
Parameters ---------- n : int Number of variables. m : int Number of residuals. uf : ndarray Computed as U.T.dot(f). s : ndarray Singular values of J. V : ndarray Transpose of VT. Delta : float Radius of a trust region. initial_alpha : float, optional Initial guess for alpha, which might be available from a previous iteration. If None, determined automatically. rtol : float, optional Stopping tolerance for the root-finding procedure. Namely, the solution ``p`` will satisfy ``abs(norm(p) - Delta) < rtol * Delta``. max_iter : int, optional Maximum allowed number of iterations for the root-finding procedure.
Returns ------- p : ndarray, shape (n,) Found solution of a trust-region problem. alpha : float Positive value such that (J.T*J + alpha*I)*p = -J.T*f. Sometimes called Levenberg-Marquardt parameter. n_iter : int Number of iterations made by root-finding procedure. Zero means that Gauss-Newton step was selected as the solution.
References ---------- .. [1] More, J. J., "The Levenberg-Marquardt Algorithm: Implementation and Theory," Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics 630, Springer Verlag, pp. 105-116, 1977. """ def phi_and_derivative(alpha, suf, s, Delta): """Function of which to find zero.
It is defined as "norm of regularized (by alpha) least-squares solution minus `Delta`". Refer to [1]_. """ denom = s**2 + alpha p_norm = norm(suf / denom) phi = p_norm - Delta phi_prime = -np.sum(suf ** 2 / denom**3) / p_norm return phi, phi_prime
suf = s * uf
# Check if J has full rank and try Gauss-Newton step. if m >= n: threshold = EPS * m * s[0] full_rank = s[-1] > threshold else: full_rank = False
if full_rank: p = -V.dot(uf / s) if norm(p) <= Delta: return p, 0.0, 0
alpha_upper = norm(suf) / Delta
if full_rank: phi, phi_prime = phi_and_derivative(0.0, suf, s, Delta) alpha_lower = -phi / phi_prime else: alpha_lower = 0.0
if initial_alpha is None or not full_rank and initial_alpha == 0: alpha = max(0.001 * alpha_upper, (alpha_lower * alpha_upper)**0.5) else: alpha = initial_alpha
for it in range(max_iter): if alpha < alpha_lower or alpha > alpha_upper: alpha = max(0.001 * alpha_upper, (alpha_lower * alpha_upper)**0.5)
phi, phi_prime = phi_and_derivative(alpha, suf, s, Delta)
if phi < 0: alpha_upper = alpha
ratio = phi / phi_prime alpha_lower = max(alpha_lower, alpha - ratio) alpha -= (phi + Delta) * ratio / Delta
if np.abs(phi) < rtol * Delta: break
p = -V.dot(suf / (s**2 + alpha))
# Make the norm of p equal to Delta, p is changed only slightly during # this. It is done to prevent p lie outside the trust region (which can # cause problems later). p *= Delta / norm(p)
return p, alpha, it + 1
"""Solve a general trust-region problem in 2 dimensions.
The problem is reformulated as a 4-th order algebraic equation, the solution of which is found by numpy.roots.
Parameters ---------- B : ndarray, shape (2, 2) Symmetric matrix, defines a quadratic term of the function. g : ndarray, shape (2,) Defines a linear term of the function. Delta : float Radius of a trust region.
Returns ------- p : ndarray, shape (2,) Found solution. newton_step : bool Whether the returned solution is the Newton step which lies within the trust region. """ try: R, lower = cho_factor(B) p = -cho_solve((R, lower), g) if np.dot(p, p) <= Delta**2: return p, True except LinAlgError: pass
a = B[0, 0] * Delta**2 b = B[0, 1] * Delta**2 c = B[1, 1] * Delta**2
d = g[0] * Delta f = g[1] * Delta
coeffs = np.array( [-b + d, 2 * (a - c + f), 6 * b, 2 * (-a + c + f), -b - d]) t = np.roots(coeffs) # Can handle leading zeros. t = np.real(t[np.isreal(t)])
p = Delta * np.vstack((2 * t / (1 + t**2), (1 - t**2) / (1 + t**2))) value = 0.5 * np.sum(p * B.dot(p), axis=0) + np.dot(g, p) i = np.argmin(value) p = p[:, i]
return p, False
step_norm, bound_hit): """Update the radius of a trust region based on the cost reduction.
Returns ------- Delta : float New radius. ratio : float Ratio between actual and predicted reductions. Zero if predicted reduction is zero. """ if predicted_reduction > 0: ratio = actual_reduction / predicted_reduction else: ratio = 0
if ratio < 0.25: Delta = 0.25 * step_norm elif ratio > 0.75 and bound_hit: Delta *= 2.0
return Delta, ratio
# Construction and minimization of quadratic functions.
"""Parameterize a multivariate quadratic function along a line.
The resulting univariate quadratic function is given as follows: :: f(t) = 0.5 * (s0 + s*t).T * (J.T*J + diag) * (s0 + s*t) + g.T * (s0 + s*t)
Parameters ---------- J : ndarray, sparse matrix or LinearOperator shape (m, n) Jacobian matrix, affects the quadratic term. g : ndarray, shape (n,) Gradient, defines the linear term. s : ndarray, shape (n,) Direction vector of a line. diag : None or ndarray with shape (n,), optional Addition diagonal part, affects the quadratic term. If None, assumed to be 0. s0 : None or ndarray with shape (n,), optional Initial point. If None, assumed to be 0.
Returns ------- a : float Coefficient for t**2. b : float Coefficient for t. c : float Free term. Returned only if `s0` is provided. """ v = J.dot(s) a = np.dot(v, v) if diag is not None: a += np.dot(s * diag, s) a *= 0.5
b = np.dot(g, s)
if s0 is not None: u = J.dot(s0) b += np.dot(u, v) c = 0.5 * np.dot(u, u) + np.dot(g, s0) if diag is not None: b += np.dot(s0 * diag, s) c += 0.5 * np.dot(s0 * diag, s0) return a, b, c else: return a, b
"""Minimize a 1-d quadratic function subject to bounds.
The free term `c` is 0 by default. Bounds must be finite.
Returns ------- t : float Minimum point. y : float Minimum value. """ t = [lb, ub] if a != 0: extremum = -0.5 * b / a if lb < extremum < ub: t.append(extremum) t = np.asarray(t) y = a * t**2 + b * t + c min_index = np.argmin(y) return t[min_index], y[min_index]
"""Compute values of a quadratic function arising in least squares.
The function is 0.5 * s.T * (J.T * J + diag) * s + g.T * s.
Parameters ---------- J : ndarray, sparse matrix or LinearOperator, shape (m, n) Jacobian matrix, affects the quadratic term. g : ndarray, shape (n,) Gradient, defines the linear term. s : ndarray, shape (k, n) or (n,) Array containing steps as rows. diag : ndarray, shape (n,), optional Addition diagonal part, affects the quadratic term. If None, assumed to be 0.
Returns ------- values : ndarray with shape (k,) or float Values of the function. If `s` was 2-dimensional then ndarray is returned, otherwise float is returned. """ if s.ndim == 1: Js = J.dot(s) q = np.dot(Js, Js) if diag is not None: q += np.dot(s * diag, s) else: Js = J.dot(s.T) q = np.sum(Js**2, axis=0) if diag is not None: q += np.sum(diag * s**2, axis=1)
l = np.dot(s, g)
return 0.5 * q + l
# Utility functions to work with bound constraints.
"""Check if a point lies within bounds.""" return np.all((x >= lb) & (x <= ub))
"""Compute a min_step size required to reach a bound.
The function computes a positive scalar t, such that x + s * t is on the bound.
Returns ------- step : float Computed step. Non-negative value. hits : ndarray of int with shape of x Each element indicates whether a corresponding variable reaches the bound:
* 0 - the bound was not hit. * -1 - the lower bound was hit. * 1 - the upper bound was hit. """ non_zero = np.nonzero(s) s_non_zero = s[non_zero] steps = np.empty_like(x) steps.fill(np.inf) with np.errstate(over='ignore'): steps[non_zero] = np.maximum((lb - x)[non_zero] / s_non_zero, (ub - x)[non_zero] / s_non_zero) min_step = np.min(steps) return min_step, np.equal(steps, min_step) * np.sign(s).astype(int)
"""Determine which constraints are active in a given point.
The threshold is computed using `rtol` and the absolute value of the closest bound.
Returns ------- active : ndarray of int with shape of x Each component shows whether the corresponding constraint is active:
* 0 - a constraint is not active. * -1 - a lower bound is active. * 1 - a upper bound is active. """ active = np.zeros_like(x, dtype=int)
if rtol == 0: active[x <= lb] = -1 active[x >= ub] = 1 return active
lower_dist = x - lb upper_dist = ub - x
lower_threshold = rtol * np.maximum(1, np.abs(lb)) upper_threshold = rtol * np.maximum(1, np.abs(ub))
lower_active = (np.isfinite(lb) & (lower_dist <= np.minimum(upper_dist, lower_threshold))) active[lower_active] = -1
upper_active = (np.isfinite(ub) & (upper_dist <= np.minimum(lower_dist, upper_threshold))) active[upper_active] = 1
return active
"""Shift a point to the interior of a feasible region.
Each element of the returned vector is at least at a relative distance `rstep` from the closest bound. If ``rstep=0`` then `np.nextafter` is used. """ x_new = x.copy()
active = find_active_constraints(x, lb, ub, rstep) lower_mask = np.equal(active, -1) upper_mask = np.equal(active, 1)
if rstep == 0: x_new[lower_mask] = np.nextafter(lb[lower_mask], ub[lower_mask]) x_new[upper_mask] = np.nextafter(ub[upper_mask], lb[upper_mask]) else: x_new[lower_mask] = (lb[lower_mask] + rstep * np.maximum(1, np.abs(lb[lower_mask]))) x_new[upper_mask] = (ub[upper_mask] - rstep * np.maximum(1, np.abs(ub[upper_mask])))
tight_bounds = (x_new < lb) | (x_new > ub) x_new[tight_bounds] = 0.5 * (lb[tight_bounds] + ub[tight_bounds])
return x_new
"""Compute Coleman-Li scaling vector and its derivatives.
Components of a vector v are defined as follows: :: | ub[i] - x[i], if g[i] < 0 and ub[i] < np.inf v[i] = | x[i] - lb[i], if g[i] > 0 and lb[i] > -np.inf | 1, otherwise
According to this definition v[i] >= 0 for all i. It differs from the definition in paper [1]_ (eq. (2.2)), where the absolute value of v is used. Both definitions are equivalent down the line. Derivatives of v with respect to x take value 1, -1 or 0 depending on a case.
Returns ------- v : ndarray with shape of x Scaling vector. dv : ndarray with shape of x Derivatives of v[i] with respect to x[i], diagonal elements of v's Jacobian.
References ---------- .. [1] M.A. Branch, T.F. Coleman, and Y. Li, "A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems," SIAM Journal on Scientific Computing, Vol. 21, Number 1, pp 1-23, 1999. """ v = np.ones_like(x) dv = np.zeros_like(x)
mask = (g < 0) & np.isfinite(ub) v[mask] = ub[mask] - x[mask] dv[mask] = -1
mask = (g > 0) & np.isfinite(lb) v[mask] = x[mask] - lb[mask] dv[mask] = 1
return v, dv
"""Compute reflective transformation and its gradient.""" if in_bounds(y, lb, ub): return y, np.ones_like(y)
lb_finite = np.isfinite(lb) ub_finite = np.isfinite(ub)
x = y.copy() g_negative = np.zeros_like(y, dtype=bool)
mask = lb_finite & ~ub_finite x[mask] = np.maximum(y[mask], 2 * lb[mask] - y[mask]) g_negative[mask] = y[mask] < lb[mask]
mask = ~lb_finite & ub_finite x[mask] = np.minimum(y[mask], 2 * ub[mask] - y[mask]) g_negative[mask] = y[mask] > ub[mask]
mask = lb_finite & ub_finite d = ub - lb 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.
print("{0:^15}{1:^15}{2:^15}{3:^15}{4:^15}{5:^15}" .format("Iteration", "Total nfev", "Cost", "Cost reduction", "Step norm", "Optimality"))
step_norm, optimality): if cost_reduction is None: cost_reduction = " " * 15 else: cost_reduction = "{0:^15.2e}".format(cost_reduction)
if step_norm is None: step_norm = " " * 15 else: step_norm = "{0:^15.2e}".format(step_norm)
print("{0:^15}{1:^15}{2:^15.4e}{3}{4}{5:^15.2e}" .format(iteration, nfev, cost, cost_reduction, step_norm, optimality))
print("{0:^15}{1:^15}{2:^15}{3:^15}{4:^15}" .format("Iteration", "Cost", "Cost reduction", "Step norm", "Optimality"))
optimality): if cost_reduction is None: cost_reduction = " " * 15 else: cost_reduction = "{0:^15.2e}".format(cost_reduction)
if step_norm is None: step_norm = " " * 15 else: step_norm = "{0:^15.2e}".format(step_norm)
print("{0:^15}{1:^15.4e}{2}{3}{4:^15.2e}".format( iteration, cost, cost_reduction, step_norm, optimality))
# Simple helper functions.
"""Compute gradient of the least-squares cost function.""" if isinstance(J, LinearOperator): return J.rmatvec(f) else: return J.T.dot(f)
"""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
"""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)
"""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)
"""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)
"""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
"""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
"""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
"""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 |