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Filename :
_differentiable_functions.py
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import numpy as np import scipy.sparse as sps from ._numdiff import approx_derivative, group_columns from ._hessian_update_strategy import HessianUpdateStrategy from scipy.sparse.linalg import LinearOperator FD_METHODS = ('2-point', '3-point', 'cs') class ScalarFunction(object): """Scalar function and its derivatives. This class defines a scalar function F: R^n->R and methods for computing or approximating its first and second derivatives. Notes ----- This class implements a memoization logic. There are methods `fun`, `grad`, hess` and corresponding attributes `f`, `g` and `H`. The following things should be considered: 1. Use only public methods `fun`, `grad` and `hess`. 2. After one of the methods is called, the corresponding attribute will be set. However, a subsequent call with a different argument of *any* of the methods may overwrite the attribute. """ def __init__(self, fun, x0, args, grad, hess, finite_diff_rel_step, finite_diff_bounds, epsilon=None): if not callable(grad) and grad not in FD_METHODS: raise ValueError("`grad` must be either callable or one of {}." .format(FD_METHODS)) if not (callable(hess) or hess in FD_METHODS or isinstance(hess, HessianUpdateStrategy)): raise ValueError("`hess` must be either callable," "HessianUpdateStrategy or one of {}." .format(FD_METHODS)) if grad in FD_METHODS and hess in FD_METHODS: raise ValueError("Whenever the gradient is estimated via " "finite-differences, we require the Hessian " "to be estimated using one of the " "quasi-Newton strategies.") self.x = np.atleast_1d(x0).astype(float) self.n = self.x.size self.nfev = 0 self.ngev = 0 self.nhev = 0 self.f_updated = False self.g_updated = False self.H_updated = False finite_diff_options = {} if grad in FD_METHODS: finite_diff_options["method"] = grad finite_diff_options["rel_step"] = finite_diff_rel_step finite_diff_options["abs_step"] = epsilon finite_diff_options["bounds"] = finite_diff_bounds if hess in FD_METHODS: finite_diff_options["method"] = hess finite_diff_options["rel_step"] = finite_diff_rel_step finite_diff_options["abs_step"] = epsilon finite_diff_options["as_linear_operator"] = True # Function evaluation def fun_wrapped(x): self.nfev += 1 return fun(x, *args) def update_fun(): self.f = fun_wrapped(self.x) self._update_fun_impl = update_fun self._update_fun() # Gradient evaluation if callable(grad): def grad_wrapped(x): self.ngev += 1 return np.atleast_1d(grad(x, *args)) def update_grad(): self.g = grad_wrapped(self.x) elif grad in FD_METHODS: def update_grad(): self._update_fun() self.ngev += 1 self.g = approx_derivative(fun_wrapped, self.x, f0=self.f, **finite_diff_options) self._update_grad_impl = update_grad self._update_grad() # Hessian Evaluation if callable(hess): self.H = hess(x0, *args) self.H_updated = True self.nhev += 1 if sps.issparse(self.H): def hess_wrapped(x): self.nhev += 1 return sps.csr_matrix(hess(x, *args)) self.H = sps.csr_matrix(self.H) elif isinstance(self.H, LinearOperator): def hess_wrapped(x): self.nhev += 1 return hess(x, *args) else: def hess_wrapped(x): self.nhev += 1 return np.atleast_2d(np.asarray(hess(x, *args))) self.H = np.atleast_2d(np.asarray(self.H)) def update_hess(): self.H = hess_wrapped(self.x) elif hess in FD_METHODS: def update_hess(): self._update_grad() self.H = approx_derivative(grad_wrapped, self.x, f0=self.g, **finite_diff_options) return self.H update_hess() self.H_updated = True elif isinstance(hess, HessianUpdateStrategy): self.H = hess self.H.initialize(self.n, 'hess') self.H_updated = True self.x_prev = None self.g_prev = None def update_hess(): self._update_grad() self.H.update(self.x - self.x_prev, self.g - self.g_prev) self._update_hess_impl = update_hess if isinstance(hess, HessianUpdateStrategy): def update_x(x): self._update_grad() self.x_prev = self.x self.g_prev = self.g self.x = np.atleast_1d(x).astype(float) self.f_updated = False self.g_updated = False self.H_updated = False self._update_hess() else: def update_x(x): self.x = np.atleast_1d(x).astype(float) self.f_updated = False self.g_updated = False self.H_updated = False self._update_x_impl = update_x def _update_fun(self): if not self.f_updated: self._update_fun_impl() self.f_updated = True def _update_grad(self): if not self.g_updated: self._update_grad_impl() self.g_updated = True def _update_hess(self): if not self.H_updated: self._update_hess_impl() self.H_updated = True def fun(self, x): if not np.array_equal(x, self.x): self._update_x_impl(x) self._update_fun() return self.f def grad(self, x): if not np.array_equal(x, self.x): self._update_x_impl(x) self._update_grad() return self.g def hess(self, x): if not np.array_equal(x, self.x): self._update_x_impl(x) self._update_hess() return self.H def fun_and_grad(self, x): if not np.array_equal(x, self.x): self._update_x_impl(x) self._update_fun() self._update_grad() return self.f, self.g class VectorFunction(object): """Vector function and its derivatives. This class defines a vector function F: R^n->R^m and methods for computing or approximating its first and second derivatives. Notes ----- This class implements a memoization logic. There are methods `fun`, `jac`, hess` and corresponding attributes `f`, `J` and `H`. The following things should be considered: 1. Use only public methods `fun`, `jac` and `hess`. 2. After one of the methods is called, the corresponding attribute will be set. However, a subsequent call with a different argument of *any* of the methods may overwrite the attribute. """ def __init__(self, fun, x0, jac, hess, finite_diff_rel_step, finite_diff_jac_sparsity, finite_diff_bounds, sparse_jacobian): if not callable(jac) and jac not in FD_METHODS: raise ValueError("`jac` must be either callable or one of {}." .format(FD_METHODS)) if not (callable(hess) or hess in FD_METHODS or isinstance(hess, HessianUpdateStrategy)): raise ValueError("`hess` must be either callable," "HessianUpdateStrategy or one of {}." .format(FD_METHODS)) if jac in FD_METHODS and hess in FD_METHODS: raise ValueError("Whenever the Jacobian is estimated via " "finite-differences, we require the Hessian to " "be estimated using one of the quasi-Newton " "strategies.") self.x = np.atleast_1d(x0).astype(float) self.n = self.x.size self.nfev = 0 self.njev = 0 self.nhev = 0 self.f_updated = False self.J_updated = False self.H_updated = False finite_diff_options = {} if jac in FD_METHODS: finite_diff_options["method"] = jac finite_diff_options["rel_step"] = finite_diff_rel_step if finite_diff_jac_sparsity is not None: sparsity_groups = group_columns(finite_diff_jac_sparsity) finite_diff_options["sparsity"] = (finite_diff_jac_sparsity, sparsity_groups) finite_diff_options["bounds"] = finite_diff_bounds self.x_diff = np.copy(self.x) if hess in FD_METHODS: finite_diff_options["method"] = hess finite_diff_options["rel_step"] = finite_diff_rel_step finite_diff_options["as_linear_operator"] = True self.x_diff = np.copy(self.x) if jac in FD_METHODS and hess in FD_METHODS: raise ValueError("Whenever the Jacobian is estimated via " "finite-differences, we require the Hessian to " "be estimated using one of the quasi-Newton " "strategies.") # Function evaluation def fun_wrapped(x): self.nfev += 1 return np.atleast_1d(fun(x)) def update_fun(): self.f = fun_wrapped(self.x) self._update_fun_impl = update_fun update_fun() self.v = np.zeros_like(self.f) self.m = self.v.size # Jacobian Evaluation if callable(jac): self.J = jac(self.x) self.J_updated = True self.njev += 1 if (sparse_jacobian or sparse_jacobian is None and sps.issparse(self.J)): def jac_wrapped(x): self.njev += 1 return sps.csr_matrix(jac(x)) self.J = sps.csr_matrix(self.J) self.sparse_jacobian = True elif sps.issparse(self.J): def jac_wrapped(x): self.njev += 1 return jac(x).toarray() self.J = self.J.toarray() self.sparse_jacobian = False else: def jac_wrapped(x): self.njev += 1 return np.atleast_2d(jac(x)) self.J = np.atleast_2d(self.J) self.sparse_jacobian = False def update_jac(): self.J = jac_wrapped(self.x) elif jac in FD_METHODS: self.J = approx_derivative(fun_wrapped, self.x, f0=self.f, **finite_diff_options) self.J_updated = True if (sparse_jacobian or sparse_jacobian is None and sps.issparse(self.J)): def update_jac(): self._update_fun() self.J = sps.csr_matrix( approx_derivative(fun_wrapped, self.x, f0=self.f, **finite_diff_options)) self.J = sps.csr_matrix(self.J) self.sparse_jacobian = True elif sps.issparse(self.J): def update_jac(): self._update_fun() self.J = approx_derivative(fun_wrapped, self.x, f0=self.f, **finite_diff_options).toarray() self.J = self.J.toarray() self.sparse_jacobian = False else: def update_jac(): self._update_fun() self.J = np.atleast_2d( approx_derivative(fun_wrapped, self.x, f0=self.f, **finite_diff_options)) self.J = np.atleast_2d(self.J) self.sparse_jacobian = False self._update_jac_impl = update_jac # Define Hessian if callable(hess): self.H = hess(self.x, self.v) self.H_updated = True self.nhev += 1 if sps.issparse(self.H): def hess_wrapped(x, v): self.nhev += 1 return sps.csr_matrix(hess(x, v)) self.H = sps.csr_matrix(self.H) elif isinstance(self.H, LinearOperator): def hess_wrapped(x, v): self.nhev += 1 return hess(x, v) else: def hess_wrapped(x, v): self.nhev += 1 return np.atleast_2d(np.asarray(hess(x, v))) self.H = np.atleast_2d(np.asarray(self.H)) def update_hess(): self.H = hess_wrapped(self.x, self.v) elif hess in FD_METHODS: def jac_dot_v(x, v): return jac_wrapped(x).T.dot(v) def update_hess(): self._update_jac() self.H = approx_derivative(jac_dot_v, self.x, f0=self.J.T.dot(self.v), args=(self.v,), **finite_diff_options) update_hess() self.H_updated = True elif isinstance(hess, HessianUpdateStrategy): self.H = hess self.H.initialize(self.n, 'hess') self.H_updated = True self.x_prev = None self.J_prev = None def update_hess(): self._update_jac() # When v is updated before x was updated, then x_prev and # J_prev are None and we need this check. if self.x_prev is not None and self.J_prev is not None: delta_x = self.x - self.x_prev delta_g = self.J.T.dot(self.v) - self.J_prev.T.dot(self.v) self.H.update(delta_x, delta_g) self._update_hess_impl = update_hess if isinstance(hess, HessianUpdateStrategy): def update_x(x): self._update_jac() self.x_prev = self.x self.J_prev = self.J self.x = np.atleast_1d(x).astype(float) self.f_updated = False self.J_updated = False self.H_updated = False self._update_hess() else: def update_x(x): self.x = np.atleast_1d(x).astype(float) self.f_updated = False self.J_updated = False self.H_updated = False self._update_x_impl = update_x def _update_v(self, v): if not np.array_equal(v, self.v): self.v = v self.H_updated = False def _update_x(self, x): if not np.array_equal(x, self.x): self._update_x_impl(x) def _update_fun(self): if not self.f_updated: self._update_fun_impl() self.f_updated = True def _update_jac(self): if not self.J_updated: self._update_jac_impl() self.J_updated = True def _update_hess(self): if not self.H_updated: self._update_hess_impl() self.H_updated = True def fun(self, x): self._update_x(x) self._update_fun() return self.f def jac(self, x): self._update_x(x) self._update_jac() return self.J def hess(self, x, v): # v should be updated before x. self._update_v(v) self._update_x(x) self._update_hess() return self.H class LinearVectorFunction(object): """Linear vector function and its derivatives. Defines a linear function F = A x, where x is N-D vector and A is m-by-n matrix. The Jacobian is constant and equals to A. The Hessian is identically zero and it is returned as a csr matrix. """ def __init__(self, A, x0, sparse_jacobian): if sparse_jacobian or sparse_jacobian is None and sps.issparse(A): self.J = sps.csr_matrix(A) self.sparse_jacobian = True elif sps.issparse(A): self.J = A.toarray() self.sparse_jacobian = False else: # np.asarray makes sure A is ndarray and not matrix self.J = np.atleast_2d(np.asarray(A)) self.sparse_jacobian = False self.m, self.n = self.J.shape self.x = np.atleast_1d(x0).astype(float) self.f = self.J.dot(self.x) self.f_updated = True self.v = np.zeros(self.m, dtype=float) self.H = sps.csr_matrix((self.n, self.n)) def _update_x(self, x): if not np.array_equal(x, self.x): self.x = np.atleast_1d(x).astype(float) self.f_updated = False def fun(self, x): self._update_x(x) if not self.f_updated: self.f = self.J.dot(x) self.f_updated = True return self.f def jac(self, x): self._update_x(x) return self.J def hess(self, x, v): self._update_x(x) self.v = v return self.H class IdentityVectorFunction(LinearVectorFunction): """Identity vector function and its derivatives. The Jacobian is the identity matrix, returned as a dense array when `sparse_jacobian=False` and as a csr matrix otherwise. The Hessian is identically zero and it is returned as a csr matrix. """ def __init__(self, x0, sparse_jacobian): n = len(x0) if sparse_jacobian or sparse_jacobian is None: A = sps.eye(n, format='csr') sparse_jacobian = True else: A = np.eye(n) sparse_jacobian = False super(IdentityVectorFunction, self).__init__(A, x0, sparse_jacobian)