def _assertions(self, x):
if not self.validate_args:
return []
shape = tf.shape(x)
is_matrix = assert_util.assert_rank_at_least(
x, 2, message="Input must have rank at least 2.")
is_square = assert_util.assert_equal(
shape[-2], shape[-1], message="Input must be a square matrix.")
above_diagonal = tf.linalg.band_part(
tf.linalg.set_diag(x, tf.zeros(shape[:-1], dtype=tf.float32)), 0, -1)
is_lower_triangular = assert_util.assert_equal(
above_diagonal,
tf.zeros_like(above_diagonal),
message="Input must be lower triangular.")
# A lower triangular matrix is nonsingular iff all its diagonal entries are
# nonzero.
diag_part = tf.linalg.diag_part(x)
is_nonsingular = assert_util.assert_none_equal(
diag_part,
tf.zeros_like(diag_part),
message="Input must have all diagonal entries nonzero.")
return [is_matrix, is_square, is_lower_triangular, is_nonsingular]
def _parameter_control_dependencies(self, is_init):
if not self.validate_args:
return []
lu, perm = None, None
assertions = []
if (is_init != tensor_util.is_ref(self.lower_upper)
or is_init != tensor_util.is_ref(self.permutation)):
lu, perm = self._broadcast_params()
assertions.extend(
lu_reconstruct_assertions(lu, perm, self.validate_args))
if is_init != tensor_util.is_ref(self.lower_upper):
lu = tf.convert_to_tensor(self.lower_upper) if lu is None else lu
assertions.append(
assert_util.assert_none_equal(
tf.linalg.diag_part(lu),
tf.zeros([], dtype=lu.dtype),
message=
'Invertible `lower_upper` must have nonzero diagonal.'))
return assertions
def __init__(self,
hinge_softness=None,
validate_args=False,
name="softplus"):
with tf.name_scope(name):
if hinge_softness is None:
self._hinge_softness = None
else:
self._hinge_softness = tf.convert_to_tensor(
value=hinge_softness, name="hinge_softness")
if validate_args:
nonzero_check = assert_util.assert_none_equal(
dtype_util.as_numpy_dtype(
self._hinge_softness.dtype)(0),
self.hinge_softness,
message="hinge_softness must be non-zero")
self._hinge_softness = distribution_util.with_dependencies(
[nonzero_check], self.hinge_softness)
super(Softplus, self).__init__(forward_min_event_ndims=0,
validate_args=validate_args,
name=name)
def _assertions(self, t):
if not self.validate_args:
return []
return [assert_util.assert_none_equal(
t, dtype_util.as_numpy_dtype(t.dtype)(0.),
message="All elements must be non-zero.")]
def _create_scale_operator(self, identity_multiplier, diag, tril,
perturb_diag, perturb_factor, shift,
validate_args, dtype):
"""Construct `scale` from various components.
Args:
identity_multiplier: floating point rank 0 `Tensor` representing a scaling
done to the identity matrix.
diag: Floating-point `Tensor` representing the diagonal matrix.`diag` has
shape `[N1, N2, ... k]`, which represents a k x k diagonal matrix.
tril: Floating-point `Tensor` representing the lower triangular matrix.
`tril` has shape `[N1, N2, ... k, k]`, which represents a k x k lower
triangular matrix.
perturb_diag: Floating-point `Tensor` representing the diagonal matrix of
the low rank update.
perturb_factor: Floating-point `Tensor` representing factor matrix.
shift: Floating-point `Tensor` representing `shift in `scale @ X + shift`.
validate_args: Python `bool` indicating whether arguments should be
checked for correctness.
dtype: `DType` for arg `Tensor` conversions.
Returns:
scale. In the case of scaling by a constant, scale is a
floating point `Tensor`. Otherwise, scale is a `LinearOperator`.
Raises:
ValueError: if all of `tril`, `diag` and `identity_multiplier` are `None`.
"""
identity_multiplier = _as_tensor(identity_multiplier,
"identity_multiplier", dtype)
diag = _as_tensor(diag, "diag", dtype)
tril = _as_tensor(tril, "tril", dtype)
perturb_diag = _as_tensor(perturb_diag, "perturb_diag", dtype)
perturb_factor = _as_tensor(perturb_factor, "perturb_factor", dtype)
# If possible, use the low rank update to infer the shape of
# the identity matrix, when scale represents a scaled identity matrix
# with a low rank update.
shape_hint = None
if perturb_factor is not None:
shape_hint = distribution_util.dimension_size(perturb_factor,
axis=-2)
if self._is_only_identity_multiplier:
if validate_args:
return distribution_util.with_dependencies([
assert_util.assert_none_equal(
identity_multiplier,
tf.zeros([], identity_multiplier.dtype),
["identity_multiplier should be non-zero."])
], identity_multiplier)
return identity_multiplier
scale = _make_tril_scale(loc=shift,
scale_tril=tril,
scale_diag=diag,
scale_identity_multiplier=identity_multiplier,
validate_args=validate_args,
assert_positive=False,
shape_hint=shape_hint)
if perturb_factor is not None:
return tf.linalg.LinearOperatorLowRankUpdate(
scale,
u=perturb_factor,
diag_update=perturb_diag,
is_diag_update_positive=perturb_diag is None,
is_non_singular=True, # Implied by is_positive_definite=True.
is_self_adjoint=True,
is_positive_definite=True,
is_square=True)
return scale
def find_root_chandrupatla(objective_fn,
low,
high,
position_tolerance=1e-8,
value_tolerance=0.,
max_iterations=50,
stopping_policy_fn=tf.reduce_all,
validate_args=False,
name='find_root_chandrupatla'):
r"""Finds root(s) of a scalar function using Chandrupatla's method.
Chandrupatla's method [1, 2] is a root-finding algorithm that is guaranteed
to converge if a root lies within the given bounds. It generalizes the
[bisection method](https://en.wikipedia.org/wiki/Bisection_method); at each
step it chooses to perform either bisection or inverse quadratic
interpolation. This makes it similar in spirit to [Brent's method](
https://en.wikipedia.org/wiki/Brent%27s_method), which also considers steps
that use the secant method, but Chandrupatla's method is simpler and often
converges at least as quickly [3].
Args:
objective_fn: Python callable for which roots are searched. It must be a
callable of a single variable. `objective_fn` must return a `Tensor` with
shape `batch_shape` and dtype matching `lower_bound` and `upper_bound`.
low: Float `Tensor` of shape `batch_shape` representing a lower
bound(s) on the value of a root(s).
high: Float `Tensor` of shape `batch_shape` representing an upper
bound(s) on the value of a root(s).
position_tolerance: Optional `Tensor` representing the maximum absolute
error in the positions of the estimated roots. Shape must broadcast with
`batch_shape`.
Default value: `1e-8`.
value_tolerance: Optional `Tensor` representing the absolute error allowed
in the value of the objective function. If the absolute value of
`objective_fn` is smaller than
`value_tolerance` at a given position, then that position is considered a
root for the function. Shape must broadcast with `batch_shape`.
Default value: `1e-8`.
max_iterations: Optional `Tensor` or Python integer specifying the maximum
number of steps to perform. Shape must broadcast with `batch_shape`.
Default value: `50`.
stopping_policy_fn: Python `callable` controlling the algorithm termination.
It must be a callable accepting a `Tensor` of booleans with the same shape
as `lower_bound` and `upper_bound` (denoting whether each search is
finished), and returning a scalar boolean `Tensor` indicating
whether the overall search should stop. Typical values are
`tf.reduce_all` (which returns only when the search is finished for all
points), and `tf.reduce_any` (which returns as soon as the search is
finished for any point).
Default value: `tf.reduce_all` (returns only when the search is finished
for all points).
validate_args: Python `bool` indicating whether to validate arguments.
Default value: `False`.
name: Python `str` name prefixed to ops created by this function.
Default value: 'find_root_chandrupatla'.
Returns:
root_search_results: A Python `namedtuple` containing the following items:
estimated_root: `Tensor` containing the last position explored. If the
search was successful within the specified tolerance, this position is
a root of the objective function.
objective_at_estimated_root: `Tensor` containing the value of the
objective function at `position`. If the search was successful within
the specified tolerance, then this is close to 0.
num_iterations: The number of iterations performed.
#### References
[1] Tirupathi R. Chandrupatla. A new hybrid quadratic/bisection algorithm for
finding the zero of a nonlinear function without using derivatives.
_Advances in Engineering Software_, 28.3:145-149, 1997.
[2] Philipp OJ Scherer. Computational Physics. _Springer Berlin_,
Heidelberg, 2010.
Section 6.1.7.3 https://books.google.com/books?id=cC-8BAAAQBAJ&pg=PA95
[3] Jason Sachs. Ten Little Algorithms, Part 5: Quadratic Extremum
Interpolation and Chandrupatla's Method (2015).
https://www.embeddedrelated.com/showarticle/855.php
"""
################################################
# Loop variables used by Chandrupatla's method:
#
# a: endpoint of an interval `[min(a, b), max(a, b)]` containing the
# root. There is no guarantee as to which of `a` and `b` is larger.
# b: endpoint of an interval `[min(a, b), max(a, b)]` containing the
# root. There is no guarantee as to which of `a` and `b` is larger.
# f_a: value of the objective at `a`.
# f_b: value of the objective at `b`.
# t: the next position to be evaluated as the coefficient of a convex
# combination of `a` and `b` (i.e., a value in the unit interval).
# num_iterations: integer number of steps taken so far.
# converged: boolean indicating whether each batch element has converged.
#
# All variables have the same shape `batch_shape`.
def _should_continue(a, b, f_a, f_b, t, num_iterations, converged):
del a, b, f_a, f_b, t # Unused.
all_converged = stopping_policy_fn(
tf.logical_or(converged, num_iterations >= max_iterations))
return ~all_converged
def _body(a, b, f_a, f_b, t, num_iterations, converged):
"""One step of Chandrupatla's method for root finding."""
previous_loop_vars = (a, b, f_a, f_b, t, num_iterations, converged)
finalized_elements = tf.logical_or(converged,
num_iterations >= max_iterations)
# Evaluate the new point.
x_new = (1 - t) * a + t * b
f_new = objective_fn(x_new)
# Tighten the bounds.
a, b, c, f_a, f_b, f_c = _structure_broadcasting_where(
tf.equal(tf.math.sign(f_new), tf.math.sign(f_a)),
(x_new, b, a, f_new, f_b, f_a), (x_new, a, b, f_new, f_a, f_b))
# Check for convergence.
f_best = tf.where(tf.abs(f_a) < tf.abs(f_b), f_a, f_b)
interval_tolerance = position_tolerance / (tf.abs(b - c))
converged = tf.logical_or(interval_tolerance > 0.5,
tf.math.abs(f_best) <= value_tolerance)
# Propose next point to evaluate.
xi = (a - b) / (c - b)
phi = (f_a - f_b) / (f_c - f_b)
t = tf.where(
# Condition for inverse quadratic interpolation.
tf.logical_and(1 - tf.math.sqrt(1 - xi) < phi,
tf.math.sqrt(xi) > phi),
# Propose a point by inverse quadratic interpolation.
(f_a / (f_b - f_a) * f_c / (f_b - f_c) + (c - a) / (b - a) * f_a /
(f_c - f_a) * f_b / (f_c - f_b)),
# Otherwise, just cut the interval in half (bisection).
0.5)
# Constrain the proposal to the current interval (0 < t < 1).
t = tf.minimum(tf.maximum(t, interval_tolerance),
1 - interval_tolerance)
# Update elements that haven't converged.
return _structure_broadcasting_where(
finalized_elements, previous_loop_vars,
(a, b, f_a, f_b, t, num_iterations + 1, converged))
with tf.name_scope(name):
max_iterations = tf.convert_to_tensor(max_iterations,
name='max_iterations',
dtype_hint=tf.int32)
a = tf.convert_to_tensor(low, name='lower_bound')
b = tf.convert_to_tensor(high, name='upper_bound')
f_a, f_b = objective_fn(a), objective_fn(b)
batch_shape = ps.broadcast_shape(ps.shape(f_a), ps.shape(f_b))
assertions = []
if validate_args:
assertions += [
assert_util.assert_none_equal(
tf.math.sign(f_a),
tf.math.sign(f_b),
message='Bounds must be on different sides of a root.')
]
with tf.control_dependencies(assertions):
initial_loop_vars = [
a, b, f_a, f_b,
tf.cast(0.5, dtype=f_a.dtype),
tf.cast(0, dtype=max_iterations.dtype), False
]
a, b, f_a, f_b, _, num_iterations, _ = tf.while_loop(
_should_continue,
_body,
loop_vars=tf.nest.map_structure(
lambda x: tf.broadcast_to(x, batch_shape),
initial_loop_vars))
x_best, f_best = _structure_broadcasting_where(
tf.abs(f_a) < tf.abs(f_b), (a, f_a), (b, f_b))
return RootSearchResults(estimated_root=x_best,
objective_at_estimated_root=f_best,
num_iterations=num_iterations)
def _maybe_assert_valid(self, t):
if not self.validate_args:
return t
is_valid = assert_util.assert_none_equal(
t, 0., message="All elements must be non-zero.")
return distribution_util.with_dependencies([is_valid], t)
