def _get_initial_state(value_and_gradients_function, initial_position,
num_correction_pairs, tolerance):
"""Create LBfgsOptimizerResults with initial state of search procedure."""
init_args = bfgs_utils.get_initial_state_args(value_and_gradients_function,
initial_position, tolerance)
empty_queue = _make_empty_queue_for(num_correction_pairs, initial_position)
init_args.update(position_deltas=empty_queue, gradient_deltas=empty_queue)
return LBfgsOptimizerResults(**init_args)
def minimize(value_and_gradients_function,
initial_position,
tolerance=1e-8,
x_tolerance=0,
f_relative_tolerance=0,
initial_inverse_hessian_estimate=None,
max_iterations=50,
parallel_iterations=1,
stopping_condition=None,
name=None):
"""Applies the BFGS algorithm to minimize a differentiable function.
Performs unconstrained minimization of a differentiable function using the
BFGS scheme. For details of the algorithm, see [Nocedal and Wright(2006)][1].
### Usage:
The following example demonstrates the BFGS optimizer attempting to find the
minimum for a simple two dimensional quadratic objective function.
```python
minimum = np.array([1.0, 1.0]) # The center of the quadratic bowl.
scales = np.array([2.0, 3.0]) # The scales along the two axes.
# The objective function and the gradient.
def quadratic(x):
value = tf.reduce_sum(scales * (x - minimum) ** 2)
return value, tf.gradients(value, x)[0]
start = tf.constant([0.6, 0.8]) # Starting point for the search.
optim_results = tfp.optimizer.bfgs_minimize(
quadratic, initial_position=start, tolerance=1e-8)
with tf.Session() as session:
results = session.run(optim_results)
# Check that the search converged
assert(results.converged)
# Check that the argmin is close to the actual value.
np.testing.assert_allclose(results.position, minimum)
# Print out the total number of function evaluations it took. Should be 6.
print ("Function evaluations: %d" % results.num_objective_evaluations)
```
### References:
[1]: Jorge Nocedal, Stephen Wright. Numerical Optimization. Springer Series in
Operations Research. pp 136-140. 2006
http://pages.mtu.edu/~struther/Courses/OLD/Sp2013/5630/Jorge_Nocedal_Numerical_optimization_267490.pdf
Args:
value_and_gradients_function: A Python callable that accepts a point as a
real `Tensor` and returns a tuple of `Tensor`s of real dtype containing
the value of the function and its gradient at that point. The function
to be minimized. The input should be of shape `[..., n]`, where `n` is
the size of the domain of input points, and all others are batching
dimensions. The first component of the return value should be a real
`Tensor` of matching shape `[...]`. The second component (the gradient)
should also be of shape `[..., n]` like the input value to the function.
initial_position: real `Tensor` of shape `[..., n]`. The starting point, or
points when using batching dimensions, of the search procedure. At these
points the function value and the gradient norm should be finite.
tolerance: Scalar `Tensor` of real dtype. Specifies the gradient tolerance
for the procedure. If the supremum norm of the gradient vector is below
this number, the algorithm is stopped.
x_tolerance: Scalar `Tensor` of real dtype. If the absolute change in the
position between one iteration and the next is smaller than this number,
the algorithm is stopped.
f_relative_tolerance: Scalar `Tensor` of real dtype. If the relative change
in the objective value between one iteration and the next is smaller
than this value, the algorithm is stopped.
initial_inverse_hessian_estimate: Optional `Tensor` of the same dtype
as the components of the output of the `value_and_gradients_function`.
If specified, the shape should broadcastable to shape `[..., n, n]`; e.g.
if a single `[n, n]` matrix is provided, it will be automatically
broadcasted to all batches. Alternatively, one can also specify a
different hessian estimate for each batch member.
For the correctness of the algorithm, it is required that this parameter
be symmetric and positive definite. Specifies the starting estimate for
the inverse of the Hessian at the initial point. If not specified,
the identity matrix is used as the starting estimate for the
inverse Hessian.
max_iterations: Scalar positive int32 `Tensor`. The maximum number of
iterations for BFGS updates.
parallel_iterations: Positive integer. The number of iterations allowed to
run in parallel.
stopping_condition: (Optional) A Python function that takes as input two
Boolean tensors of shape `[...]`, and returns a Boolean scalar tensor.
The input tensors are `converged` and `failed`, indicating the current
status of each respective batch member; the return value states whether
the algorithm should stop. The default is tfp.optimizer.converged_all
which only stops when all batch members have either converged or failed.
An alternative is tfp.optimizer.converged_any which stops as soon as one
batch member has converged, or when all have failed.
name: (Optional) Python str. The name prefixed to the ops created by this
function. If not supplied, the default name 'minimize' is used.
Returns:
optimizer_results: A namedtuple containing the following items:
converged: boolean tensor of shape `[...]` indicating for each batch
member whether the minimum was found within tolerance.
failed: boolean tensor of shape `[...]` indicating for each batch
member whether a line search step failed to find a suitable step size
satisfying Wolfe conditions. In the absence of any constraints on the
number of objective evaluations permitted, this value will
be the complement of `converged`. However, if there is
a constraint and the search stopped due to available
evaluations being exhausted, both `failed` and `converged`
will be simultaneously False.
num_objective_evaluations: The total number of objective
evaluations performed.
position: A tensor of shape `[..., n]` containing the last argument value
found during the search from each starting point. If the search
converged, then this value is the argmin of the objective function.
objective_value: A tensor of shape `[...]` with the value of the
objective function at the `position`. If the search converged, then
this is the (local) minimum of the objective function.
objective_gradient: A tensor of shape `[..., n]` containing the gradient
of the objective function at the `position`. If the search converged
the max-norm of this tensor should be below the tolerance.
inverse_hessian_estimate: A tensor of shape `[..., n, n]` containing the
inverse of the estimated Hessian.
"""
with tf.compat.v1.name_scope(
name, 'minimize',
[initial_position, tolerance, initial_inverse_hessian_estimate]):
initial_position = tf.convert_to_tensor(
value=initial_position, name='initial_position')
dtype = initial_position.dtype.base_dtype
tolerance = tf.convert_to_tensor(
value=tolerance, dtype=dtype, name='grad_tolerance')
f_relative_tolerance = tf.convert_to_tensor(
value=f_relative_tolerance, dtype=dtype, name='f_relative_tolerance')
x_tolerance = tf.convert_to_tensor(
value=x_tolerance, dtype=dtype, name='x_tolerance')
max_iterations = tf.convert_to_tensor(
value=max_iterations, name='max_iterations')
input_shape = distribution_util.prefer_static_shape(initial_position)
batch_shape, domain_size = input_shape[:-1], input_shape[-1]
if stopping_condition is None:
stopping_condition = bfgs_utils.converged_all
# Control inputs are an optional list of tensors to evaluate before
# the start of the search procedure. These can be used to assert the
# validity of inputs to the search procedure.
control_inputs = None
if initial_inverse_hessian_estimate is None:
# Create a default initial inverse Hessian.
initial_inv_hessian = tf.eye(domain_size,
batch_shape=batch_shape,
dtype=dtype,
name='initial_inv_hessian')
else:
# If an initial inverse Hessian is supplied, compute some control inputs
# to ensure that it is positive definite and symmetric.
initial_inv_hessian = tf.convert_to_tensor(
value=initial_inverse_hessian_estimate,
dtype=dtype,
name='initial_inv_hessian')
control_inputs = _inv_hessian_control_inputs(initial_inv_hessian)
hessian_shape = tf.concat([batch_shape, [domain_size, domain_size]], 0)
initial_inv_hessian = tf.broadcast_to(initial_inv_hessian, hessian_shape)
# The `state` here is a `BfgsOptimizerResults` tuple with values for the
# current state of the algorithm computation.
def _cond(state):
"""Continue if iterations remain and stopping condition is not met."""
return ((state.num_iterations < max_iterations) &
tf.logical_not(stopping_condition(state.converged, state.failed)))
def _body(state):
"""Main optimization loop."""
search_direction = _get_search_direction(state.inverse_hessian_estimate,
state.objective_gradient)
derivative_at_start_pt = tf.reduce_sum(
input_tensor=state.objective_gradient * search_direction, axis=-1)
# If the derivative at the start point is not negative, recompute the
# search direction with the initial inverse Hessian.
needs_reset = (~state.failed & ~state.converged &
(derivative_at_start_pt >= 0))
search_direction_reset = _get_search_direction(
initial_inv_hessian, state.objective_gradient)
actual_serch_direction = tf.compat.v1.where(needs_reset,
search_direction_reset,
search_direction)
actual_inv_hessian = tf.compat.v1.where(needs_reset, initial_inv_hessian,
state.inverse_hessian_estimate)
# Replace the hessian estimate in the state, in case it had to be reset.
current_state = bfgs_utils.update_fields(
state, inverse_hessian_estimate=actual_inv_hessian)
next_state = bfgs_utils.line_search_step(
current_state,
value_and_gradients_function, actual_serch_direction,
tolerance, f_relative_tolerance, x_tolerance, stopping_condition)
# Update the inverse Hessian if needed and continue.
return [_update_inv_hessian(current_state, next_state)]
kwargs = bfgs_utils.get_initial_state_args(
value_and_gradients_function,
initial_position,
tolerance,
control_inputs)
kwargs['inverse_hessian_estimate'] = initial_inv_hessian
initial_state = BfgsOptimizerResults(**kwargs)
return tf.while_loop(
cond=_cond,
body=_body,
loop_vars=[initial_state],
parallel_iterations=parallel_iterations)[0]
def minimize(value_and_gradients_function,
initial_position,
tolerance=1e-8,
x_tolerance=0,
f_relative_tolerance=0,
initial_inverse_hessian_estimate=None,
max_iterations=50,
parallel_iterations=1,
name=None):
"""Applies the BFGS algorithm to minimize a differentiable function.
Performs unconstrained minimization of a differentiable function using the
BFGS scheme. For details of the algorithm, see [Nocedal and Wright(2006)][1].
### Usage:
The following example demonstrates the BFGS optimizer attempting to find the
minimum for a simple two dimensional quadratic objective function.
```python
minimum = np.array([1.0, 1.0]) # The center of the quadratic bowl.
scales = np.array([2.0, 3.0]) # The scales along the two axes.
# The objective function and the gradient.
def quadratic(x):
value = tf.reduce_sum(scales * (x - minimum) ** 2)
return value, tf.gradients(value, x)[0]
start = tf.constant([0.6, 0.8]) # Starting point for the search.
optim_results = tfp.optimizer.bfgs_minimize(
quadratic, initial_position=start, tolerance=1e-8)
with tf.Session() as session:
results = session.run(optim_results)
# Check that the search converged
assert(results.converged)
# Check that the argmin is close to the actual value.
np.testing.assert_allclose(results.position, minimum)
# Print out the total number of function evaluations it took. Should be 6.
print ("Function evaluations: %d" % results.num_objective_evaluations)
```
### References:
[1]: Jorge Nocedal, Stephen Wright. Numerical Optimization. Springer Series in
Operations Research. pp 136-140. 2006
http://pages.mtu.edu/~struther/Courses/OLD/Sp2013/5630/Jorge_Nocedal_Numerical_optimization_267490.pdf
Args:
value_and_gradients_function: A Python callable that accepts a point as a
real `Tensor` and returns a tuple of `Tensor`s of real dtype containing
the value of the function and its gradient at that point. The function
to be minimized. The first component of the return value should be a
real scalar `Tensor`. The second component (the gradient) should have the
same shape as the input value to the function.
initial_position: `Tensor` of real dtype. The starting point of the search
procedure. Should be a point at which the function value and the gradient
norm are finite.
tolerance: Scalar `Tensor` of real dtype. Specifies the gradient tolerance
for the procedure. If the supremum norm of the gradient vector is below
this number, the algorithm is stopped.
x_tolerance: Scalar `Tensor` of real dtype. If the absolute change in the
position between one iteration and the next is smaller than this number,
the algorithm is stopped.
f_relative_tolerance: Scalar `Tensor` of real dtype. If the relative change
in the objective value between one iteration and the next is smaller
than this value, the algorithm is stopped.
initial_inverse_hessian_estimate: Optional `Tensor` of the same dtype
as the components of the output of the `value_and_gradients_function`.
If specified, the shape should be `initial_position.shape` * 2.
For example, if the shape of `initial_position` is `[n]`, then the
acceptable shape of `initial_inverse_hessian_estimate` is as a square
matrix of shape `[n, n]`.
If the shape of `initial_position` is `[n, m]`, then the required shape
is `[n, m, n, m]`.
For the correctness of the algorithm, it is required that this parameter
be symmetric and positive definite. Specifies the starting estimate for
the inverse of the Hessian at the initial point. If not specified,
the identity matrix is used as the starting estimate for the
inverse Hessian.
max_iterations: Scalar positive int32 `Tensor`. The maximum number of
iterations for BFGS updates.
parallel_iterations: Positive integer. The number of iterations allowed to
run in parallel.
name: (Optional) Python str. The name prefixed to the ops created by this
function. If not supplied, the default name 'minimize' is used.
Returns:
optimizer_results: A namedtuple containing the following items:
converged: Scalar boolean tensor indicating whether the minimum was
found within tolerance.
failed: Scalar boolean tensor indicating whether a line search
step failed to find a suitable step size satisfying Wolfe
conditions. In the absence of any constraints on the
number of objective evaluations permitted, this value will
be the complement of `converged`. However, if there is
a constraint and the search stopped due to available
evaluations being exhausted, both `failed` and `converged`
will be simultaneously False.
num_objective_evaluations: The total number of objective
evaluations performed.
position: A tensor containing the last argument value found
during the search. If the search converged, then
this value is the argmin of the objective function.
objective_value: A tensor containing the value of the objective
function at the `position`. If the search converged, then this is
the (local) minimum of the objective function.
objective_gradient: A tensor containing the gradient of the objective
function at the `position`. If the search converged the
max-norm of this tensor should be below the tolerance.
inverse_hessian_estimate: A tensor containing the inverse of the
estimated Hessian.
"""
with tf.name_scope(
name, 'minimize',
[initial_position, tolerance, initial_inverse_hessian_estimate]):
initial_position = tf.convert_to_tensor(initial_position,
name='initial_position')
dtype = initial_position.dtype.base_dtype
tolerance = tf.convert_to_tensor(tolerance,
dtype=dtype,
name='grad_tolerance')
f_relative_tolerance = tf.convert_to_tensor(
f_relative_tolerance, dtype=dtype, name='f_relative_tolerance')
x_tolerance = tf.convert_to_tensor(x_tolerance,
dtype=dtype,
name='x_tolerance')
max_iterations = tf.convert_to_tensor(max_iterations,
name='max_iterations')
if initial_inverse_hessian_estimate is None:
# Control inputs are an optional list of tensors to evaluate before
# the start of the search procedure. These can be used to assert the
# validity of inputs to the search procedure.
control_inputs = None
domain_shape = distribution_util.prefer_static_shape(
initial_position)
inv_hessian_shape = tf.concat([domain_shape, domain_shape], 0)
initial_inv_hessian = tf.eye(tf.size(initial_position),
dtype=dtype)
initial_inv_hessian = tf.reshape(initial_inv_hessian,
inv_hessian_shape,
name='initial_inv_hessian')
else:
# If an initial inverse Hessian is supplied, these control inputs ensure
# that it is positive definite and symmetric.
initial_inv_hessian = tf.convert_to_tensor(
initial_inverse_hessian_estimate,
dtype=dtype,
name='initial_inv_hessian')
control_inputs = _inv_hessian_control_inputs(
initial_inv_hessian, initial_position)
# The `state` here is a `BfgsOptimizerResults` tuple with values for the
# current state of the algorithm computation.
def _cond(state):
"""Stopping condition for the algorithm."""
keep_going = tf.logical_not(state.converged | state.failed | (
state.num_iterations >= max_iterations))
return keep_going
def _body(state):
"""Main optimization loop."""
search_direction = _get_search_direction(
state.inverse_hessian_estimate, state.objective_gradient)
derivative_at_start_pt = tf.reduce_sum(state.objective_gradient *
search_direction)
# If the derivative at the start point is not negative, reset the
# Hessian estimate and recompute the search direction.
needs_reset = derivative_at_start_pt >= 0
def _reset_search_dirn():
search_direction = _get_search_direction(
initial_inv_hessian, state.objective_gradient)
return search_direction, initial_inv_hessian
search_direction, inv_hessian_estimate = tf.contrib.framework.smart_cond(
needs_reset,
true_fn=_reset_search_dirn,
false_fn=lambda:
(search_direction, state.inverse_hessian_estimate))
# Replace the hessian estimate in the state, in case it had to be reset.
current_state = bfgs_utils.update_fields(
state, inverse_hessian_estimate=inv_hessian_estimate)
next_state = bfgs_utils.line_search_step(
current_state, value_and_gradients_function, search_direction,
tolerance, f_relative_tolerance, x_tolerance)
# If not failed or converged, update the Hessian.
state_after_inv_hessian_update = tf.contrib.framework.smart_cond(
next_state.converged | next_state.failed, lambda: next_state,
lambda: _update_inv_hessian(current_state, next_state))
return [state_after_inv_hessian_update]
kwargs = bfgs_utils.get_initial_state_args(
value_and_gradients_function, initial_position, tolerance,
control_inputs)
kwargs['inverse_hessian_estimate'] = initial_inv_hessian
initial_state = BfgsOptimizerResults(**kwargs)
return tf.while_loop(_cond,
_body, [initial_state],
parallel_iterations=parallel_iterations)[0]
