TensorFlow定义实现梯度计算的图生成
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“”“实现梯度计算的图生成.”“”
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
import collections
import contextlib
import warnings
import numpy as np
import six
from six.moves import xrange # pylint: disable=redefined-builtin
from tensorflow.core.framework import attr_value_pb2
from tensorflow.python.framework import constant_op
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import ops
from tensorflow.python.framework import tensor_shape
from tensorflow.python.framework import tensor_util
from tensorflow.python.ops import array_grad # pylint: disable=unused-import
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import control_flow_grad # pylint: disable=unused-import
from tensorflow.python.ops import control_flow_ops
from tensorflow.python.ops import functional_ops
from tensorflow.python.ops import image_grad # pylint: disable=unused-import
from tensorflow.python.ops import linalg_grad # pylint: disable=unused-import
from tensorflow.python.ops import linalg_ops # pylint: disable=unused-import
from tensorflow.python.ops import logging_ops # pylint: disable=unused-import
from tensorflow.python.ops import math_grad # pylint: disable=unused-import
from tensorflow.python.ops import math_ops
from tensorflow.python.ops import resource_variable_ops
from tensorflow.python.ops import spectral_grad # pylint: disable=unused-import
from tensorflow.python.platform import tf_logging as logging
# Warn the user if we convert a sparse representation to dense with at
# least this number of elements.
_LARGE_SPARSE_NUM_ELEMENTS = 100000000
def _IndexedSlicesToTensor(value, dtype=None, name=None, as_ref=False):
"""Converts an IndexedSlices object `value` to a Tensor.
NOTE(mrry): This function is potentially expensive.
Args:
value: An ops.IndexedSlices object.
dtype: The dtype of the Tensor to be returned.
name: Optional name to use for the returned Tensor.
as_ref: True if a ref is requested.
Returns:
A dense Tensor representing the values in the given IndexedSlices.
Raises:
ValueError: If the IndexedSlices does not have the same dtype.
"""
_ = as_ref
if dtype and not dtype.is_compatible_with(value.dtype):
raise ValueError(
"Tensor conversion requested dtype %s for IndexedSlices with dtype %s" %
(dtype.name, value.dtype.name))
if value.dense_shape is None:
raise ValueError(
"Tensor conversion requested for IndexedSlices without dense_shape: %s"
% str(value))
# TODO(mrry): Consider adding static shape information to
# IndexedSlices, to avoid using numpy here.
dense_shape_value = tensor_util.constant_value(value.dense_shape)
if dense_shape_value is not None:
num_elements = np.prod(dense_shape_value)
if num_elements >= _LARGE_SPARSE_NUM_ELEMENTS:
warnings.warn(
"Converting sparse IndexedSlices to a dense Tensor with %d elements. "
"This may consume a large amount of memory." % num_elements)
else:
warnings.warn(
"Converting sparse IndexedSlices to a dense Tensor of unknown shape. "
"This may consume a large amount of memory.")
return math_ops.unsorted_segment_sum(
value.values, value.indices, value.dense_shape[0], name=name)
ops.register_tensor_conversion_function(ops.IndexedSlices,
_IndexedSlicesToTensor)
def _MarkReachedOps(from_ops, reached_ops):
"""Mark all ops reached from "from_ops".
Args:
from_ops: list of Operations.
reached_ops: list of booleans, indexed by operation id.
"""
queue = collections.deque()
queue.extend(from_ops)
while queue:
op = queue.popleft()
if not reached_ops[op._id]:
reached_ops[op._id] = True
for output in op.outputs:
queue.extend(output.consumers())
def _GatherInputs(to_ops, reached_ops):
"""List all inputs of to_ops that are in reached_ops.
Args:
to_ops: list of Operations.
reached_ops: list of booleans, indexed by operation id.
Returns:
The list of all inputs of to_ops that are in reached_ops.
That list includes all elements of to_ops.
"""
inputs = []
queue = collections.deque()
queue.extend(to_ops)
while queue:
op = queue.popleft()
# We are interested in this op.
if reached_ops[op._id]:
inputs.append(op)
# Clear the boolean so we won't add the inputs again.
reached_ops[op._id] = False
for inp in op.inputs:
queue.append(inp.op)
return inputs
def _PendingCount(graph, to_ops, from_ops, colocate_gradients_with_ops):
"""Initialize the pending count for ops between two lists of Operations.
'pending_count[op._id]' indicates the number of backprop inputs
to this operation.
Args:
graph: a Graph.
to_ops: list of Operations.
from_ops: list of Operations.
colocate_gradients_with_ops: Python bool. See docstring of gradients().
Returns:
A tuple containing: (1) a list of integers indexed by operation id,
indicating the number of backprop inputs to this operation, and (2)
a ControlFlowState object which is not None if the ops between from_ops
and to_ops contain control flow loops.
"""
# Mark reachable ops from from_ops.
reached_ops = [False] * (graph._last_id + 1)
for op in to_ops:
reached_ops[op._id] = True
_MarkReachedOps(from_ops, reached_ops)
# Mark between ops.
between_ops = [False] * (graph._last_id + 1)
between_op_list = []
queue = collections.deque()
queue.extend(to_ops)
while queue:
op = queue.popleft()
# We are interested in this op.
if reached_ops[op._id]:
between_ops[op._id] = True
between_op_list.append(op)
# Clear the boolean so we won't add the inputs again.
reached_ops[op._id] = False
for inp in op.inputs:
queue.append(inp.op)
# 'loop_state' is None if there are no while loops.
loop_state = control_flow_ops.MaybeCreateControlFlowState(
between_op_list, between_ops, colocate_gradients_with_ops)
# Initialize pending count for between ops.
pending_count = [0] * (graph._last_id + 1)
for op in between_op_list:
for x in op.inputs:
if between_ops[x.op._id]:
pending_count[x.op._id] += 1
return pending_count, loop_state
def _AsList(x):
return x if isinstance(x, (list, tuple)) else [x]
def _DefaultGradYs(grad_ys, ys, colocate_gradients_with_ops):
"""Fill in default values for grad_ys.
Args:
grad_ys: List of gradients, can contain None.
ys: List of tensors.
colocate_gradients_with_ops: If True, try colocating gradients with
the corresponding op.
Returns:
A list of gradients to use, without None.
Raises:
ValueError: If sizes of gradients and inputs don't match
TypeError: If type of any gradient is not valid for its input.
"""
if len(grad_ys) != len(ys):
raise ValueError("Passed %d grad_ys for %d ys" % (len(grad_ys), len(ys)))
grad_ys = ops.convert_n_to_tensor_or_indexed_slices(grad_ys, name="grad_y")
for i in xrange(len(grad_ys)):
grad_y = grad_ys[i]
y = ys[i]
if grad_y is None:
if y.dtype.is_complex:
raise TypeError(
"Gradients of complex tensors must set grad_ys (y.dtype = %r)" %
y.dtype)
with _maybe_colocate_with(y.op, colocate_gradients_with_ops):
grad_ys[i] = array_ops.fill(
array_ops.shape(y), constant_op.constant(
1, dtype=y.dtype))
continue
if y.dtype.is_floating or y.dtype.is_integer:
if not grad_y.dtype.is_floating and not grad_y.dtype.is_integer:
raise TypeError("Gradient type %s generated for real or "
"integer-valued tensor %s with type %s must be "
"real or integer" %
(dtypes.as_dtype(grad_y.dtype).name, y,
dtypes.as_dtype(y.dtype).name))
elif y.dtype.is_complex:
if not grad_y.dtype.is_complex:
raise TypeError("Gradient type %s generated for complex-valued "
"tensor %s with type %s must be real" %
(dtypes.as_dtype(grad_y.dtype).name, y,
dtypes.as_dtype(y.dtype).name))
else:
raise TypeError("Tensor %s with type %s must be numeric "
"to obtain a default gradient" %
(y, dtypes.as_dtype(y.dtype).name))
return grad_ys
def _IsTrainable(tensor):
dtype = dtypes.as_dtype(tensor.dtype)
return dtype.base_dtype in (dtypes.float16, dtypes.float32, dtypes.float64,
dtypes.complex64, dtypes.complex128)
def _VerifyGeneratedGradients(grads, op):
"""Verify that gradients are valid in number and type.
Args:
grads: List of generated gradients.
op: Operation for which the gradients where generated.
Raises:
ValueError: if sizes of gradients and inputs don't match.
TypeError: if type of any gradient is not valid for its input.
"""
if len(grads) != len(op.inputs):
raise ValueError("Num gradients %d generated for op %s do not match num "
"inputs %d" % (len(grads), op.node_def, len(op.inputs)))
def _StopOps(from_ops, pending_count):
"""The set of ops that terminate the gradient computation.
This computes the frontier of the forward graph *before* which backprop
should stop. Operations in the returned set will not be differentiated.
This set is defined as the subset of `from_ops` containing ops that have
no predecessor in `from_ops`. `pending_count` is the result of
`_PendingCount(g, xs, from_ops)`. An 'op' has predecessors in `from_ops`
iff pending_count[op._id] > 0.
Args:
from_ops: list of Operations.
pending_count: List of integers, indexed by operation id.
Returns:
The set of operations.
"""
stop_ops = set()
for op in from_ops:
is_stop_op = True
for inp in op.inputs:
if pending_count[inp.op._id] > 0:
is_stop_op = False
break
if is_stop_op:
stop_ops.add(op._id)
return stop_ops
@contextlib.contextmanager
def _maybe_colocate_with(op, colocate_gradients_with_ops):
"""Context to colocate with `op` if `colocate_gradients_with_ops`."""
if colocate_gradients_with_ops:
with ops.colocate_with(op):
yield
else:
yield
def _SymGrad(op, out_grads):
"""Backprop through a function call node op given its outputs' gradients."""
f_in = [x for x in op.inputs] + out_grads
f_types = [x.dtype for x in op.inputs]
f = attr_value_pb2.NameAttrList()
f.name = op.type
for k in op.node_def.attr:
f.attr[k].CopyFrom(op.node_def.attr[k])
# pylint: disable=protected-access
in_grads = functional_ops._symbolic_gradient(input=f_in, Tout=f_types, f=f)
# pylint: enable=protected-access
return in_grads
def _MaybeCompile(scope, op, func, grad_fn):
"""Compile the calculation in grad_fn if op was marked as compiled."""
scope = scope.rstrip("/").replace("/", "_")
if func is not None:
xla_compile = func.definition.attr["_XlaCompile"].b
xla_separate_compiled_gradients = func.definition.attr[
"_XlaSeparateCompiledGradients"].b
xla_scope = func.definition.attr["_XlaScope"].s.decode()
else:
try:
xla_compile = op.get_attr("_XlaCompile")
xla_separate_compiled_gradients = op.get_attr(
"_XlaSeparateCompiledGradients")
xla_scope = op.get_attr("_XlaScope").decode()
except ValueError:
return grad_fn() # Exit early
if not xla_compile:
return grad_fn() # Exit early
# If the gradients are supposed to be compiled separately, we give them a
# _XlaScope name that is based on the name_scope of the gradients. Otherwise
# they just inherit the existing _XlaScope name, which lets them be merged
# together with the non-gradient computation.
if xla_separate_compiled_gradients:
xla_grad_scope = "%s_grad_%s" % (xla_scope, scope)
else:
xla_grad_scope = xla_scope
attrs = {
"_XlaCompile": attr_value_pb2.AttrValue(b=xla_compile),
"_XlaScope": attr_value_pb2.AttrValue(s=xla_grad_scope.encode())
}
with ops.get_default_graph()._attr_scope(attrs): # pylint: disable=protected-access
return grad_fn()
def gradients(ys,
xs,
grad_ys=None,
name="gradients",
colocate_gradients_with_ops=False,
gate_gradients=False,
aggregation_method=None):
"""Constructs symbolic partial derivatives of sum of `ys` w.r.t. x in `xs`.
`ys` and `xs` are each a `Tensor` or a list of tensors. `grad_ys`
is a list of `Tensor`, holding the gradients received by the
`ys`. The list must be the same length as `ys`.
`gradients()` adds ops to the graph to output the partial
derivatives of `ys` with respect to `xs`. It returns a list of
`Tensor` of length `len(xs)` where each tensor is the `sum(dy/dx)`
for y in `ys`.
`grad_ys` is a list of tensors of the same length as `ys` that holds
the initial gradients for each y in `ys`. When `grad_ys` is None,
we fill in a tensor of '1's of the shape of y for each y in `ys`. A
user can provide their own initial `grad_ys` to compute the
derivatives using a different initial gradient for each y (e.g., if
one wanted to weight the gradient differently for each value in
each y).
Args:
ys: A `Tensor` or list of tensors to be differentiated.
xs: A `Tensor` or list of tensors to be used for differentiation.
grad_ys: Optional. A `Tensor` or list of tensors the same size as
`ys` and holding the gradients computed for each y in `ys`.
name: Optional name to use for grouping all the gradient ops together.
defaults to 'gradients'.
colocate_gradients_with_ops: If True, try colocating gradients with
the corresponding op.
gate_gradients: If True, add a tuple around the gradients returned
for an operations. This avoids some race conditions.
aggregation_method: Specifies the method used to combine gradient terms.
Accepted values are constants defined in the class `AggregationMethod`.
Returns:
A list of `sum(dy/dx)` for each x in `xs`.
Raises:
LookupError: if one of the operations between `x` and `y` does not
have a registered gradient function.
ValueError: if the arguments are invalid.
"""
ys = _AsList(ys)
xs = _AsList(xs)
if grad_ys is None:
grad_ys = [None] * len(ys)
else:
grad_ys = _AsList(grad_ys)
with ops.name_scope(name, "gradients", ys + xs + grad_ys) as grad_scope:
ys = ops.convert_n_to_tensor_or_indexed_slices(ys, name="y")
xs = [x.handle if isinstance(x, resource_variable_ops.ResourceVariable)
else x
for x in xs]
xs = ops.internal_convert_n_to_tensor_or_indexed_slices(xs, name="x",
as_ref=True)
grad_ys = _DefaultGradYs(grad_ys, ys, colocate_gradients_with_ops)
# The approach we take here is as follows: Create a list of all ops in the
# subgraph between the ys and xs. Visit these ops in reverse order of ids
# to ensure that when we visit an op the gradients w.r.t its outputs have
# been collected. Then aggregate these gradients if needed, call the op's
# gradient function, and add the generated gradients to the gradients for
# its input.
# Initialize the pending count for ops in the connected subgraph from ys
# to the xs.
if len(ys) > 1:
ys = [array_ops.identity(y) if y.consumers() else y for y in ys]
to_ops = [t.op for t in ys]
from_ops = [t.op for t in xs]
pending_count, loop_state = _PendingCount(ops.get_default_graph(), to_ops,
from_ops,
colocate_gradients_with_ops)
# Iterate over the collected ops.
#
# grads: op => list of gradients received on each output endpoint of the
# op. The gradients for each endpoint are initially collected as a list.
# When it is time to call the op's gradient function, for each endpoint we
# aggregate the list of received gradients into a Add() Operation if there
# is more than one.
grads = {}
# Add the initial gradients for the ys.
for y, grad_y in zip(ys, grad_ys):
_SetGrad(grads, y, grad_y)
# Initialize queue with to_ops.
queue = collections.deque()
# Add the ops in 'to_ops' into the queue.
to_ops_set = set()
for op in to_ops:
# 'ready' handles the case where one output gradient relies on
# another output's gradient.
# pylint: disable=protected-access
ready = (pending_count[op._id] == 0)
if ready and op._id not in to_ops_set:
to_ops_set.add(op._id)
queue.append(op)
# pylint: enable=protected-access
if loop_state:
loop_exits = loop_state.ProcessUnusedLoopExits(pending_count, to_ops_set)
for y in loop_exits:
if _IsTrainable(y):
_SetGrad(grads, y, loop_state.ZerosLikeForExit(y))
queue.append(y.op)
# The set of 'from_ops'.
stop_ops = _StopOps(from_ops, pending_count)
while queue:
# generate gradient subgraph for op.
op = queue.popleft()
with _maybe_colocate_with(op, colocate_gradients_with_ops):
if loop_state:
loop_state.EnterGradWhileContext(op, before=True)
out_grads = _AggregatedGrads(grads, op, loop_state, aggregation_method)
if loop_state:
loop_state.ExitGradWhileContext(op, before=True)
grad_fn = None
# pylint: disable=protected-access
func_call = None
is_func_call = ops.get_default_graph()._is_function(op.type)
has_out_grads = any(isinstance(g, ops.Tensor) or g for g in out_grads)
if has_out_grads and (op._id not in stop_ops):
if is_func_call:
func_call = ops.get_default_graph()._get_function(op.type)
grad_fn = func_call.python_grad_func
# pylint: enable=protected-access
else:
# A grad_fn must be defined, either as a function or as None
# for ops that do not have gradients.
try:
grad_fn = ops.get_gradient_function(op)
except LookupError:
raise LookupError(
"No gradient defined for operation '%s' (op type: %s)" %
(op.name, op.type))
if loop_state:
loop_state.EnterGradWhileContext(op, before=False)
if (grad_fn or is_func_call) and has_out_grads:
# NOTE: If _AggregatedGrads didn't compute a value for the i'th
# output, it means that the cost does not depend on output[i],
# therefore dC/doutput[i] is 0.
for i, out_grad in enumerate(out_grads):
if (not isinstance(out_grad, ops.Tensor) and
not out_grad) and _IsTrainable(op.outputs[i]):
# Only floating-point outputs get a zero gradient. Gradient
# functions should ignore the gradient for other outputs.
# TODO(apassos) gradients of resource handles might be an
# issue here because of zeros.
if loop_state:
out_grads[i] = loop_state.ZerosLike(op, i)
else:
out_grads[i] = control_flow_ops.ZerosLikeOutsideLoop(op, i)
with ops.name_scope(op.name + "_grad"):
# pylint: disable=protected-access
with ops.get_default_graph()._original_op(op):
# pylint: enable=protected-access
if grad_fn:
# If grad_fn was found, do not use SymbolicGradient even for
# functions.
in_grads = _MaybeCompile(
grad_scope, op, func_call, lambda: grad_fn(op, *out_grads))
else:
# For function call ops, we add a 'SymbolicGradient'
# node to the graph to compute gradients.
in_grads = _MaybeCompile(
grad_scope, op, func_call, lambda: _SymGrad(op, out_grads))
in_grads = _AsList(in_grads)
_VerifyGeneratedGradients(in_grads, op)
if gate_gradients and len(
[x for x in in_grads if x is not None]) > 1:
in_grads = control_flow_ops.tuple(in_grads)
_LogOpGradients(op, out_grads, in_grads)
else:
# If no grad_fn is defined or none of out_grads is available,
# just propagate a list of None backwards.
in_grads = [None] * len(op.inputs)
for t_in, in_grad in zip(op.inputs, in_grads):
if in_grad is not None:
if (isinstance(in_grad, ops.Tensor) and
t_in.dtype != dtypes.resource):
in_grad.set_shape(t_in.get_shape())
_SetGrad(grads, t_in, in_grad)
if loop_state:
loop_state.ExitGradWhileContext(op, before=False)
# Update pending count for the inputs of op and enqueue ready ops.
_UpdatePendingAndEnqueueReady(grads, op, queue, pending_count, loop_state)
if loop_state:
loop_state.PostProcessing()
return [_GetGrad(grads, x) for x in xs]
def _HasAnyNotNoneGrads(grads, op):
"""Return true iff op has real gradient."""
out_grads = _GetGrads(grads, op)
for out_grad in out_grads:
if isinstance(out_grad, (ops.Tensor, ops.IndexedSlices)):
return True
if out_grad and isinstance(out_grad, collections.Sequence):
if any([g is not None for g in out_grad]):
return True
return False
def _UpdatePendingAndEnqueueReady(grads, op, queue, pending_count, loop_state):
"""Update pending count for the inputs of op and enqueue ready ops."""
for x in op.inputs:
# pylint: disable=protected-access
pending_count[x.op._id] -= 1
ready = (pending_count[x.op._id] == 0)
if loop_state and not ready:
ready = (pending_count[x.op._id] > 0 and
control_flow_ops.IsLoopSwitch(x.op))
# pylint: enable=protected-access
if ready:
if control_flow_ops.IsLoopExit(x.op):
# if x is an exit without real gradient, defer processing them.
grad_state = loop_state.GetGradState(x.op, before=False)
grad_state.deferred_exits.append(x)
grad_state.pending_exits_count -= 1
if grad_state.pending_exits_count == 0:
# We now have all the exits so process them.
has_real_grad = False
for y in grad_state.deferred_exits:
if _HasAnyNotNoneGrads(grads, y.op):
has_real_grad = True
queue.append(y.op)
else:
grad_state.unused_exits.append(y)
if has_real_grad:
# For an unused exit, if it has floating-point outputs, backprop
# a zero gradient. Otherwise, just ignore it.
for y in grad_state.unused_exits:
if _IsTrainable(y):
_SetGrad(grads, y, loop_state.ZerosLikeForExit(y))
queue.append(y.op)
else:
# All exits are "unused" so use None as gradient.
for y in grad_state.unused_exits:
queue.append(y.op)
else:
queue.append(x.op)
def _SetGrad(grads, t, grad):
"""Sets gradient "grad" in "grads" for tensor "t"."""
op = t.op
op_grads = grads.get(op)
if not op_grads:
op_grads = [[] for _ in xrange(len(op.outputs))]
grads[op] = op_grads
t_grads = op_grads[t.value_index]
if isinstance(t_grads, list):
t_grads.append(grad)
else:
assert control_flow_ops.IsLoopSwitch(op)
op_grads[t.value_index] = grad
def _GetGrad(grads, t):
"""Gets gradient for tensor "t"."""
op = t.op
op_grads = grads.get(op)
if not op_grads:
return None
t_grad = op_grads[t.value_index]
assert not isinstance(t_grad, list), (
"gradients list should have been aggregated by now.")
return t_grad
def _GetGrads(grads, op):
"""Gets all gradients for op."""
if op in grads:
return grads[op]
else:
return [[] for _ in xrange(len(op.outputs))]
def _HandleNestedIndexedSlices(grad):
assert isinstance(grad, ops.IndexedSlices)
if isinstance(grad.values, ops.Tensor):
return grad
else:
assert isinstance(grad.values, ops.IndexedSlices)
g = _HandleNestedIndexedSlices(grad.values)
return ops.IndexedSlices(g.values,
array_ops.gather(grad.indices, g.indices),
g.dense_shape)
def _AccumulatorShape(inputs):
shape = tensor_shape.unknown_shape()
for i in inputs:
if isinstance(i, ops.Tensor):
shape = shape.merge_with(i.get_shape())
return shape
def _LogOpGradients(op, out_grads, in_grads):
"""Log the in and out grads of an op."""
logging.vlog(1, "Gradient for '" + op.name + "'")
def _FilterGrad(x):
if x is None:
return False
if isinstance(x, (list, tuple)):
return bool(x)
else:
return True
logging.vlog(1, " in --> %s",
", ".join([x.name for x in out_grads if _FilterGrad(x)]))
logging.vlog(1, " out --> %s",
", ".join([x.name for x in in_grads if _FilterGrad(x)]))
def _MultiDeviceAddN(tensor_list):
"""Adds tensors from potentially multiple devices."""
# Basic function structure comes from control_flow_ops.group().
# Sort tensors according to their devices.
tensors_on_device = collections.defaultdict(lambda: [])
for tensor in tensor_list:
tensors_on_device[tensor.device].append(tensor)
# For each device, add the tensors on that device first.
# Then gather the partial sums from multiple devices.
# TODO(sjhwang): Create hierarchical aggregation tree as pbar's suggestion.
# E.g., aggregate per GPU, then per task, and so on.
summands = []
def DeviceKey(dev):
return "" if dev is None else dev
for dev in sorted(six.iterkeys(tensors_on_device), key=DeviceKey):
tensors = tensors_on_device[dev]
with ops.colocate_with(tensors[0].op, ignore_existing=True):
summands.append(math_ops.add_n(tensors))
return math_ops.add_n(summands)
class AggregationMethod(object):
"""A class listing aggregation methods used to combine gradients.
Computing partial derivatives can require aggregating gradient
contributions. This class lists the various methods that can
be used to combine gradients in the graph:
* `ADD_N`: All of the gradient terms are summed as part of one
operation using the "AddN" op. It has the property that all
gradients must be ready before any aggregation is performed.
* `DEFAULT`: The system-chosen default aggregation method.
"""
ADD_N = 0
DEFAULT = ADD_N
# The following are experimental and may not be supported in future releases.
EXPERIMENTAL_TREE = 1
EXPERIMENTAL_ACCUMULATE_N = 2
def _AggregatedGrads(grads, op, loop_state, aggregation_method=None):
"""Get the aggregated gradients for op.
Args:
grads: The map of memoized gradients.
op: The op to get gradients for.
loop_state: An object for maintaining the state of the while loops in the
graph. It is of type ControlFlowState. None if the graph
contains no while loops.
aggregation_method: Specifies the method used to combine gradient terms.
Accepted values are constants defined in the class `AggregationMethod`.
Returns:
A list of gradients, one per each output of `op`. If the gradients
for a particular output is a list, this function aggregates it
before returning.
Raises:
TypeError: if the incoming grads are not Tensors or IndexedSlices.
ValueError: if the arguments are invalid.
"""
if aggregation_method is None:
aggregation_method = AggregationMethod.DEFAULT
if aggregation_method not in [
AggregationMethod.ADD_N, AggregationMethod.EXPERIMENTAL_TREE,
AggregationMethod.EXPERIMENTAL_ACCUMULATE_N
]:
raise ValueError("Invalid aggregation_method specified %s." %
aggregation_method)
out_grads = _GetGrads(grads, op)
for i, out_grad in enumerate(out_grads):
if loop_state:
if isinstance(out_grad, (ops.Tensor, ops.IndexedSlices)):
assert control_flow_ops.IsLoopSwitch(op)
continue
# Grads have to be Tensors or IndexedSlices
if (isinstance(out_grad, collections.Sequence) and not all([
isinstance(g, (ops.Tensor, ops.IndexedSlices)) for g in out_grad
if g is not None
])):
raise TypeError("gradients have to be either all Tensors "
"or all IndexedSlices")
# Aggregate multiple gradients, and convert [] to None.
if out_grad:
if len(out_grad) < 2:
used = "nop"
out_grads[i] = out_grad[0]
elif all([isinstance(g, ops.Tensor) for g in out_grad if g is not None]):
tensor_shape = _AccumulatorShape(out_grad)
if (aggregation_method == AggregationMethod.EXPERIMENTAL_ACCUMULATE_N
and len(out_grad) > 2 and tensor_shape.is_fully_defined()):
# The benefit of using AccumulateN is that its inputs can be combined
# in any order and this can allow the expression to be evaluated with
# a smaller memory footprint. When used with gpu_allocator_retry,
# it is possible to compute a sum of terms which are much larger than
# total GPU memory.
# AccumulateN can currently only be used if we know the shape for
# an accumulator variable. If this is not known, or if we only have
# 2 grads then we fall through to the "tree" case below.
used = "accumulate_n"
out_grads[i] = math_ops.accumulate_n(out_grad)
elif aggregation_method in [
AggregationMethod.EXPERIMENTAL_TREE,
AggregationMethod.EXPERIMENTAL_ACCUMULATE_N
]:
# Aggregate all gradients by doing pairwise sums: this may
# reduce performance, but it can improve memory because the
# gradients can be released earlier.
#
# TODO(vrv): Consider replacing this with a version of
# tf.AddN() that eagerly frees its inputs as soon as they are
# ready, so the order of this tree does not become a problem.
used = "tree"
with ops.name_scope(op.name + "_gradient_sum"):
running_sum = out_grad[0]
for grad in out_grad[1:]:
running_sum = math_ops.add_n([running_sum, grad])
out_grads[i] = running_sum
else:
used = "add_n"
out_grads[i] = _MultiDeviceAddN(out_grad)
logging.vlog(2, " _AggregatedGrads %d x %s using %s",
len(out_grad), tensor_shape, used)
else:
out_grad = math_ops._as_indexed_slices_list(
[g for g in out_grad if g is not None])
out_grad = [_HandleNestedIndexedSlices(x) for x in out_grad]
# Form IndexedSlices out of the concatenated values and
# indices.
out_grads[i] = ops.IndexedSlices(
array_ops.concat([x.values for x in out_grad], 0),
array_ops.concat([x.indices for x in out_grad], 0),
out_grad[0].dense_shape)
else: # not out_grad
# out_grads[i] is [], thus its aggregation is simply None.
out_grads[i] = None
return out_grads
# TODO(vrv): Make this available when we want to make it public.
def _hessian_vector_product(ys, xs, v):
"""Multiply the Hessian of `ys` wrt `xs` by `v`.
This is an efficient construction that uses a backprop-like approach
to compute the product between the Hessian and another vector. The
Hessian is usually too large to be explicitly computed or even
represented, but this method allows us to at least multiply by it
for the same big-O cost as backprop.
Implicit Hessian-vector products are the main practical, scalable way
of using second derivatives with neural networks. They allow us to
do things like construct Krylov subspaces and approximate conjugate
gradient descent.
Example: if `y` = 1/2 `x`^T A `x`, then `hessian_vector_product(y,
x, v)` will return an expression that evaluates to the same values
as (A + A.T) `v`.
Args:
ys: A scalar value, or a tensor or list of tensors to be summed to
yield a scalar.
xs: A list of tensors that we should construct the Hessian over.
v: A list of tensors, with the same shapes as xs, that we want to
multiply by the Hessian.
Returns:
A list of tensors (or if the list would be length 1, a single tensor)
containing the product between the Hessian and `v`.
Raises:
ValueError: `xs` and `v` have different length.
"""
# Validate the input
length = len(xs)
if len(v) != length:
raise ValueError("xs and v must have the same length.")
# First backprop
grads = gradients(ys, xs)
assert len(grads) == length
elemwise_products = [
math_ops.multiply(grad_elem, array_ops.stop_gradient(v_elem))
for grad_elem, v_elem in zip(grads, v) if grad_elem is not None
]
# Second backprop
return gradients(elemwise_products, xs)
def hessians(ys, xs, name="hessians", colocate_gradients_with_ops=False,
gate_gradients=False, aggregation_method=None):
"""Constructs the Hessian of sum of `ys` with respect to `x` in `xs`.
`hessians()` adds ops to the graph to output the Hessian matrix of `ys`
with respect to `xs`. It returns a list of `Tensor` of length `len(xs)`
where each tensor is the Hessian of `sum(ys)`. This function currently
only supports evaluating the Hessian with respect to (a list of) one-
dimensional tensors.
The Hessian is a matrix of second-order partial derivatives of a scalar
tensor (see https://en.wikipedia.org/wiki/Hessian_matrix for more details).
Args:
ys: A `Tensor` or list of tensors to be differentiated.
xs: A `Tensor` or list of tensors to be used for differentiation.
name: Optional name to use for grouping all the gradient ops together.
defaults to 'hessians'.
colocate_gradients_with_ops: See `gradients()` documentation for details.
gate_gradients: See `gradients()` documentation for details.
aggregation_method: See `gradients()` documentation for details.
Returns:
A list of Hessian matrices of `sum(y)` for each `x` in `xs`.
Raises:
LookupError: if one of the operations between `xs` and `ys` does not
have a registered gradient function.
ValueError: if the arguments are invalid or not supported. Currently,
this function only supports one-dimensional `x` in `xs`.
"""
xs = _AsList(xs)
kwargs = {
'colocate_gradients_with_ops': colocate_gradients_with_ops,
'gate_gradients': gate_gradients,
'aggregation_method': aggregation_method
}
# Compute a hessian matrix for each x in xs
hessians = []
for i, x in enumerate(xs):
# Check dimensions
ndims = x.get_shape().ndims
if ndims is None:
raise ValueError('Cannot compute Hessian because the dimensionality of '
'element number %d of `xs` cannot be determined' % i)
elif ndims != 1:
raise ValueError('Computing hessians is currently only supported for '
'one-dimensional tensors. Element number %d of `xs` has '
'%d dimensions.' % (i, ndims))
with ops.name_scope(name + '_first_derivative'):
# Compute the partial derivatives of the input with respect to all
# elements of `x`
_gradients = gradients(ys, x, **kwargs)[0]
# Unpack the gradients into a list so we can take derivatives with
# respect to each element
_gradients = array_ops.unstack(_gradients)
with ops.name_scope(name + '_second_derivative'):
# Compute the partial derivatives with respect to each element of the list
_hess = [gradients(_gradient, x, **kwargs)[0] for _gradient in _gradients]
# Pack the list into a matrix and add to the list of hessians
hessians.append(array_ops.stack(_hess, name=name))
return hessians
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