def _is_finished(i, unused_alive_seq, alive_log_probs, unused_finished_seq,
finished_scores, finished_in_finished, *unused_states):
"""Checking termination condition.
We terminate when we decoded up to decode_length or the lowest scoring item
in finished has a greater score that the highest prob item in alive divided
by the max length penalty
Args:
i: loop index
alive_log_probs: probabilities of the beams. [batch_size, beam_size]
finished_scores: scores for each of these sequences.
[batch_size, beam_size]
finished_in_finished: finished bools for each of these sequences.
[batch_size, beam_size]
Returns:
Bool.
"""
# TODO(noam): support a different decode length...
# decode_length = mtf.constant(mesh, length_dim.size, dtype=tf.int32)
# del alive_log_probs, finished_scores, finished_in_finished
# return mtf.less(i, length_dim.size)
if not stop_early:
return mtf.less(i, decode_length)
max_length_penalty = mtf.pow(((5. + mtf.to_float(decode_length)) / 6.),
alpha)
# The best possible score of the most likely alive sequence.
lower_bound_alive_scores = mtf.gather(
alive_log_probs, mtf.constant(mesh, 0, dtype=tf.int32),
beam_dim) / max_length_penalty
# Now to compute the lowest score of a finished sequence in finished
# If the sequence isn't finished, we multiply it's score by 0. since
# scores are all -ve, taking the min will give us the score of the lowest
# finished item.
lowest_score_of_finished_in_finished = mtf.reduce_min(
finished_scores * mtf.to_float(finished_in_finished),
reduced_dim=beam_dim)
# If none of the sequences have finished, then the min will be 0 and
# we have to replace it by -ve INF if it is. The score of any seq in alive
# will be much higher than -ve INF and the termination condition will not
# be met.
lowest_score_of_finished_in_finished += ((1. - mtf.to_float(
mtf.reduce_any(finished_in_finished, reduced_dim=beam_dim))) *
-INF)
bound_is_met = mtf.reduce_all(
mtf.greater(lowest_score_of_finished_in_finished,
lower_bound_alive_scores))
return mtf.logical_and(mtf.less(i, decode_length),
mtf.logical_not(bound_is_met))
def attention_bias_local_block(mesh,
block_length,
memory_length,
dtype=tf.int32):
"""Bias for attention for local blocks where attention to right is disallowed.
Create the bias matrix by using two separate masks, one for the memory part
which doesn't overlap with the query and second which interacts with the query
and should be disallowed to look to the right of the current query position.
Args:
mesh: a MeshTensorflow object
block_length: a mtf.Dimension
memory_length: a mtf.Dimension
dtype: a tf.dtype
Returns:
a mtf.Tensor with shape [block_length, memory_length]
"""
memory_length = mtf.Dimension(memory_length.name, block_length.size)
memory_mask = mtf.zeros(mesh, [block_length, memory_length], dtype=dtype)
mask = mtf.cast(mtf.less(mtf.range(mesh, block_length, dtype=dtype),
mtf.range(mesh, memory_length, dtype=dtype)),
dtype=dtype)
mask = mtf.cast(mtf.concat([memory_mask, mask], memory_length.name),
dtype=tf.float32) * -1e9
return mask
def attention_mask_autoregressive(query_pos, dtype=tf.float32):
"""Bias for self-attention where attention to the right is disallowed.
Args:
query_pos: a mtf.Tensor with shape [..., length_dim]
dtype: a tf.dtype
Returns:
a mtf.Tensor with shape [..., length_dim, memory_length_dim]
"""
memory_pos = rename_length_to_memory_length(query_pos)
return mtf.cast(mtf.less(query_pos, memory_pos), dtype) * -1e9
def attention_bias_local_block(mesh, block_length, memory_length,
dtype=tf.int32):
"""Bias for attention for local blocks where attention to right is disallowed.
Args:
mesh: a MeshTensorflow object
block_length: a mtf.Dimension
memory_length: a mtf.Dimension
dtype: a tf.dtype
Returns:
a mtf.Tensor with shape [rows, cols]
"""
mask = mtf.cast(mtf.less(mtf.range(mesh, block_length, dtype=dtype),
mtf.range(mesh, memory_length, dtype=dtype)),
dtype=dtype)
mask = mtf.cast(mask, dtype=tf.float32) * -1e9
return mask
def _truncated_top_2_gating_mtf(
gates, group_dim, experts_dim, expert_capacity_dim):
"""Compute gating for mixture-of-experts in TensorFlow.
gates is usually the output of a softmax function.
The return value is a dense representation of the mapping between
the input positions in the positions in the batches sent to the experts.
TODO(noam): this function contains code factored out of
expert_utils.local_moe_tpu. Move this function to that file and
call it from both places.
Args:
gates: a Tensor
group_dim: one dimension of gates
experts_dim: one dimension of gates
expert_capacity_dim: a Dimension not in gates
Returns:
a Tensor with shape gates.shape + expert_capacity_dim
Raises:
ValueError: if group_dim has size >256
"""
gates = mtf.to_float(gates)
expert_capacity_f = float(expert_capacity_dim.size)
# Find the top expert for each position. shape=[batch, group]
index_1, gate_1 = mtf.top_1(gates, experts_dim)
# [batch, group, experts]
mask_1 = mtf.one_hot(index_1, experts_dim, dtype=gates.dtype)
if expert_capacity_dim.size > 256:
# using mtf.cumsum (implemented on TPU as bfloat16 matmul) to compute
# position in the mini-batch sent to the expert. This will cause
# very bad things to happen if expert_capacity_dim > 256.
raise ValueError(
"expert_capacity_dim.size must be <=256 to avoid roundoff errors in"
" indices - got %s" % (expert_capacity_dim,))
# [batch, group, experts]
# This is the position within the expert's mini-batch for this sequence
position_in_expert_1 = mtf.cumsum(mask_1, group_dim, exclusive=True) * mask_1
# Remove the elements that don't fit. [batch, group, experts]
mask_1 *= mtf.to_float(mtf.less(position_in_expert_1, expert_capacity_f))
# [batch, experts]
# How many examples in this sequence go to this expert
mask_1_count = mtf.reduce_sum(mask_1, reduced_dim=group_dim)
# [batch, group] - mostly ones, but zeros where something didn't fit
mask_1_flat = mtf.reduce_sum(mask_1, reduced_dim=experts_dim)
# [batch, group]
position_in_expert_1 = mtf.reduce_sum(
position_in_expert_1, reduced_dim=experts_dim)
# Weight assigned to first expert. [batch, group]
gate_1 *= mask_1_flat
# Pick a second-place expert for each position.
# We first mask out the experts that we expect to be over-capacity
# [batch, experts]
space_remaining = expert_capacity_f - mask_1_count
use_rate = (mask_1_count + 1.0) / float(group_dim.size)
# At what point in the sequence do we expect the expert to be full.
# [batch, experts]
expected_exhaustion_pos = space_remaining / use_rate
# A Tensor with shape [batch, group, experts] representing a boolean
# - whether we expect that the expert will already be full.
expected_exhausted = mtf.to_float(mtf.greater(
mtf.range(gates.mesh, group_dim, tf.float32), expected_exhaustion_pos))
masked_gates = gates - mask_1 - expected_exhausted
# This section is similar to the section above.
# [batch, group]
index_2, gate_2 = mtf.top_1(masked_gates, experts_dim)
# [batch, group, experts]
mask_2 = mtf.one_hot(index_2, experts_dim, dtype=gates.dtype)
# [batch, group, experts]
position_in_expert_2 = (
mtf.cumsum(mask_2, group_dim, exclusive=True) + mask_1_count)
position_in_expert_2 *= mask_2
mask_2 *= mtf.to_float(mtf.less(position_in_expert_2, expert_capacity_f))
# mask_2_count = mtf.reduce_sum(mask_2, reduced_dim=experts_dim)
mask_2_flat = mtf.reduce_sum(mask_2, reduced_dim=experts_dim)
position_in_expert_2 = mtf.reduce_sum(
position_in_expert_2, reduced_dim=experts_dim)
gate_2 *= mask_2_flat
# renormalize the two gate values to add up to 1
denom = gate_1 + gate_2 + 1e-9
gate_1 /= denom
gate_2 /= denom
# [batch, group, experts, expert_capacity]
assignment = (
gate_1 * mask_1_flat
* mtf.one_hot(index_1, experts_dim)
* mtf.one_hot(mtf.to_int32(position_in_expert_1), expert_capacity_dim) +
gate_2 * mask_2_flat
* mtf.one_hot(index_2, experts_dim)
* mtf.one_hot(mtf.to_int32(position_in_expert_2), expert_capacity_dim))
return assignment
def _top_2_gating(inputs,
outer_expert_dims,
experts_dim,
expert_capacity_dim,
hparams,
train,
importance=None):
"""Compute gating for mixture-of-experts in TensorFlow.
Note: until the algorithm and inferface solidify, we pass in a hyperparameters
dictionary in order not to complicate the interface in mtf_transformer.py .
Once this code moves out of "research", we should pass the hyperparameters
separately.
Hyperparameters used:
hparams.moe_use_second_place_loss: a boolean
hparams.moe_second_policy_train: a string
hparams.moe_second_policy_eval: a string
hparams.moe_second_threshold: a float
The returned forward assignment is a tensor used to map (via einsum) from the
inputs to the expert_inputs. Likewise, the returned combine_tensor is
used to map (via einsum) from the expert outputs to the outputs. Both the
forward and backward assignments are mostly zeros. The shapes of the tensors
are as follows.
inputs: [<batch_dims>, group_size_dim, input_dim]
importance: [<batch_dims>, group_size_dim]
dispatch_tensor:
[<batch_dims>, group_size_dim, experts_dim, expert_capacity_dim]
expert_inputs:
[<batch_dims>, experts_dim, expert_capacity_dim, input_dim]
expert_outputs: [<batch_dims>, experts_dim, expert_capacity_dim, output_dim]
combine_tensor:
[<batch_dims>, group_size_dim, experts_dim, expert_capacity_dim]
outputs: [<batch_dims>, group_size_dim, output_dim]
"importance" is an optional tensor with one floating-point value for each
input vector. If the importance of an input is 1.0, then we send it to
up to 2 experts. If 0.0 < importance < 1.0, then we send it to at most
one expert. If importance == 0.0, then we send it to no experts.
We use "importance" at the second-level gating function of a hierarchical
mixture of experts. Inputs to the first-choice expert-group get importance
1.0. Inputs to the second-choice expert group get importance 0.5.
Inputs that represent padding get importance 0.0.
Args:
inputs: a mtf.Tensor with shape [<batch_dims>, group_size_dim, input_dim]
outer_expert_dims: an optional list of dimensions. This is for the case
where we are at an inner level of a hierarchical MoE.
experts_dim: a Dimension (the number of experts)
expert_capacity_dim: a Dimension (number of examples per group per expert)
hparams: model hyperparameters.
train: a boolean
importance: an optional tensor with shape [<batch_dims>, group_size_dim]
Returns:
dispatch_tensor: a Tensor with shape
[<batch_dims>, group_size_dim, experts_dim, expert_capacity_dim]
combine_tensor: a Tensor with shape
[<batch_dims>, group_size_dim, experts_dim, expert_capacity_dim]
loss: a mtf scalar
Raises:
ValueError: on illegal hyperparameters
"""
group_size_dim, unused_input_dim = inputs.shape.dims[-2:]
raw_gates = mtf.softmax(
mtf_layers.dense(inputs,
experts_dim,
use_bias=False,
expert_dims=outer_expert_dims), experts_dim)
# The internals of this function run in float32.
# bfloat16 seems to reduce quality.
raw_gates = mtf.to_float(raw_gates)
expert_capacity_f = float(expert_capacity_dim.size)
# FIND TOP 2 EXPERTS PER POSITON
# Find the top expert for each position. shape=[batch, group]
index_1, gate_1 = mtf.top_1(raw_gates, experts_dim)
# [batch, group, experts]
mask_1 = mtf.one_hot(index_1, experts_dim, dtype=raw_gates.dtype)
density_1_proxy = raw_gates
if importance is not None:
mask_1 *= mtf.to_float(mtf.equal(importance, 1.0))
gate_1 *= mtf.to_float(mtf.equal(importance, 1.0))
density_1_proxy *= mtf.to_float(mtf.equal(importance, 1.0))
gates_without_top_1 = raw_gates * (1.0 - mask_1)
# [batch, group]
index_2, gate_2 = mtf.top_1(gates_without_top_1, experts_dim)
# [batch, group, experts]
mask_2 = mtf.one_hot(index_2, experts_dim, dtype=raw_gates.dtype)
if importance is not None:
mask_2 *= mtf.to_float(mtf.greater(importance, 0.0))
denom = gate_1 + gate_2 + 1e-9
gate_1 /= denom
gate_2 /= denom
# BALANCING LOSSES
# shape = [batch, experts]
# We want to equalize the fraction of the batch assigned to each expert
density_1 = mtf.reduce_mean(mask_1, reduced_dim=group_size_dim)
# Something continuous that is correlated with what we want to equalize.
density_1_proxy = mtf.reduce_mean(density_1_proxy,
reduced_dim=group_size_dim)
density_1 = mtf.Print(
density_1, [mtf.reduce_mean(density_1, output_shape=[experts_dim])],
"density_1",
summarize=1000)
loss = (mtf.reduce_mean(density_1_proxy * density_1) *
float(experts_dim.size * experts_dim.size))
if hparams.moe_use_second_place_loss:
# Also add a loss to encourage all experts to be used equally also as the
# second-place expert. Experimentally, this seems to be a wash.
# We want to equalize the fraction of the batch assigned to each expert:
density_2 = mtf.reduce_mean(mask_2, reduced_dim=group_size_dim)
# As a proxy for density_2, we renormalize the raw gates after the top one
# has been removed.
normalized = gates_without_top_1 / (mtf.reduce_sum(
gates_without_top_1, reduced_dim=experts_dim) + 1e-9)
density_2_proxy = mtf.reduce_mean(normalized,
reduced_dim=group_size_dim)
loss_2 = (mtf.reduce_mean(density_2_proxy * density_2) *
float(experts_dim.size * experts_dim.size))
loss += loss_2 * 0.5
# Depending on the policy in the hparams, we may drop out some of the
# second-place experts.
policy = (hparams.moe_second_policy_train
if train else hparams.moe_second_policy_eval)
threshold = (hparams.moe_second_threshold_train
if train else hparams.moe_second_threshold_eval)
if policy == "all":
# Use second-place experts for all examples.
pass
elif policy == "none":
# Never use second-place experts for all examples.
mask_2 = mtf.zeros_like(mask_2)
elif policy == "threshold":
# Use second-place experts if gate_2 > threshold.
mask_2 *= mtf.to_float(mtf.greater(gate_2, threshold))
elif policy == "random":
# Use second-place experts with probablity min(1.0, gate_2 / threshold).
mask_2 *= mtf.to_float(
mtf.less(mtf.random_uniform(gate_2.mesh, gate_2.shape),
gate_2 / max(threshold, 1e-9)))
else:
raise ValueError("Unknown policy %s" % policy)
mask_2 = mtf.Print(mask_2,
[mtf.reduce_mean(mask_2, output_shape=[experts_dim])],
"density_2",
summarize=1000)
# COMPUTE ASSIGNMENT TO EXPERTS
# [batch, group, experts]
# This is the position within the expert's mini-batch for this sequence
position_in_expert_1 = mtf.cumsum(mask_1, group_size_dim,
exclusive=True) * mask_1
# Remove the elements that don't fit. [batch, group, experts]
mask_1 *= mtf.to_float(mtf.less(position_in_expert_1, expert_capacity_f))
# [batch, experts]
# How many examples in this sequence go to this expert
mask_1_count = mtf.reduce_sum(mask_1, reduced_dim=group_size_dim)
# [batch, group] - mostly ones, but zeros where something didn't fit
mask_1_flat = mtf.reduce_sum(mask_1, reduced_dim=experts_dim)
# [batch, group]
position_in_expert_1 = mtf.reduce_sum(position_in_expert_1,
reduced_dim=experts_dim)
# Weight assigned to first expert. [batch, group]
gate_1 *= mask_1_flat
# [batch, group, experts]
position_in_expert_2 = (
mtf.cumsum(mask_2, group_size_dim, exclusive=True) + mask_1_count)
position_in_expert_2 *= mask_2
mask_2 *= mtf.to_float(mtf.less(position_in_expert_2, expert_capacity_f))
# mask_2_count = mtf.reduce_sum(mask_2, reduced_dim=experts_dim)
mask_2_flat = mtf.reduce_sum(mask_2, reduced_dim=experts_dim)
gate_2 *= mask_2_flat
position_in_expert_2 = mtf.reduce_sum(position_in_expert_2,
reduced_dim=experts_dim)
# [batch, group, experts, expert_capacity]
combine_tensor = (
gate_1 * mask_1_flat * mtf.one_hot(index_1, experts_dim) *
mtf.one_hot(mtf.to_int32(position_in_expert_1), expert_capacity_dim) +
gate_2 * mask_2_flat * mtf.one_hot(index_2, experts_dim) *
mtf.one_hot(mtf.to_int32(position_in_expert_2), expert_capacity_dim))
combine_tensor = mtf.cast(combine_tensor, inputs.dtype)
loss = mtf.cast(loss, inputs.dtype)
dispatch_tensor = mtf.cast(mtf.cast(combine_tensor, tf.bool),
combine_tensor.dtype)
return dispatch_tensor, combine_tensor, loss
def _top_2_gating(inputs, experts_dim, expert_capacity_dim, max_experts,
hparams, train):
"""Compute gating for mixture-of-experts in TensorFlow.
Note: until the algorithm and inferface solidify, we pass in a hyperparameters
dictionary in order not to complicate the interface in mtf_transformer.py .
Once this code moves out of "research", we should pass the hyperparameters
separately.
Hyperparameters used:
hparams.moe_use_second_place_loss: a boolean
hparams.moe_second_policy_train: a string
hparams.moe_second_policy_eval: a string
hparams.moe_second_threshold: a float
max_experts is an float tensor with shape [batch_dim, group_dim]
indicating at most how many experts to use per example. This can be
used to prevent padding from going to experts.
The returned forward assignment is a tensor used to map (via einsum) from the
inputs to the expert_inputs. Likewise, the returned backward_assignment is
used to map (via einsum) from the expert outputs to the outputs. Both the
forward and backward assignments are mostly zeros. The shapes of all of these
are as follows.
inputs: [batch_dim, group_dim, input_dim]
forward_assignment: [batch_dim, group_dim, experts_dim, expert_capacity_dim]
expert_inputs: [batch_dim, experts_dim, expert_capacity_dim, input_dim]
expert_outputs: [batch_dim, experts_dim, expert_capacity_dim, output_dim]
backward_assignment: [batch_dim, group_dim, experts_dim, expert_capacity_dim]
outputs: [batch_dim, group_dim, output_dim]
Args:
inputs: a mtf.Tensor with shape [batch_dim, group_dim, input_dim]
experts_dim: a Dimension (the number of experts)
expert_capacity_dim: a Dimension (number of examples per group per expert)
max_experts: optional mtf.Tensor with shape [batch_dim, group_dim]
hparams: model hyperparameters.
train: a boolean
Returns:
forward_assignment: a Tensor with shape
[batch_dim, group_dim, experts_dim, expert_capacity_dim]
backward_assignment: a Tensor with shape
[batch_dim, group_dim, experts_dim, expert_capacity_dim]
loss: a mtf scalar
Raises:
ValueError: on illegal hyperparameters
"""
unused_batch_dim, group_dim, unused_input_dim = inputs.shape.dims
raw_gates = mtf.softmax(
mtf_layers.dense(inputs, experts_dim, use_bias=False), experts_dim)
expert_capacity_f = float(expert_capacity_dim.size)
# FIND TOP 2 EXPERTS PER POSITON
# Find the top expert for each position. shape=[batch, group]
index_1, gate_1 = mtf.top_1(raw_gates, experts_dim)
# [batch, group, experts]
mask_1 = mtf.one_hot(index_1, experts_dim, dtype=raw_gates.dtype)
gates_without_top_1 = raw_gates * (1.0 - mask_1)
# [batch, group]
index_2, gate_2 = mtf.top_1(gates_without_top_1, experts_dim)
# [batch, group, experts]
mask_2 = mtf.one_hot(index_2, experts_dim, dtype=raw_gates.dtype)
if max_experts is not None:
geq1 = mtf.to_float(mtf.greater_equal(max_experts, 1.0))
geq2 = mtf.to_float(mtf.greater_equal(max_experts, 2.0))
mask_1 *= geq1
mask_2 *= geq2
raw_gates *= geq1
gates_without_top_1 *= geq2
# BALANCING LOSSES
# shape = [batch, experts]
# We want to equalize the fraction of the batch assigned to each expert
density_1 = mtf.reduce_mean(mask_1, reduced_dim=group_dim)
# Something continuous that is correlated with what we want to equalize.
density_1_proxy = mtf.reduce_mean(raw_gates, reduced_dim=group_dim)
density_1 = mtf.Print(
density_1, [mtf.reduce_mean(density_1, output_shape=[experts_dim])],
"density_1",
summarize=1000)
loss = (mtf.reduce_mean(density_1_proxy * density_1) *
float(experts_dim.size * experts_dim.size))
if hparams.moe_use_second_place_loss:
# Also add a loss to encourage all experts to be used equally also as the
# second-place expert. Experimentally, this seems to be a wash.
# We want to equalize the fraction of the batch assigned to each expert:
density_2 = mtf.reduce_mean(mask_2, reduced_dim=group_dim)
# As a proxy for density_2, we renormalize the raw gates after the top one
# has been removed.
normalized = gates_without_top_1 / (mtf.reduce_sum(
gates_without_top_1, reduced_dim=experts_dim) + 1e-9)
density_2_proxy = mtf.reduce_mean(normalized, reduced_dim=group_dim)
loss_2 = (mtf.reduce_mean(density_2_proxy * density_2) *
float(experts_dim.size * experts_dim.size))
loss += loss_2 * 0.5
# Depending on the policy in the hparams, we may drop out some of the
# second-place experts.
policy = (hparams.moe_second_policy_train
if train else hparams.moe_second_policy_eval)
threshold = (hparams.moe_second_threshold_train
if train else hparams.moe_second_threshold_eval)
if policy == "all":
# Use second-place experts for all examples.
pass
elif policy == "none":
# Never use second-place experts for all examples.
mask_2 = mtf.zeros_like(mask_2)
elif policy == "threshold":
# Use second-place experts if gate_2 > threshold.
mask_2 *= mtf.to_float(mtf.greater(gate_2, threshold))
elif policy == "random":
# Use second-place experts with probablity min(1.0, gate_2 / threshold).
mask_2 *= mtf.to_float(
mtf.less(mtf.random_uniform(gate_2.mesh, gate_2.shape),
gate_2 / max(threshold, 1e-9)))
else:
raise ValueError("Unknown policy %s" % policy)
mask_2 = mtf.Print(mask_2,
[mtf.reduce_mean(mask_2, output_shape=[experts_dim])],
"density_2",
summarize=1000)
# COMPUTE ASSIGNMENT TO EXPERTS
# [batch, group, experts]
# This is the position within the expert's mini-batch for this sequence
position_in_expert_1 = mtf.cumsum(mask_1, group_dim,
exclusive=True) * mask_1
# Remove the elements that don't fit. [batch, group, experts]
mask_1 *= mtf.to_float(mtf.less(position_in_expert_1, expert_capacity_f))
# [batch, experts]
# How many examples in this sequence go to this expert
mask_1_count = mtf.reduce_sum(mask_1, reduced_dim=group_dim)
# [batch, group] - mostly ones, but zeros where something didn't fit
mask_1_flat = mtf.reduce_sum(mask_1, reduced_dim=experts_dim)
# [batch, group]
position_in_expert_1 = mtf.reduce_sum(position_in_expert_1,
reduced_dim=experts_dim)
# Weight assigned to first expert. [batch, group]
gate_1 *= mask_1_flat
# [batch, group, experts]
position_in_expert_2 = (mtf.cumsum(mask_2, group_dim, exclusive=True) +
mask_1_count)
position_in_expert_2 *= mask_2
mask_2 *= mtf.to_float(mtf.less(position_in_expert_2, expert_capacity_f))
# mask_2_count = mtf.reduce_sum(mask_2, reduced_dim=experts_dim)
mask_2_flat = mtf.reduce_sum(mask_2, reduced_dim=experts_dim)
gate_2 *= mask_2_flat
position_in_expert_2 = mtf.reduce_sum(position_in_expert_2,
reduced_dim=experts_dim)
# renormalize the two gate values to add up to 1
denom = gate_1 + gate_2 + 1e-9
gate_1 /= denom
gate_2 /= denom
# [batch, group, experts, expert_capacity]
backward_assignment = (
gate_1 * mask_1_flat * mtf.one_hot(index_1, experts_dim) *
mtf.one_hot(mtf.to_int32(position_in_expert_1), expert_capacity_dim) +
gate_2 * mask_2_flat * mtf.one_hot(index_2, experts_dim) *
mtf.one_hot(mtf.to_int32(position_in_expert_2), expert_capacity_dim))
forward_assignment = mtf.cast(mtf.cast(backward_assignment, tf.bool),
backward_assignment.dtype)
return forward_assignment, backward_assignment, loss