def masked_mean(inputs, targets, mask_id=None):
"""Mean of the inputs but counting only those where targets != mask_id."""
x = inputs.astype(np.float32)
if mask_id is None:
return np.mean(x)
unmask = 1.0 - np.equal(targets, mask_id).astype(np.float32)
return np.sum(x * unmask) / np.sum(unmask)
def masked_mean(inputs, targets, mask_id=None):
"""Mean of the inputs but counting only those where targets != mask_id."""
inputs = [x.astype(np.float32) for x in inputs]
# We assume all elements in the list contribute equally.
# TODO(lukaszkaiser): remove this assumption (e.g., when masks differ).
length = len(inputs)
if mask_id is None:
# TODO(lukaszkaiser): can we just divide the sum by length? XLA optimizes?
return sum([np.mean(x) / length for x in inputs])
unmask = [1.0 - np.equal(t, mask_id).astype(np.float32) for t in targets]
return sum([np.sum(x * m) / (length * np.sum(m))
for x, m in zip(inputs, unmask)])
def masked_mean(inputs, targets, weights, mask_id=None):
"""Weighted mean of the inputs, excluding where targets == mask_id."""
inputs = [x.astype(np.float32) for x in inputs]
# We assume all elements in the list contribute equally.
# TODO(lukaszkaiser): remove this assumption (e.g., when masks differ).
length = len(inputs)
if mask_id is not None:
weights = [w * (1.0 - np.equal(t, mask_id).astype(np.float32))
for t, w in zip(targets, weights)]
weight_sums = [np.float32(t.size) if np.isscalar(w) else np.sum(w)
for w, t in zip(weights, targets)]
return sum([np.sum(x * w) / (length * s)
for x, w, s in zip(inputs, weights, weight_sums)])
def neg_log_perplexity(batch, model_predictions):
"""Calculate negative log perplexity."""
_, targets = batch
model_predictions, targets = _make_list(model_predictions, targets)
xent = []
for (prediction, target) in zip(model_predictions, targets):
hot_target = layers.one_hot(target, prediction.shape[-1])
xent.append(np.sum(prediction * hot_target, axis=-1))
return masked_mean(xent, targets)
def loss(params, batch, model_predict, rng):
"""Calculate loss."""
inputs, targets = batch
predictions = model_predict(inputs, params, rng=rng)
predictions, targets = _make_list(predictions, targets)
xent = []
for (pred, target) in zip(predictions, targets):
xent.append(np.sum(pred * layers.one_hot(target, pred.shape[-1]), axis=-1))
return - masked_mean(xent, targets)
def log_gaussian_diag_pdf(x, mu, diag_sigma): # pylint: disable=invalid-name
"""Compute log N(x | mu, eye(diag_sigma))."""
a = mu.shape[-1] * np.log(2 * np.pi)
b = np.sum(np.log(diag_sigma), axis=-1)
y = x - mu / diag_sigma
y = np.expand_dims(y, axis=-1)
xm = np.expand_dims(x - mu, axis=-2)
c = np.matmul(xm, y)
c = np.squeeze(np.squeeze(c, axis=-1), axis=-1)
return -0.5 * (a + b + c)
def loss(params, batch, model_predict, state, rng, has_weights):
"""Calculate loss."""
inputs, targets, weights = unpack_batch(batch, has_weights)
model_input, get_preds = _stack_inputs_targets_and_get_predictions(
[inputs, targets])
# Call model, predictions will be the returned stack, usually consisting of
# the prediction tensor and the targets.
predictions, state = model_predict(model_input, params, state, rng=rng)
predictions = get_preds(predictions)
predictions, targets, weights = _make_list(predictions, targets, weights)
xent = []
for (pred, target) in zip(predictions, targets):
xent.append(
np.sum(pred * layers.one_hot(target, pred.shape[-1]), axis=-1))
return -masked_mean(xent, targets, weights), state
def PreparePairedSequenceBatch(source, target_in, pad=0):
"""Build masks for this batch.
Args:
source: (batch, source_len) array of integer-coded symbols for inputs
target_in: (batch, batch_len) array of integer-coded symbols for targets
pad: int: the padding symbol used to pad the above
Returns:
Prepared batch of tuple of arrays: source, input-target, shifted-target,
source mask, target mask, source-target "memory" mask, minibatch token count
"""
target = target_in[:, :-1]
target_y = target_in[:, 1:]
source_mask = np.reshape(source != pad,
(source.shape[0], 1, 1, source.shape[-1]))
target_mask = MakeTargetMask(target, pad)
memory_mask = (np.reshape(
np.arange(target.shape[-1]) < source.shape[-1], [-1, 1]))
ntokens = np.sum(target_y != pad)
return (source, target, target_y, source_mask, target_mask, memory_mask,
ntokens)
def forward_slice(query_slice, q_loop_idx, key, value): # pylint: disable=invalid-name
"""Forward pass for a subset of the query vectors."""
if self._share_qk:
key = self.make_unit_length(key)
dots = np.matmul(
query_slice, np.swapaxes(key, -1, -2)) / np.sqrt(depth)
# Causal masking
mask = make_mask(dots.shape[-2], dots.shape[-1], q_loop_idx)
dots = dots - 1e9 * mask
# Mask out attention to self except when no other targets are available.
if self._share_qk:
self_mask = make_self_mask(dots.shape[-2], dots.shape[-1], q_loop_idx)
dots = dots - 1e5 * self_mask
# Softmax.
dots = np.exp(dots - backend.logsumexp(dots, axis=-1, keepdims=True))
if self.dropout is not None and self.dropout > 0.0:
# Dropout is broadcast across the batch+head dimension
dropout_shape = (1, dots.shape[-2], dots.shape[-1])
slice_rng = jax.random.fold_in(rng, q_loop_idx)
keep_prob = jax.lax.tie_in(dots, 1.0 - self.dropout)
keep = backend.random.bernoulli(slice_rng, keep_prob, dropout_shape)
multiplier = keep.astype(dots.dtype) / jax.lax.tie_in(keep, keep_prob)
dots = dots * multiplier
if self._hard_k > 0:
top_k = np.sort(dots)[..., -self._hard_k] # Get the top-kth weight.
top_k = jax.lax.stop_gradient(top_k)
dots -= top_k[..., np.newaxis] # Subtract (be 0 for lower ones).
dots = np.maximum(dots, 0)
dots_sum = np.sum(dots, axis=-1, keepdims=True) # Re-normalize.
dots /= dots_sum # Re-normalize.
out_slice = np.matmul(dots, value)
return out_slice
def loss(params, batch, model_predict, rng): """Calculate loss.""" inputs, targets = batch preds = model_predict(params, inputs, rng=rng) xent = np.sum(preds * stax.one_hot(targets, preds.shape[-1]), axis=-1) return - masked_mean(xent, targets)
def neg_log_perplexity(batch, model_predictions): """Calculate negative log perplexity.""" _, targets = batch hot_targets = stax.one_hot(targets, model_predictions.shape[-1]) xent = np.sum(model_predictions * hot_targets, axis=-1) return masked_mean(xent, targets)
def crossentropy_loss(logpred, target):
"""Calculate crossentropy loss."""
return -np.mean(
np.sum(logpred * slax.one_hot(target, logpred.shape[-1]), axis=-1))
def L2(x, axis=-1, **kw):
del kw
prediction, target = x
return np.sum((prediction - target)**2, axis=axis)
def CrossEntropy(x, axis=-1, **kw):
del kw
prediction, target = x
return np.sum(prediction * core.one_hot(target, prediction.shape[-1]),
axis=axis)
def single_call(self, qk, v, buckets, hash_rng=None):
# We use the same vector as both a query and a key.
seqlen = qk.shape[-2]
assert int(buckets.shape[0]) == self.n_hashes * seqlen
ticker = jax.lax.tie_in(qk, np.arange(self.n_hashes * seqlen))
buckets_and_t = seqlen * buckets + (ticker % seqlen)
buckets_and_t = jax.lax.stop_gradient(buckets_and_t)
# Hash-based sort ("s" at the start of variable names means "sorted")
sbuckets_and_t, sticker = jax.lax.sort_key_val(
buckets_and_t, ticker, dimension=-1)
_, undo_sort = jax.lax.sort_key_val(sticker, ticker, dimension=-1)
sbuckets_and_t = jax.lax.stop_gradient(sbuckets_and_t)
sticker = jax.lax.stop_gradient(sticker)
undo_sort = jax.lax.stop_gradient(undo_sort)
st = (sticker % seqlen)
sqk = np.take(qk, st, axis=0)
sv = np.take(v, st, axis=0)
# Split off a "bin" axis so that attention only occurs within chunks.
bq_t = bkv_t = np.reshape(st, (self.n_hashes * self.n_bins, -1))
bqk = np.reshape(sqk, (self.n_hashes * self.n_bins, -1, sqk.shape[-1]))
bv = np.reshape(sv, (self.n_hashes * self.n_bins, -1, sv.shape[-1]))
bq_buckets = bkv_buckets = np.reshape(
sbuckets_and_t // seqlen, (self.n_hashes * self.n_bins, -1))
# Hashing operates on unit-length vectors. Unnormalized query vectors are
# fine because they effectively provide a learnable temperature for the
# attention softmax, but normalizing keys is needed so that similarity for
# the purposes of attention correctly corresponds to hash locality.
bq = bqk
bk = self.make_unit_length(bqk)
# Allow each chunk to attend within itself, and also one chunk back. Chunk
# boundaries might occur in the middle of a sequence of items from the
# same bucket, so this increases the chances of attending to relevant items.
# TODO(kitaev): benchmark whether XLA pad operation is noticeably faster.
def look_one_back(x):
if len(x.shape) == 2:
x_extra = np.concatenate([x[-1:, :], x[:-1, :]], axis=0)
else:
x_extra = np.concatenate([x[-1:, :, :], x[:-1, :, :]], axis=0)
return np.concatenate([x, x_extra], axis=1)
bk = look_one_back(bk)
bv = look_one_back(bv)
bkv_t = look_one_back(bkv_t)
bkv_buckets = look_one_back(bkv_buckets)
# Dot-product attention.
dots = np.matmul(bq, np.swapaxes(bk, -1, -2)) / np.sqrt(bq.shape[-1])
# Causal masking
mask = jax.lax.convert_element_type(
jax.lax.lt(bq_t[:, :, None], bkv_t[:, None, :]),
np.float32)
dots = dots - 1e9 * mask
# Mask out attention to self except when no other targets are available.
self_mask = jax.lax.convert_element_type(
jax.lax.eq(bq_t[:, :, None], bkv_t[:, None, :]),
np.float32)
dots = dots - 1e5 * self_mask
# Mask out attention to other hash buckets.
if not self._attend_across_buckets:
bucket_mask = jax.lax.convert_element_type(
jax.lax.ne(bq_buckets[:, :, None], bkv_buckets[:, None, :]),
np.float32)
dots = dots - 1e7 * bucket_mask
# Don't double-count query-key pairs across multiple rounds of hashing.
# There are two possible strategies here. (1) The default is to count how
# many times a query-key pair is repeated, and to lower its log-prob
# correspondingly at each repetition. (2) When hard_k is set, the code
# instead masks all but the first occurence of each query-key pair.
# TODO(kitaev): is one strategy faster or more numerically stable?
if not self._allow_duplicate_attention:
locs1 = undo_sort // bq_t.shape[-1]
locs2 = (locs1 + 1) % (self.n_hashes * self.n_bins)
if not self._attend_across_buckets:
locs1 = buckets * (self.n_hashes * self.n_bins) + locs1
locs2 = buckets * (self.n_hashes * self.n_bins) + locs2
locs = np.moveaxis(np.concatenate([
np.reshape(locs1, (self.n_hashes, seqlen)),
np.reshape(locs2, (self.n_hashes, seqlen)),
], 0), 0, -1) # produces shape (seqlen, 2 * self.n_hashes)
slocs = np.take(locs, st, axis=0)
b_locs = np.reshape(
slocs, (self.n_hashes * self.n_bins, -1, 2 * self.n_hashes))
# Queries always use the primary location (based on locs1).
b_locs1 = b_locs[:, :, None, :self.n_hashes]
if self._hard_k > 0:
range_n_hashes = jax.lax.tie_in(b_locs, np.arange(self.n_hashes))
nouse_locs = (range_n_hashes[:, None] > range_n_hashes[None, :])
nouse_locs = 2 * nouse_locs - 1 # 1 = use, -1 = don't use
nouse_locs = np.reshape(
np.broadcast_to(nouse_locs[:, None, :],
(self.n_hashes, self.n_bins, self.n_hashes)),
(self.n_hashes * self.n_bins, 1, 1, self.n_hashes))
b_locs1 = b_locs1 * nouse_locs
bq_locs = np.broadcast_to(
b_locs1,
b_locs.shape[:2] + (2, self.n_hashes))
bq_locs = np.reshape(bq_locs, b_locs.shape)
bkv_locs = look_one_back(b_locs)
dup_counts = np.sum(
jax.lax.convert_element_type(
jax.lax.eq(bq_locs[:, :, None, :], bkv_locs[:, None, :, :]),
np.float32),
axis=-1)
assert dup_counts.shape == dots.shape
if self._hard_k > 0:
dots = dots - 1e7 * jax.lax.stop_gradient(dup_counts)
else:
dots = dots - jax.lax.stop_gradient(np.log(dup_counts + 1e-9))
# Each query only attends to the top k most relevant keys.
if self._hard_k > 0:
b_top_dots = np.sort(dots)[..., -self._hard_k:] # Get the top k dots.
b_top_dots = jax.lax.stop_gradient(b_top_dots)
s_top_dots = np.reshape(b_top_dots, (-1, self._hard_k))
top_dots = np.take(s_top_dots, undo_sort, axis=0)
merged_top_dots = np.moveaxis(
np.reshape(top_dots, (self.n_hashes, seqlen, self._hard_k)), 0, -1)
merged_top_dots = np.reshape(merged_top_dots, (seqlen, -1))
dots_thresh = np.sort(merged_top_dots)[:, -self._hard_k]
# It's possible to compute the partition function at this point, but right
# now this codepath isn't set up for backprop, and there might also be
# issues computing it this way if two dot-products are exactly equal.
sdots_thresh = dots_thresh[st]
bdots_thresh = np.reshape(sdots_thresh, (self.n_hashes * self.n_bins, -1))
bdots_thresh = jax.lax.stop_gradient(bdots_thresh)
top_k_mask = jax.lax.convert_element_type(
dots < bdots_thresh[..., None], np.float32)
dots = dots - 1e7 * jax.lax.stop_gradient(top_k_mask)
# Softmax.
dots_logsumexp = backend.logsumexp(dots, axis=-1, keepdims=True)
dots = np.exp(dots - dots_logsumexp)
bo = np.matmul(dots, bv)
so = np.reshape(bo, (-1, bo.shape[-1]))
slogits = np.reshape(dots_logsumexp, (-1,))
def unsort_for_output_impl(so, slogits):
o = np.take(so, undo_sort, axis=0)
# Sorting is considerably faster than gather, but first we need to get the
# XLA compiler to abandon the idea of fusing this sort with the input sort
# (which introduces a computation cycle and leads to a crash).
# TODO(kitaev): remove "sticker_" variable if XLA is fixed.
sticker_ = sticker + jax.lax.convert_element_type(
slogits[0] > 0, sticker.dtype)
_, logits = jax.lax.sort_key_val(sticker_, slogits, dimension=-1)
return o, logits
def unsort_for_output_vjp(so, slogits):
"""Custom gradient for unsort_for_output."""
so = jax.lax.stop_gradient(so)
slogits = jax.lax.stop_gradient(slogits)
o, logits = unsort_for_output_impl(so, slogits)
def vjpfun(o_logits_grads):
so_grad = np.take(o_logits_grads[0], sticker, axis=0)
# TODO(kitaev): this exists to match the forward pass, but I'm not sure
# if it's actually required.
buckets_and_t_ = buckets_and_t + jax.lax.convert_element_type(
o_logits_grads[1][0] > 0, buckets_and_t.dtype)
_, slogits_grad = jax.lax.sort_key_val(
buckets_and_t_, o_logits_grads[1], dimension=-1)
return (so_grad, slogits_grad)
return (o, logits), vjpfun
unsort_for_output = jax.custom_transforms(unsort_for_output_impl)
jax.defvjp_all(unsort_for_output, unsort_for_output_vjp)
o, logits = unsort_for_output_impl(so, slogits)
if self.n_hashes == 1:
out = o
else:
o = np.reshape(o, (self.n_hashes, seqlen, o.shape[-1]))
logits = np.reshape(logits, (self.n_hashes, seqlen, 1))
probs = np.exp(logits - backend.logsumexp(logits, axis=0, keepdims=True))
out = np.sum(o * probs, axis=0)
assert out.shape == v.shape
return out
def call(self, inputs, params=(), state=(), rng=None, **kwargs):
del params, kwargs
# We use the same vector as both a query and a key. For now we haven't
# adjusted any of the surrounding code, so we still get a separate "key"
# input that we ignore.
qk, _, v = inputs
seqlen = qk.shape[-2]
# qk/v are n_hashes*n_batch*n_heads, seqlen, d_head
# TODO(kitaev): is it faster to fuse this tiling into gather/scatter ops?
qk = np.tile(qk, (self.n_hashes, 1, 1))
v = np.tile(v, (self.n_hashes, 1, 1))
# bins are n_hashes*n_batch*n_heads, seqlen
# They specify which hash bucket the query/key/value vectors fall in.
bins = self.hash_vectors(qk, rng=rng)
# joint_t is n_hashes*n_batch*n_heads, seqlen
joint_t = jax.lax.tie_in(qk, np.arange(seqlen))
joint_t = np.reshape(joint_t, (1, seqlen))
joint_t = np.broadcast_to(joint_t, qk.shape[:-1])
assert int(
(self.n_buckets_per_bin * self.n_bins + 1) * seqlen
) < 2**31, (
'Potential 32-bit integer overflow; please double-check the code.')
joint_bins_and_t = seqlen * bins + joint_t
def chunk_scalars(x): # pylint: disable=invalid-name
return np.reshape(x, (x.shape[0], self.n_bins, -1))
def chunk_vectors(x): # pylint: disable=invalid-name
return np.reshape(x, (x.shape[0], self.n_bins, -1, x.shape[-1]))
def unchunk_vectors(x): # pylint: disable=invalid-name
return np.reshape(x, (x.shape[0], -1, x.shape[-1]))
# Sort everything by bin number, with a secondary sort by time
# (variables starting with "s" are sorted)
_, sjoint_t = jax.lax.sort_key_val(joint_bins_and_t,
joint_t,
dimension=-1)
_, undo_sort = jax.lax.sort_key_val(sjoint_t, joint_t, dimension=-1)
# TODO(kitaev): why does jax flag integer indices as differentiable?
# If we don't call stop_gradient here, custom gradients below won't work
# because the primitive functions close over "differentiable" variables.
sjoint_t = jax.lax.stop_gradient(sjoint_t)
undo_sort = jax.lax.stop_gradient(undo_sort)
# The backward pass of gather is in general a scatter operation, but we know
# we're dealing with permutations so we use gather for the backward pass
# too. This custom gradient should be about 2x faster than having jax infer
# one that uses scatter ops instead.
def permute_impl(vecs):
assert len(vecs.shape) == 3
return np.take_along_axis(vecs, sjoint_t[:, :, None], axis=-2)
def unpermute_impl(vecs):
assert len(vecs.shape) == 3
return np.take_along_axis(vecs, undo_sort[:, :, None], axis=-2)
@jax.custom_transforms
def permute(vecs):
return permute_impl(vecs)
def permute_vjp(vecs):
out_vecs = permute_impl(vecs)
def vjpfun(grad):
return (unpermute_impl(grad), )
return out_vecs, vjpfun
@jax.custom_transforms
def unpermute(vecs):
return unpermute_impl(vecs)
def unpermute_vjp(vecs):
out_vecs = unpermute_impl(vecs)
def vjpfun(grad):
return (permute_impl(grad), )
return out_vecs, vjpfun
jax.defvjp_all(permute, permute_vjp)
jax.defvjp_all(unpermute, unpermute_vjp)
sqk = permute(qk)
sv = permute(v)
# Split off a "bin" axis so that attention only occurs within chunks.
bq_t = bkv_t = chunk_scalars(sjoint_t)
bqk = chunk_vectors(sqk)
bv = chunk_vectors(sv)
# Hashing operates on unit-length vectors. Unnormalized query vectors are
# fine because they effectively provide a learnable temperature for the
# attention softmax, but normalizing keys is needed so that similarity for
# the purposes of attention correctly corresponds to hash locality.
bq = bqk
bk = self.make_unit_length(bqk)
# Allow each chunk to attend within itself, and also one chunk back. Chunk
# boundaries might occur in the middle of a sequence of items from the
# same bin, so this increases the chances of attending to relevant items.
# TODO(kitaev): benchmark whether XLA pad operation is noticeably faster.
bk_extra = np.concatenate([bk[:, -1:, :, :], bk[:, :-1, :, :]], axis=1)
bk = np.concatenate([bk, bk_extra], axis=2)
bv_extra = np.concatenate([bv[:, -1:, :, :], bv[:, :-1, :, :]], axis=1)
bv = np.concatenate([bv, bv_extra], axis=2)
bkv_t_extra = np.concatenate([bkv_t[:, -1:, :], bkv_t[:, :-1, :]],
axis=1)
bkv_t = np.concatenate([bkv_t, bkv_t_extra], axis=2)
# Dot-product attention.
dots = np.matmul(bq, np.swapaxes(bk, -1, -2)) / np.sqrt(bq.shape[-1])
# Causal masking
mask = jax.lax.convert_element_type(
jax.lax.lt(bq_t[:, :, :, None], bkv_t[:, :, None, :]), np.float32)
dots = dots - 1e9 * mask
# Mask out attention to self except when no other targets are available.
self_mask = jax.lax.broadcasted_eye(dots.dtype, dots.shape, (2, 3))
self_mask = jax.lax.tie_in(dots, self_mask)
dots = dots - 32 * self_mask
# Softmax.
dots_logsumexp = backend.logsumexp(dots, axis=-1, keepdims=True)
dots = np.exp(dots - dots_logsumexp)
if self._hard_k > 0:
top_k = np.sort(dots)[...,
-self._hard_k] # Get the top-kth weight.
top_k = jax.lax.stop_gradient(top_k)
dots -= top_k[..., np.newaxis] # Subtract (be 0 for lower ones).
dots = np.maximum(dots, 0)
dots_sum = np.sum(dots, axis=-1,
keepdims=True) # Sum to re-normalize.
dots_logsumexp += np.log(dots_sum) # Add it to the weight.
dots /= dots_sum # Re-normalize.
bo = np.matmul(dots, bv)
so = unchunk_vectors(bo)
slogits = unchunk_vectors(dots_logsumexp)
o = unpermute(so)
logits = unpermute(slogits)
o = np.reshape(o, (self.n_hashes, -1, seqlen, o.shape[-1]))
logits = np.reshape(logits, (self.n_hashes, -1, seqlen, 1))
probs = np.exp(logits -
backend.logsumexp(logits, axis=0, keepdims=True))
out = np.sum(o * probs, axis=0)
assert out.shape == inputs[2].shape
return out, state
def WeightedMean(x, **kw):
del kw
metric, weights = x
weights_sum = np.sum(weights)
return np.sum(metric * weights) / weights_sum
def kl_div(logpred, target, eps=np.finfo(np.float32).eps):
"""Calculate KL-divergence."""
return np.sum(target * (np.log(target + eps) - logpred))