def make_mask(N, M, k):
x = np.arange(N, dtype=np.int32)
y = np.arange(M, dtype=np.int32)
mask = jax.lax.lt((jax.lax.broadcast_in_dim(
x, shape=(N, M), broadcast_dimensions=(0, )) + k),
jax.lax.broadcast(y, [N]))
mask = jax.lax.convert_element_type(mask, np.float32)
return mask
def one_hot(x, size, dtype=np.float32): # pylint: disable=invalid-name
"""Make a n+1 dim one-hot array from n dim int-categorical array."""
arange_size = np.arange(size)
if backend.get_name() == 'jax':
# Work around a jax broadcasting issue.
arange_size = jax.lax.tie_in(x, arange_size)
return np.array(x[..., np.newaxis] == arange_size, dtype)
def make_mask(N, M, k): # pylint: disable=invalid-name
"""Constructs a slice of the causal attention mask.
Args:
N: number of query positions
M: number of key positions
k: position of the initial query element
Returns:
N x M mask, where 1.0 indicates that attention is not allowed.
"""
x = np.arange(N, dtype=np.int32)
y = np.arange(M, dtype=np.int32)
mask = jax.lax.lt((jax.lax.broadcast_in_dim(
x, shape=(N, M), broadcast_dimensions=(0, )) + k),
jax.lax.broadcast(y, [N]))
mask = jax.lax.convert_element_type(mask, np.float32)
return mask
def bin_vectors_by_time(self, vecs):
seqlen = vecs.shape[-2]
assert seqlen % self.n_bins == 0
bin_size = int(seqlen // self.n_bins)
bins = np.arange(seqlen, dtype=np.int32) // bin_size
bins = jax.lax.tie_in(vecs, bins)
bins = bins[None, :]
bins = np.broadcast_to(bins, vecs.shape[:-1])
return bins
def make_self_mask(N, M, k): # pylint: disable=invalid-name
"""Masks out elements attending to self.
Args:
N: number of query positions
M: number of key positions
k: position of the initial query element
Returns:
N x M mask, where 1.0 indicates that attention is not allowed.
"""
x = jax.lax.tie_in(k, np.arange(N, dtype=np.int32))
y = jax.lax.tie_in(k, np.arange(M, dtype=np.int32))
mask = jax.lax.eq(
(jax.lax.broadcast_in_dim(
x, shape=(N, M), broadcast_dimensions=(0,)) + k),
jax.lax.broadcast(y, [N]))
mask = jax.lax.convert_element_type(mask, np.float32)
return mask
def label_smoothed_loss(logpred, target, size, padding_idx=0, smoothing=0.0):
"""Returns a label-smoothing loss-criterion function."""
confidence = 1.0 - smoothing
zerosmoothed = smoothing / (size - 2)
delta = confidence - zerosmoothed
assert logpred.shape[1] == size
truedist = (np.full_like(logpred, zerosmoothed) +
delta * slax.one_hot(target, size))
truedist *= (1 - (np.arange(size) == padding_idx))
truedist *= (1 - (target == padding_idx))[:, np.newaxis]
return kl_div(logpred, truedist, eps=1e-6)
def hash_vectors(self, vecs, rng):
# See https://arxiv.org/pdf/1509.02897.pdf
# We sample a different random rotation for each round of hashing to
# decrease the probability of hash misses.
assert self.n_buckets % 2 == 0
random_rotations_shape = (
vecs.shape[-1],
self.n_hashes if self._rehash_each_round else 1,
self.n_buckets // 2)
rng = jax.lax.tie_in(vecs, rng)
rng, subrng = backend.random.split(rng)
random_rotations = jax.random.normal(
rng, random_rotations_shape).astype('float32')
# TODO(lukaszkaiser): the dropout mask will be used for all rounds of
# hashing, so it's shared between them. Check if that's what we want.
dropped_vecs = self.drop_for_hash(vecs, subrng)
rotated_vecs = np.einsum('tf,fhb->htb', dropped_vecs, random_rotations)
rotated_vecs = np.concatenate([rotated_vecs, -rotated_vecs], axis=-1)
if self._rehash_each_round:
buckets = np.argmax(rotated_vecs, axis=-1)
# buckets is now (self.n_hashes, seqlen). Next we add offsets so that
# bucket numbers from different hashing rounds don't overlap.
offsets = jax.lax.tie_in(buckets, np.arange(self.n_hashes))
offsets = np.reshape(offsets * self.n_buckets, (-1, 1))
buckets = np.reshape(buckets + offsets, (-1,))
else:
# In this configuration, we map each item to the top self.n_hashes buckets
rotated_vecs = np.squeeze(rotated_vecs, 0)
bucket_range = jax.lax.tie_in(vecs, np.arange(rotated_vecs.shape[-1]))
bucket_range = np.reshape(bucket_range, (1, -1))
bucket_range = np.broadcast_to(bucket_range, rotated_vecs.shape)
_, buckets = jax.lax.sort_key_val(
rotated_vecs, bucket_range, dimension=-1)
buckets = buckets[:, -self.n_hashes:]
buckets = np.reshape(np.moveaxis(buckets, 0, -1), (-1,))
return buckets
def test_batch_norm(self):
input_shape = (2, 3, 4)
input_dtype = np.float32
eps = 1e-5
rng = backend.random.get_prng(0)
inp1 = np.reshape(np.arange(np.prod(input_shape), dtype=input_dtype),
input_shape)
m1 = 11.5 # Mean of this random input.
v1 = 47.9167 # Variance of this random input.
layer = normalization.BatchNorm(axis=(0, 1, 2))
params, state = layer.initialize(input_shape, input_dtype, rng)
onp.testing.assert_allclose(state[0], 0)
onp.testing.assert_allclose(state[1], 1)
self.assertEqual(state[2], 0)
out, state = layer(inp1, params, state)
onp.testing.assert_allclose(state[0], m1 * 0.001)
onp.testing.assert_allclose(state[1], 0.999 + v1 * 0.001, rtol=1e-6)
self.assertEqual(state[2], 1)
onp.testing.assert_allclose(out, (inp1 - m1) / np.sqrt(v1 + eps),
rtol=1e-6)
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 test_batch_norm(self):
input_shape = (2, 3, 4)
input_dtype = np.float32
eps = 1e-5
rng = backend.random.get_prng(0)
inp1 = np.reshape(np.arange(np.prod(input_shape), dtype=input_dtype),
input_shape)
m1 = 11.5
v1 = 47.9167
layer = normalization.BatchNorm(axis=(0, 1, 2))
params, state = layer.initialize(input_shape, input_dtype, rng)
onp.testing.assert_allclose(state[0], 0)
onp.testing.assert_allclose(state[1], 0)
self.assertEqual(state[2], 0)
out, state = layer(inp1, params, state)
onp.testing.assert_allclose(state[0], m1)
onp.testing.assert_allclose(state[1], v1, rtol=1e-6)
self.assertEqual(state[2], 1)
onp.testing.assert_allclose(out, (inp1 - m1) / np.sqrt(v1 + eps),
rtol=1e-6)
inp2 = inp1 * 2 + 3
m2 = m1 * 2 + 3
v2 = v1 * 4
m12 = (m1 + m2) / 2
v12 = (v1 + v2) / 2
out, state = layer(inp2, params, state)
onp.testing.assert_allclose(state[0], m12)
onp.testing.assert_allclose(state[1], v12, rtol=1e-6)
self.assertEqual(state[2], 2)
onp.testing.assert_allclose(out, (inp2 - m2) / np.sqrt(v2 + eps),
rtol=1e-6)
layer = normalization.BatchNorm(axis=(0, 1, 2), mode="eval")
inp3 = inp1 * 5 + 7
out, state_unchanged = layer(inp3, params, state)
for i in range(3):
onp.testing.assert_allclose(state_unchanged[i], state[i])
onp.testing.assert_allclose(out, (inp3 - m12) / np.sqrt(v12 + eps),
rtol=1e-6)
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 call_and_grad(self, inputs, ct, rng=None, **kwargs):
del 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, ignored_k, v = inputs
seqlen = qk.shape[-2]
# qk/v are n_batch*n_heads, seqlen, d_head
# bins are 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_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_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)
sqk = np.take_along_axis(qk, sjoint_t[:, :, None], axis=-2)
sv = np.take_along_axis(v, sjoint_t[:, :, None], axis=-2)
if ct is not None:
so_ct = np.take_along_axis(ct, sjoint_t[:, :, None], axis=-2)
@jax.jit
def binned_attn(sqk, sv): # pylint: disable=invalid-name
"""Performs attention on sorted queries/keys/values."""
# Split off a "bin" axis so that attention only occurs whithin 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 = np.exp(dots -
backend.logsumexp(dots, axis=-1, keepdims=True))
bo = np.matmul(dots, bv)
so = unchunk_vectors(bo)
return so
@jax.jit
def binned_attn_vjp(sqk, sv, so_ct): # pylint: disable=invalid-name
so, vjpfun = jax.vjp(binned_attn, sqk, sv)
sqkv_ct = vjpfun(so_ct)
return so, sqkv_ct
if ct is None:
so = binned_attn(sqk, sv)
_, undo_sort = jax.lax.sort_key_val(sjoint_t,
joint_t,
dimension=-1)
out = np.take_along_axis(so, undo_sort[:, :, None], axis=-2)
return out, None
else:
# Jax can construct a backward pass automatically, but it's about 2x
# slower than writing our own. The main reason is that 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.
so, (sqk_ct, sv_ct) = binned_attn_vjp(sqk, sv, so_ct)
_, undo_sort = jax.lax.sort_key_val(sjoint_t,
joint_t,
dimension=-1)
out = np.take_along_axis(so, undo_sort[:, :, None], axis=-2)
qk_ct = np.take_along_axis(sqk_ct, undo_sort[:, :, None], axis=-2)
v_ct = np.take_along_axis(sv_ct, undo_sort[:, :, None], axis=-2)
return out, (qk_ct, np.zeros_like(ignored_k), v_ct)
def forward_and_vjp(self, inputs, ct, params=(), **kwargs):
del params, kwargs
q, k, v = inputs
# q/k/v are n_batch, n_heads, seqlen, d_head
assert k.shape[2] % self.n_bins == 0
bin_size = int(k.shape[2] // self.n_bins)
# q_bins/kv_bins are n_batch, n_heads, seqlen
# They specify which hash bucket the query/key/value vectors fall in. For
# now, instead of hashing we just put consecutive items in the same bucket.
q_bins = np.arange(q.shape[2], dtype=np.int32) // bin_size
q_bins = jax.lax.tie_in(q, q_bins)
q_bins = q_bins[None, None, :]
q_bins = np.broadcast_to(q_bins, q.shape[:-1])
q_bins = -q_bins
kv_bins = q_bins * 2
# q_t/kv_t are n_batch, n_heads, seqlen
q_t = jax.lax.tie_in(q, np.arange(q.shape[2]))
q_t = np.reshape(q_t, (1, 1, q_t.shape[0]))
q_t = np.broadcast_to(q_t, q.shape[:-1])
kv_t = q_t
def chunk_rank3(x):
return np.reshape(x, (x.shape[0], x.shape[1], self.n_bins, -1))
def chunk_rank4(x):
return np.reshape(
x, (x.shape[0], x.shape[1], self.n_bins, -1, x.shape[-1]))
def unchunk_rank4(x):
return np.reshape(x, (x.shape[0], x.shape[1], -1, x.shape[-1]))
# Sort everything by bin number (variables starting with "s" are sorted)
_, sq_t = jax.lax.sort_key_val(q_bins, q_t, dimension=2)
sq = np.take_along_axis(q, sq_t[:, :, :, None], axis=2)
if ct is not None:
so_ct = np.take_along_axis(ct, sq_t[:, :, :, None], axis=2)
_, skv_t = jax.lax.sort_key_val(kv_bins, kv_t, dimension=2)
sk = np.take_along_axis(k, skv_t[:, :, :, None], axis=2)
sv = np.take_along_axis(v, skv_t[:, :, :, None], axis=2)
@jax.jit
def binned_attn(sq, sk, sv):
"""Performs attention on sorted queries/keys/values."""
# Split off a "bin" axis so that attention only occurs whithin chunks.
bq_t = chunk_rank3(sq_t)
bkv_t = chunk_rank3(skv_t)
bq = chunk_rank4(sq)
bk = chunk_rank4(sk)
bv = chunk_rank4(sv)
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
# Softmax.
dots = np.exp(dots - dots.max(axis=-1, keepdims=True))
dots = dots / dots.sum(axis=-1, keepdims=True)
bo = np.matmul(dots, bv)
so = unchunk_rank4(bo)
return so
@jax.jit
def binned_attn_vjp(sq, sk, sv, so_ct):
so, vjpfun = jax.vjp(binned_attn, sq, sk, sv)
sqkv_ct = vjpfun(so_ct)
return so, sqkv_ct
if ct is None:
so = binned_attn(sq, sk, sv)
_, undo_q_sort = jax.lax.sort_key_val(sq_t, q_t, dimension=2)
out = np.take_along_axis(so, undo_q_sort[:, :, :, None], axis=2)
return out, None
else:
# Jax can construct a backward pass automatically, but it's about 2x
# slower than writing our own. The main reason is that 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.
so, (sq_ct, sk_ct, sv_ct) = binned_attn_vjp(sq, sk, sv, so_ct)
_, undo_q_sort = jax.lax.sort_key_val(sq_t, q_t, dimension=2)
out = np.take_along_axis(so, undo_q_sort[:, :, :, None], axis=2)
q_ct = np.take_along_axis(sq_ct,
undo_q_sort[:, :, :, None],
axis=2)
_, undo_kv_sort = jax.lax.sort_key_val(skv_t, kv_t, dimension=2)
k_ct = np.take_along_axis(sk_ct,
undo_kv_sort[:, :, :, None],
axis=2)
v_ct = np.take_along_axis(sv_ct,
undo_kv_sort[:, :, :, None],
axis=2)
return out, (q_ct, k_ct, v_ct)
def one_hot(x, size, dtype=np.float32):
"""Make a n+1 dim one-hot array from n dim int-categorical array."""
return np.array(x[..., np.newaxis] == np.arange(size), dtype)
def _forward_train_eval(self, inputs, rng):
(inputs, original_len, n_bins) = self._pad_inputs(inputs)
q, k, v = inputs
seqlen = q.shape[-2]
# q/k/v are n_batch*n_heads, seqlen, d_head
# Time indices for causal masking.
t = jax.lax.tie_in(q, np.arange(seqlen))
# Split off a "bin" axis for chunks of consecutive items.
bq_t = np.reshape(t, (n_bins, -1))
bq = np.reshape(q, (q.shape[0], n_bins, -1, q.shape[-1]))
if self._share_qk:
bk = self.make_unit_length(bq)
else:
bk = np.reshape(k, (k.shape[0], n_bins, -1, k.shape[-1]))
bv = np.reshape(v, (v.shape[0], n_bins, -1, v.shape[-1]))
# Allow each chunk to attend within itself, and also one chunk back.
def look_one_back(x):
# Output: pairs [ bin_i bin_{i-1} ] concatenated on the time axis.
if len(x.shape) == 2:
x_extra = np.concatenate([x[-1:, :], x[:-1, :]], axis=0)
return np.concatenate([x, x_extra], axis=1)
else:
assert len(x.shape) == 4
x_extra = np.concatenate([x[:, -1:, :, :], x[:, :-1, :, :]], axis=1)
return np.concatenate([x, x_extra], axis=2)
bkv_t = look_one_back(bq_t)
bk = look_one_back(bk)
bv = look_one_back(bv)
# Dot-product attention.
dots = np.matmul(bq, np.swapaxes(bk, -1, -2)) / np.sqrt(bq.shape[-1])
# Causal masking based on the time indices.
mask = jax.lax.convert_element_type(
jax.lax.lt(bq_t[None, :, :, None], bkv_t[None, :, None, :]),
np.float32)
dots = dots - 1e9 * mask
# Mask out attention to self except when no other targets are available.
if self._share_qk:
self_mask = jax.lax.broadcasted_eye(dots.dtype, dots.shape, (2, 3))
self_mask = jax.lax.tie_in(dots, self_mask)
dots = dots - 1e5 * self_mask
if self.dropout > 0.0:
# Dropout is broadcast across the batch+head dimension
dropout_shape = (1, dots.shape[-3], dots.shape[-2], dots.shape[-1])
keep_prob = jax.lax.tie_in(dots, 1.0 - self.dropout)
keep = backend.random.bernoulli(rng, keep_prob, dropout_shape)
multiplier = keep.astype(dots.dtype) / jax.lax.tie_in(keep, keep_prob)
dots = dots * multiplier
# Softmax.
dots = np.exp(dots - backend.logsumexp(dots, axis=-1, keepdims=True))
bo = np.matmul(dots, bv)
output = np.reshape(bo, (bo.shape[0], -1, bo.shape[-1]))
assert output.shape == v.shape
return output[..., :original_len, :]
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
