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_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 test_batch_norm(self): input_shape = (2, 3, 4) input_dtype = np.float32 input_signature = ShapeDtype(input_shape, input_dtype) eps = 1e-5 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)) _, _ = layer.initialize_once(input_signature) state = layer.state onp.testing.assert_allclose(state[0], 0) onp.testing.assert_allclose(state[1], 1) self.assertEqual(state[2], 0) out = layer(inp1) state = layer.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 single_call(self, qk, v, buckets, 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) if self._dropout > 0.0: # Dropout is broadcast across the bin dimension dropout_shape = (1, 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 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 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 # If we factorize the hash, find a factor dividing n_buckets nicely. rot_size, factor_list = self.n_buckets, [self.n_buckets] if self._factorize_hash: # If we are given a list of factors, verify it and use later. if isinstance(self._factorize_hash, list): rot_size, product = 0, 1 factor_list = self._factorize_hash for factor in factor_list: assert factor % 2 == 0 product *= factor rot_size += factor assert product == self.n_buckets else: # Find one factor if just set to True. # We want to represent self.n_buckets = factor * rest so that # (1) both factor and rest are even, and (2) factor + rest is minimal. # To compute this we start from factor = sqrt(n_buckets) and go down # with it until we find one that satisfies the constraints above. factor = int(math.sqrt(self.n_buckets)) while factor > 0 and not (self.n_buckets % factor == 0 and factor % 2 == 0 and (self.n_buckets // factor) % 2 == 0): factor -= 1 if factor > 2: # Factor of 2 does not warrant the effort. rot_size = factor + (self.n_buckets // factor) factor_list = [factor, self.n_buckets // factor] rotations_shape = (vecs.shape[-1], self.n_hashes if self._rehash_each_round else 1, rot_size // 2) rng = jax.lax.tie_in(vecs, rng) rng, subrng = backend.random.split(rng) random_rotations = self._sample_rotation(rotations_shape, vecs, rng) # 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) if self._rehash_each_round: if self._factorize_hash and len(factor_list) > 1: # We factorized self.n_buckets as the product of factor_list. # Get the buckets for them and combine. buckets, cur_sum, cur_product = None, 0, 1 for factor in factor_list: rv = rotated_vecs[..., cur_sum:cur_sum + (factor // 2)] cur_sum += factor // 2 rv = np.concatenate([rv, -rv], axis=-1) if buckets is None: buckets = np.argmax(rv, axis=-1) else: buckets += cur_product * np.argmax(rv, axis=-1) cur_product *= factor else: rotated_vecs = np.concatenate([rotated_vecs, -rotated_vecs], axis=-1) 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: assert not self._factorize_hash rotated_vecs = np.concatenate([rotated_vecs, -rotated_vecs], axis=-1) # 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 _sample_rotation(self, shape, vecs, rng): """Samples a rotation matrix, either randomly or based on `vecs`.""" if not self._data_rotation: return jax.random.normal(rng, shape).astype('float32') assert len(shape) == 3 unused_n_dim, n_hashes, r_div_2 = shape assert len(vecs.shape) == 2 n_vecs = vecs.shape[0] rng1, rng2 = backend.random.split(rng, num=2) # We need to sample 2 * n_hashes * r_div_2 vectors from `vecs` at random. num_needed = 2 * n_hashes * r_div_2 if n_vecs < num_needed: # shape = (n_hashes, r_div_2) random_idxs_1 = jax.random.randint(rng1, (n_hashes, r_div_2), 0, n_vecs) random_idxs_2 = jax.random.randint(rng2, (n_hashes, r_div_2), 0, n_vecs) else: # Sample without replacement. shuffled_indices = jax.random.shuffle(rng1, np.arange(n_vecs)) random_idxs = np.reshape(shuffled_indices[:num_needed], (2, n_hashes, r_div_2)) random_idxs_1 = random_idxs[0] random_idxs_2 = random_idxs[1] if self._data_rotation_farthest: # shape = (n_hashes * r_div_2, ) random_idxs_1 = np.reshape(random_idxs_1, (-1, )) random_vecs_1 = vecs[random_idxs_1] # Sample candidates for vec2s. rng, subrng = backend.random.split(rng) # shape = (self._data_rotation_farthest_num, n_hashes * r_div_2) candidate_idxs_2 = jax.random.randint( subrng, (self._data_rotation_farthest_num, n_hashes * r_div_2), 0, n_vecs) candidate_vecs_2 = vecs[candidate_idxs_2] # shape = candidate_idxs_2.shape distances = -np.abs( np.einsum('hd,chd->ch', random_vecs_1, candidate_vecs_2)) # shape = (n_hashes * r_div_2,) farthest_idxs = np.argmax(distances, axis=0) # candidate_vecs_2.shape random_vecs_2 = candidate_vecs_2[farthest_idxs, np.arange(n_hashes * r_div_2)] # reshape to (n_hashes, r_div_2, n_dim) random_vecs_1 = np.reshape(random_vecs_1, (n_hashes, r_div_2, -1)) random_vecs_2 = np.reshape(random_vecs_2, (n_hashes, r_div_2, -1)) else: # shape = (n_hashes, r_div_2, n_dim) random_vecs_1 = vecs[random_idxs_1] random_vecs_2 = vecs[random_idxs_2] # shape = (n_dim, n_hashes, r_div_2) return np.transpose(random_vecs_2 - random_vecs_1, axes=[2, 0, 1])
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 # Softmax. dots = np.exp(dots - backend.logsumexp(dots, axis=-1, keepdims=True)) 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 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, :]