""" NestedTensors =============================================================== NestedTensors are similar to regular tensors, except for their shape: * for a regular tensor, each dimension has a size * for a nestedtensor, not all dimensions have regular sizes; some of them are jagged Nestedtensors are a natural solution for representing sequential data within various domains: * in NLP, sentences can have variable lengths, so a batch of sentences forms a nestedtensor * in CV, images can have variable shapes, so a batch of images forms a nestedtensor In this tutorial, we will demonstrate basic usage of nestedtensors and motivate their usefulness for operating on sequential data of varying lengths with a real-world example. NestedTensor are currently a prototype feature and are subject to change. """ import torch import torch.nn.functional as F device = torch.device('cuda' if torch.cuda.is_available() else 'cpu') ###################################################################### # NestedTensor Initialization # ---------------- # ###################################################################### # From the Python frontend, a nestedtensor can be created from a list of tensors. # We denote nt[i] as the ith tensor component of a nestedtensor. nt = torch.nested.nested_tensor([torch.arange(12).reshape( 2, 6), torch.arange(18).reshape(3, 6)], dtype=torch.float, device=device) print(f"{nt=}") ###################################################################### # By padding every underlying tensor to the same shape, # a nestedtensor can be converted to a regular tensor. padded_out_tensor = torch.nested.to_padded_tensor(nt, padding=0.0) print(f"{padded_out_tensor=}") ###################################################################### # All tensors posses an attribute for determining if they are nested; print(f"nt is nested: {nt.is_nested}") print(f"padded_out_tensor is nested: {padded_out_tensor.is_nested}") ###################################################################### # It is common to construct nestedtensors from batches of irregularly shaped tensors. # i.e. dimension 0 is assumed to be the batch dimension. # Indexing dimension 0 gives back the first underlying tensor component. print("First underlying tensor component:", nt[0], sep='\n') print("last column of 2nd underlying tensor component:", nt[1, :, -1], sep='\n') # When indexing a nestedtensor's 0th dimension, the result is a regular tensor. print(f"First underlying tensor component is nested: {nt[0].is_nested}") ###################################################################### # An important note is that slicing in dimension 0 has not been supported yet. # Which means it not currently possible to construct a view that combines the underlying # tensor components. ###################################################################### # Nested Tensor Operations # ---------------- # ###################################################################### # As each operation must be explicitly implemented for nestedtensors, # operation coverage for nestedtensors is currently narrower than that of regular tensors. # For now, only basic operations such as index, dropout, softmax, transpose, reshape, linear, bmm are covered. # However, coverage is being expanded. # If you need certain operations, please file an `issue `__ # to help us prioritize coverage. # # **reshape** # # The reshape op is for changing the shape of a tensor. # Its full semantics for regular tensors can be found # `here `__. # For regular tensors, when specifying the new shape, # a single dimension may be -1, in which case it is inferred # from the remaining dimensions and the number of elements. # # The semantics for nestedtensors are similar, except that -1 no longer infers. # Instead, it inherits the old size (here 2 for ``nt[0]`` and 3 for ``nt[1]``). # -1 is the only legal size to specify for a jagged dimension. nt_reshaped = nt.reshape(2, -1, 2, 3) print(f"{nt_reshaped=}") ###################################################################### # **transpose** # # The transpose op is for swapping two dimensions of a tensor. # Its full semantics can be found # `here `__. # Note that for nestedtensors dimension 0 is special; # it is assumed to be the batch dimension, # so transposes involving nestedtensor dimension 0 are not supported. nt_transposed = nt_reshaped.transpose(1, 2) print(f"{nt_transposed=}") ###################################################################### # **others** # # Other operations have the same semantics as for regular tensors. # Applying the operation on a nestedtensor is equivalent to # applying the operation to the underlying tensor components, # with the result being a nestedtensor as well. nt_mm = torch.nested.nested_tensor([torch.randn((2, 3, 4)), torch.randn((2, 3, 5))], device=device) nt3 = torch.matmul(nt_transposed, nt_mm) print(f"Result of Matmul:\n {nt3}") nt4 = F.dropout(nt3, 0.1) print(f"Result of Dropout:\n {nt4}") nt5 = F.softmax(nt4, -1) print(f"Result of Softmax:\n {nt5}") ###################################################################### # Why Nested Tensor # ---------------- # ###################################################################### # When data is sequential, it is often the case that each sample has a different length. # For example, in a batch of sentences, each sentence has a different number of words. # A common technique for handling varying sequences is to manually pad each data tensor # to the same shape in order to form a batch. # For example, we have 2 sentences with different lengths and a vocabulary # In order to represent his as single tensor we pad with 0 to the max length in the batch. sentences = [["goodbye", "padding"], ["embrace", "nested", "tensor"]] vocabulary = {"goodbye": 1.0, "padding": 2.0, "embrace": 3.0, "nested": 4.0, "tensor": 5.0} padded_sentences = torch.tensor([[1.0, 2.0, 0.0], [3.0, 4.0, 5.0]]) nested_sentences = torch.nested.nested_tensor([torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0, 5.0])]) print(f"{padded_sentences=}") print(f"{nested_sentences=}") ###################################################################### # This techinque of padding a batch of data to its max length is not optimal. # The padded data is not needed for computation and wastes memory by allocating # larger tensors than necessary. # Further, not all operations have the same semnatics when applied to padded data. # For matrix multiplications in order to ignore the padded entries, one needs to pad # with 0 while for softmax one has to pad with -inf to ignore specific entries. padded_sentences_for_softmax = torch.tensor([[1.0, 2.0, float("-inf")], [3.0, 4.0, 5.0]]) print(F.softmax(padded_sentences_for_softmax, -1)) print(F.softmax(nested_sentences, -1)) ###################################################################### # Let us take a look at a practical example: the multi-head attention component # utilized in `Transformers `__. # The nestedtensor version is straightforward. import math def mha_nested(query: torch.Tensor, key: torch.Tensor, value: torch.Tensor, nheads: int, W_q: torch.Tensor, W_k: torch.Tensor, W_v: torch.Tensor, W_out: torch.Tensor, b_q: torch.Tensor = None, b_k: torch.Tensor = None, b_v: torch.Tensor = None, b_out: torch.Tensor = None, dropout_p: float = 0.0) -> torch.Tensor: """Compute multi-head attention with nested tensors. Args: query (torch.Tensor): query of shape (N, L_t, E_q) key (torch.Tensor): key of shape (N, L_s, E_k) value (torch.Tensor): value of shape (N, L_s, E_v) nheads (int): number of heads in multi-head attention W_q (torch.Tensor): Weight for query input projection of shape (E_total, E_q) W_k (torch.Tensor): Weight for key input projection of shape (E_total, E_k) W_v (torch.Tensor): Weight for value input projection of shape (E_total, E_v) W_out (torch.Tensor): Weight for output projection of shape (E_out, E_total) b_q (torch.Tensor, optional): Bias for query input projection of shape E_total. Default: None. Defaults to None. b_k (torch.Tensor, optional): Bias for key input projection of shape E_total. Default: None. Defaults to None. b_v (torch.Tensor, optional): Bias for value input projection of shape E_total. Default: None. Defaults to None. b_out (torch.Tensor, optional): Bias for output projection of shape E_out. Default: None. Defaults to None. dropout_p (float, optional): Dropout probability. Defaults to 0.0. Where: N is the batch size L_t is the target sequence length (jagged) L_s is the source sequence length (jagged) E_q is the embedding size for query E_k is the embedding size for key E_v is the embedding size for value E_total is the embedding size for all heads combined E_out is the output embedding size Returns: torch.Tensor: Output of shape (N, L_t, E_out) """ N = query.size(0) E_total = W_q.size(0) assert E_total % nheads == 0, "Embedding dim is not divisible by nheads" E_head = E_total // nheads # apply input projection # (N, L_t, E_q) -> (N, L_t, E_total) query = F.linear(query, W_q, b_q) # (N, L_s, E_k) -> (N, L_s, E_total) key = F.linear(key, W_k, b_k) # (N, L_s, E_v) -> (N, L_s, E_total) value = F.linear(value, W_v, b_v) # reshape query, key, value to separate by head # (N, L_t, E_total) -> (N, L_t, nheads, E_head) -> (N, nheads, L_t, E_head) query = query.reshape(N, -1, nheads, E_head).transpose(1, 2) # (N, L_s, E_total) -> (N, L_s, nheads, E_head) -> (N, nheads, L_s, E_head) key = key.reshape(N, -1, nheads, E_head).transpose(1, 2) # (N, L_s, E_total) -> (N, L_s, nheads, E_head) -> (N, nheads, L_s, E_head) value = value.reshape(N, -1, nheads, E_head).transpose(1, 2) # query matmul key^T # (N, nheads, L_t, E_head) x (N, nheads, L_s, E_head)^T -> (N, nheads, L_t, L_s) keyT = key.transpose(-1, -2) attn_weights = torch.matmul(query, keyT) # scale down attn_weights = attn_weights * (1.0 / math.sqrt(E_head)) # softmax attn_weights = F.softmax(attn_weights, dim=-1) # dropout if dropout_p > 0.0: attn_weights = F.dropout(attn_weights, p=dropout_p) # attention_weights matmul value # (N, nheads, L_t, L_s) x (N, nheads, L_s, E_head) -> (N, nheads, L_t, E_head) attn_output = torch.matmul(attn_weights, value) # merge heads # (N, nheads, L_t, E_head) -> (N, L_t, nheads, E_head) -> (N, L_t, E_total) attn_output = attn_output.transpose(1, 2).reshape(N, -1, E_total) # apply output projection # (N, L_t, E_total) -> (N, L_t, E_out) attn_output = F.linear(attn_output, W_out, b_out) return attn_output ###################################################################### # The 0-padded tensor version additionally requires masks # for more complicated treatments at padded entries. def mha_padded(query: torch.Tensor, key: torch.Tensor, value: torch.Tensor, nheads: int, attn_mask_q: torch.Tensor, attn_mask_kv: torch.Tensor, W_q: torch.Tensor, W_k: torch.Tensor, W_v: torch.Tensor, W_out: torch.Tensor, b_q: torch.Tensor = None, b_k: torch.Tensor = None, b_v: torch.Tensor = None, b_out: torch.Tensor = None, dropout_p: float = 0.0) -> torch.Tensor: """Compute multi-head attention for padded out dense tensors. Args: query (torch.Tensor): query of shape (N, L_t, E_q) key (torch.Tensor): key of shape (N, L_s, E_k) value (torch.Tensor): value of shape (N, L_s, E_v) nheads (int): number of heads in multi-head attention attn_mask_q (torch.Tensor): boolean mask indicating locations that should not take part in attention for query, shape (N, L_t) attn_mask_kv (torch.Tensor): boolean mask indicating locations that should not take part in attention for key and value, shape (N, L_s) W_q (torch.Tensor): Weight for query input projection of shape (E_total, E_q) W_k (torch.Tensor): Weight for key input projection of shape (E_total, E_k) W_v (torch.Tensor): Weight for value input projection of shape (E_total, E_v) W_out (torch.Tensor): Weight for output projection of shape (E_out, E_total) b_q (torch.Tensor, optional): Bias for query input projection of shape E_total.. Defaults to None. b_k (torch.Tensor, optional): Bias for key input projection of shape E_total.. Defaults to None. b_v (torch.Tensor, optional): Bias for value input projection of shape E_total.. Defaults to None. b_out (torch.Tensor, optional): Bias for output projection of shape E_out. Defaults to None. dropout_p (float, optional): Dropout probability. Defaults to 0.0. Where: N is the batch size L_t is the target sequence length (padded) L_s is the source sequence length (padded) E_q is the embedding size for query E_k is the embedding size for key E_v is the embedding size for value E_total is the embedding size for all heads combined E_out is the output embedding size Returns: torch.Tensor: Output of shape (N, L_t, E_out) """ N = query.size(0) L_t = query.size(1) L_s = key.size(1) E_total = W_q.size(0) assert E_total % nheads == 0, "Embedding dim is not divisible by nheads" assert L_t == L_s, "This implementation assumes equal query and key sequence lengths" E_head = E_total // nheads # apply input projection # (N, L_t, E_q) -> (N, L_t, E_total) query = F.linear(query, W_q, b_q) # (N, L_s, E_k) -> (N, L_s, E_total) key = F.linear(key, W_k, b_k) # (N, L_s, E_v) -> (N, L_s, E_total) value = F.linear(value, W_v, b_v) # reshape query, key, value to separate by head # (N, L_t, E_total) -> (N, L_t, nheads, E_head) -> (N, nheads, L_t, E_head) -> (N * nheads, L_t, E_head) query = query.reshape(N, -1, nheads, E_head).transpose(1, 2).reshape(N * nheads, -1, E_head) # (N, L_s, E_total) -> (N, L_s, nheads, E_head) -> (N, nheads, L_s, E_head) -> (N * nheads, L_s, E_head) key = key.reshape(N, -1, nheads, E_head).transpose(1, 2).reshape(N * nheads, -1, E_head) # (N, L_s, E_total) -> (N, L_s, nheads, E_head) -> (N, nheads, L_s, E_head) -> (N * nheads, L_s, E_head) value = value.reshape(N, -1, nheads, E_head).transpose(1, 2).reshape(N * nheads, -1, E_head) # query bmm key^T # (N * nheads, L_t, E_head) x (N * nheads, L_s, E_head)^T -> (N * nheads, L_t, L_s) keyT = key.transpose(-1, -2) attn_weights = torch.bmm(query, keyT) # scale down attn_weights = attn_weights * (1.0 / math.sqrt(E_head)) # Have to manipulate masks in order to apply them to the attention weights key_padding_mask = attn_mask_q.view(N, 1, 1, L_t).expand(-1, nheads, -1, -1).reshape(N*nheads, 1, L_t).to(device=device) attn_mask = torch.zeros(key_padding_mask.shape, device=device, dtype=torch.float32) attn_mask = attn_mask.masked_fill_(key_padding_mask, float("-inf")) # Zero out the attention weights where the mask is True by adding -inf prior to softmax attn_weights.add_(attn_mask) # softmax attn_weights = F.softmax(attn_weights, dim=-1).nan_to_num_(0.0) # dropout if dropout_p > 0.0: attn_weights = F.dropout(attn_weights, p=dropout_p) # attention_weights bmm value # (N * nheads, L_t, L_s) x (N * nheads, L_s, E_head) -> (N * nheads, L_t, E_head) attn_output = attn_weights.bmm(value) # merge heads # (N * nheads, L_t, E_head) -> (N, nheads, L_t, E_head) -> (N, L_t, nheads, E_head) -> (N, L_t, E_total) attn_output = attn_output.reshape(N, nheads, -1, E_head).transpose(1, 2).reshape(N, -1, E_total) # apply output projection # (N, L_t, E_total) -> (N, L_t, E_out) attn_output = F.linear(attn_output, W_out, b_out) # padding-specific step: remove output projection bias from padded entries attn_output[attn_mask_q, :] = 0.0 return attn_output ###################################################################### # set hyperparameters following `the Transformer paper `__ N = 512 E_q, E_k, E_v, E_total, E_out = 512, 512, 512, 512, 512 nheads = 8 ###################################################################### # except for dropout probability: set to 0 for correctness check dropout_p = 0.0 ###################################################################### # Let us generate some realistic fake data from Zipf's law. import numpy as np def zipf_sentence_lengths(alpha: float, batch_size: int) -> np.ndarray: # generate fake corpus by unigram Zipf distribution # from wikitext-2 corpus, we get rank "." = 3, "!" = 386, "?" = 858 sentence_lengths = np.empty(batch_size, dtype=int) for ibatch in range(batch_size): sentence_lengths[ibatch] = 1 word = np.random.zipf(alpha) while word != 3 and word != 386 and word != 858: sentence_lengths[ibatch] += 1 word = np.random.zipf(alpha) return sentence_lengths alpha = 1.2 sentence_lengths = zipf_sentence_lengths(alpha, N) L_t = np.max(sentence_lengths) L_s = L_t ###################################################################### # create inputs # create parameters W_q, b_q = torch.randn((E_total, E_q), device=device), torch.randn(E_total, device=device) W_k, b_k = torch.randn((E_total, E_k), device=device), torch.randn(E_total, device=device) W_v, b_v = torch.randn((E_total, E_v), device=device), torch.randn(E_total, device=device) W_out, b_out = torch.randn((E_out, E_total), device=device), torch.randn(E_out, device=device) # create nested input queries = [] keys = [] values = [] for i in range(N): l = sentence_lengths[i] s = l queries.append(torch.randn((l, E_q), device=device)) keys .append(torch.randn((s, E_k), device=device)) values .append(torch.randn((s, E_v), device=device)) query = torch.nested.nested_tensor(queries) key = torch.nested.nested_tensor(keys) value = torch.nested.nested_tensor(values) # pad input padded_query = torch.nested.to_padded_tensor(query, 0.0, (N, L_t, E_q)) padded_key = torch.nested.to_padded_tensor(key, 0.0, (N, L_s, E_k)) padded_value = torch.nested.to_padded_tensor(value, 0.0, (N, L_s, E_v)) # create attention masks attn_mask_q = torch.zeros((N, L_t), dtype=torch.bool) attn_mask_kv = torch.zeros((N, L_s), dtype=torch.bool) # We need to mask out the padding entries in the attention weights. for i, entry_length in enumerate(sentence_lengths): attn_mask_q[i, entry_length:] = True attn_mask_kv[i, entry_length:] = True ###################################################################### # check correctness and performance import timeit t0 = timeit.default_timer() out_nested = mha_nested( query, key, value, nheads, W_q, W_k, W_v, W_out, b_q=b_q, b_k=b_k, b_v=b_v, b_out=b_out, dropout_p=dropout_p) t1 = timeit.default_timer() out_padded = mha_padded( padded_query, padded_key, padded_value, nheads, attn_mask_q, attn_mask_kv, W_q, W_k, W_v, W_out, b_q=b_q, b_k=b_k, b_v=b_v, b_out=b_out, dropout_p=dropout_p) t2 = timeit.default_timer() print("nested and padded calculations differ by", (torch.nested.to_padded_tensor(out_nested, 0.0, (N, L_t, E_out)) - out_padded).abs().max().item()) print("nestedtensor multi-head attention takes", t1 - t0, "seconds") print("padded tensor multi-head attention takes", t2 - t1, "seconds") ###################################################################### # Although the nestedtensor version avoids wasted computation on padding, it is not faster # then the equivalent padded tensor version. This is because the nestedtensor version # has implemented a few of the kernels, like softmax, in a non optimal way. # # There are plans to implement performance critical operations using the new Pytorch 2.0 stack # For now, some performant kernels are provided for specific use cases, e.g. # self-attention evaluation by multi-head attention formula. # embeddings are assumed to be the same E = E_total mha_lib = torch.nn.MultiheadAttention(E, nheads, batch_first=True, device=device) mha_lib.eval() ###################################################################### # extract parameters for correctness check mha_lib.in_proj_weight.requires_grad_(False) mha_lib.in_proj_bias.requires_grad_(False) mha_lib.out_proj.weight.requires_grad_(False) mha_lib.out_proj.bias.requires_grad_(False) W_q, b_q = mha_lib.in_proj_weight[: E, :], mha_lib.in_proj_bias[: E] W_k, b_k = mha_lib.in_proj_weight[E : 2 * E, :], mha_lib.in_proj_bias[E : 2 * E] W_v, b_v = mha_lib.in_proj_weight[2 * E :, :], mha_lib.in_proj_bias[2 * E :] W_out, b_out = mha_lib.out_proj.weight, mha_lib.out_proj.bias ###################################################################### # If we set need_weights to False this will enable the fast path in the library. # Under the hood this will call _scaled_dot_product_attention. If your tensors # are on CUDA, than a fused, efficient attention kernel will be used. For # more detailed performance characteristics look at the benchmark in # pytorch/benchmarks/transformer/sdp.py with torch.inference_mode(): t0 = timeit.default_timer() out_lib, out_lib_weights = mha_lib(query, query, query, need_weights=False) t1 = timeit.default_timer() padded_out = mha_padded( padded_query, padded_query, padded_query, nheads, attn_mask_q, attn_mask_q, W_q, W_k, W_v, W_out, b_q=b_q, b_k=b_k, b_v=b_v, b_out=b_out, dropout_p=dropout_p) t2 = timeit.default_timer() nested_time = t1 - t0 padded_time = t2 - t1 print("Nested and padded calculations differ by", (torch.nested.to_padded_tensor(out_lib, 0.0) - padded_out).abs().max().item()) print("Nested library multi-head attention takes", nested_time, "seconds") print("Padded tensor multi-head attention takes", padded_time, "seconds") print(f"Nested Speedup: {padded_time / nested_time:.3f}")