动手学深度学习
这里写自定义目录标题
- 注意力
- 加性注意力
- 缩放点积注意力
- 多头注意力
- 自注意力
- 自注意力+缩放点积注意力：案例
- Transformer
注意力
注意力汇聚的输出为值的加权和
查询的长度为q，键的长度为k，值的长度为v。
q ∈ 1 × q , k ∈ 1 × k , v ∈ R 1 × v {\bf{q}} \in {^{1 \times q}},{{\bf{k}}} \in {^{1 \times k}},{{\bf{v}}} \in {\mathbb{R}^{1 \times v}} q∈1×q,k∈1×k,v∈R1×v
n个查询和m个键-值对
Q ∈ n × q , K ∈ m × k , V ∈ R m × v {\bf{Q}} \in {^{n \times q}},{\bf{K}} \in {^{m \times k}},{\bf{V}} \in {\mathbb{R}^{m \times v}} Q∈n×q,K∈m×k,V∈Rm×v
a ( Q , K ) ∈ R n × m {\bf{a}}\left( {{\bf{Q}},{\bf{K}}} \right) \in {\mathbb{R}^{n \times m}} a(Q,K)∈Rn×m是注意力评分函数
α ( Q , K ) = s o f t m a x ( a ( Q , K ) ) = exp ( a ( Q , K ) ) ∑ j = 1 m exp ( a ( Q , K ) ) ∈ R n × m {\boldsymbol{\alpha}} \left( {{\bf{Q}},{\bf{K}}} \right) = {\rm{softmax}}\left( {{\bf{a}}\left( {{\bf{Q}},{\bf{K}}} \right)} \right) = \frac{{\exp \left( {{\bf{a}}\left( {{\bf{Q}},{\bf{K}}} \right)} \right)}}{{\sum\limits_{j = 1}^m {\exp \left( {{\bf{a}}\left( {{\bf{Q}},{\bf{K}}} \right)} \right)} }} \in {\mathbb{R}^{n \times m}} α(Q,K)=softmax(a(Q,K))=j=1∑mexp(a(Q,K))exp(a(Q,K))∈Rn×m是注意力权重
f ( Q , K , V ) = α ( Q , K ) ⊤ V ∈ R n × v f({\bf{Q}},{\bf{K}},{\bf{V}}) = {\boldsymbol{\alpha}} {\left( {{\bf{Q}},{\bf{K}}} \right)^ \top }{\bf{V}} \in {\mathbb{R}^{n \times v}} f(Q,K,V)=α(Q,K)⊤V∈Rn×v是注意力汇聚函数
加性注意力
q ∈ R 1 × q , k ∈ R 1 × k {\bf{q}} \in {\mathbb {R}^{1 \times q}},{\bf{k}} \in {\mathbb {R}^{1 \times k}} q∈R1×q,k∈R1×k
W q ∈ R h × q , W k ∈ R h × k , w v ∈ R h × 1 {{\bf{W}}_q} \in {{\mathbb R}^{h \times q}},{{\bf{W}}_k} \in {{\mathbb R}^{h \times k}},{{\bf{w}}_v} \in {{\mathbb R}^{h \times 1}} Wq∈Rh×q,Wk∈Rh×k,wv∈Rh×1
a ( q , k ) = w v ⊤ t a n h ( W q q ⊤ + W k k ⊤ ) ∈ R a({\bf{q}},{\bf{k}}) = {\bf{w}}_v^ \top {\rm{tanh}}({{\bf{W}}_q}{{\bf{q}}^ \top } + {{\bf{W}}_k}{{\bf{k}}^ \top }) \in \mathbb {R} a(q,k)=wv⊤tanh(Wqq⊤+Wkk⊤)∈R是注意力评分函数
缩放点积注意力
q ∈ R 1 × d , k ∈ R 1 × d , v ∈ R 1 × v {\bf{q}} \in \mathbb{R}{^{1 \times d}},{\bf{k}} \in \mathbb{R}{^{1 \times d}},{\bf{v}} \in {{\mathbb R}^{1 \times v}} q∈R1×d,k∈R1×d,v∈R1×v
a ( q , k ) = 1 d q k ⊤ ∈ R a\left( {{\bf{q}},{\bf{k}}} \right) = \frac{1}{{\sqrt d }}{\bf{q}}{{\bf{k}}^ \top } \in \mathbb{R} a(q,k)=d1qk⊤∈R是注意力评分函数
f ( q , k , v ) = α ( q , k ) ⊤ v = s o f t m a x ( 1 d q k ⊤ ) v ∈ R 1 × v f({\bf{q}},{\bf{k}},{\bf{v}}) = \alpha {\left( {{\bf{q}},{\bf{k}}} \right)^ \top }{\bf{v}} = {\rm{softmax}}\left( {\frac{1}{{\sqrt d }}{\bf{q}}{{\bf{k}}^ \top }} \right){\bf{v}} \in {{\mathbb R}^{1 \times v}} f(q,k,v)=α(q,k)⊤v=softmax(d1qk⊤)v∈R1×v是注意力汇聚函数
n个查询和m个键-值对
Q ∈ R n × d , K ∈ R m × d , V ∈ R m × v \mathbf Q\in\mathbb R^{n\times d}, \mathbf K\in\mathbb R^{m\times d}, \mathbf V\in\mathbb R^{m\times v} Q∈Rn×d,K∈Rm×d,V∈Rm×v
a ( Q , K ) = 1 d Q K ⊤ ∈ R n × m {\bf{a}}\left( {{\bf{Q}},{\bf{K}}} \right) = \frac{1}{{\sqrt d }}{\bf{Q}}{{\bf{K}}^ \top } \in {\mathbb{R}^{n \times m}} a(Q,K)=d1QK⊤∈Rn×m是注意力评分函数
f ( Q , K , V ) = α ( Q , K ) ⊤ V = s o f t m a x ( 1 d Q K ⊤ ) V ∈ R n × v f({\bf{Q}},{\bf{K}},{\bf{V}}) = {\boldsymbol{\alpha}} {\left( {{\bf{Q}},{\bf{K}}} \right)^ \top }{\bf{V}} ={\rm{softmax}}\left( {\frac{1}{{\sqrt d }}{\bf{Q}}{{\bf{K}}^ \top }} \right){\bf{V}} \in {\mathbb{R}^{n \times v}} f(Q,K,V)=α(Q,K)⊤V=softmax(d1QK⊤)V∈Rn×v是注意力汇聚函数
Attention Is All You Need
多头注意力
q ∈ R 1 × d q , k ∈ R 1 × d k , v ∈ R 1 × d v {\bf{q}} \in {{\mathbb R}^{1 \times {d_q}}},{\bf{k}} \in {{\mathbb R}^{1 \times {d_k}}},{\bf{v}} \in {{\mathbb R}^{1 \times {d_v}}} q∈R1×dq,k∈R1×dk,v∈R1×dv
W i ( q ) ∈ R p q × d q , W i ( k ) ∈ R p k × d k , W i ( v ) ∈ R p v × d v {\bf{W}}_i^{(q)} \in {{\mathbb R}^{{p_q} \times {d_q}}},{\bf{W}}_i^{(k)} \in {{\mathbb R}^{{p_k} \times {d_k}}},{\bf{W}}_i^{(v)} \in {{\mathbb R}^{{p_v} \times {d_v}}} Wi(q)∈Rpq×dq,Wi(k)∈Rpk×dk,Wi(v)∈Rpv×dv
q W i ( q ) ⊤ ∈ R 1 × p q , k W i ( k ) ⊤ ∈ R 1 × p k , v W i ( v ) ⊤ ∈ R 1 × p v {\bf{qW}}{_i^{(q)\top} } \in {{\mathbb R}^{1 \times {p_q}}},{\bf{kW}}{_i^{(k)\top} } \in {{\mathbb R}^{1 \times {p_k}}},{\bf{vW}}{_i^{(v)\top} } \in {{\mathbb R}^{1 \times {p_v}}} qWi(q)⊤∈R1×pq,kWi(k)⊤∈R1×pk,vWi(v)⊤∈R1×pv
h i = f ( q W i ( q ) ⊤ , k W i ( k ) ⊤ , v W i ( v ) ⊤ ) ∈ R 1 × p v {{\bf{h}}_i} = f\left( {{\bf{qW}}{{_i^{(q)\top}} },{\bf{kW}}{{_i^{(k)\top}} },{\bf{vW}}{{_i^{(v)\top}} }} \right) \in {{\mathbb R}^{1 \times {p_v}}} hi=f(qWi(q)⊤,kWi(k)⊤,vWi(v)⊤)∈R1×pv是注意力头
多个注意力头连结然后线性变换
W o ∈ R p o × h p v {{\bf{W}}_o} \in {{\mathbb R}^{{p_o} \times h{p_v}}} Wo∈Rpo×hpv
W o [ h 1 ⊤ ⋮ h h ⊤ ] ∈ R p o {{\bf{W}}_o}\left[ {\begin{array}{c} {{{\bf{h}}_1^ \top}}\\ \vdots \\ {{{\bf{h}}_h^ \top}} \end{array}} \right] \in {{\mathbb R}^{{p_o}}} Wo h1⊤⋮hh⊤ ∈Rpo
p q h = p k h = p v h = p o p_q h = p_k h = p_v h = p_o pqh=pkh=pvh=po
多头注意力：多个注意力头连结然后线性变换
自注意力
x i ∈ R 1 × d , X = [ x 1 ⋯ x n ] ∈ R n × d {{\bf{x}}_i} \in {{\mathbb R}^{1 \times d}},{\bf{X}} = \left[ {\begin{array}{c} {{{\bf{x}}_1}}\\ \cdots \\ {{{\bf{x}}_n}} \end{array}} \right] \in {{\mathbb R}^{n \times d}} xi∈R1×d,X= x1⋯xn ∈Rn×d
Q = X , K = X , V = X {\bf{Q}} = {\bf{X}},{\bf{K}} = {\bf{X}},{\bf{V}} = {\bf{X}} Q=X,K=X,V=X
f ( Q , K , V ) = α ( Q , K ) ⊤ V = s o f t m a x ( 1 d Q K ⊤ ) V ∈ R n × d f({\bf{Q}},{\bf{K}},{\bf{V}}) = {\boldsymbol{\alpha}} {\left( {{\bf{Q}},{\bf{K}}} \right)^ \top }{\bf{V}} ={\rm{softmax}}\left( {\frac{1}{{\sqrt d }}{\bf{Q}}{{\bf{K}}^ \top }} \right){\bf{V}} \in {\mathbb{R}^{n \times d}} f(Q,K,V)=α(Q,K)⊤V=softmax(d1QK⊤)V∈Rn×d
y i = f ( x i , ( x 1 , x 1 ) , … , ( x n , x n ) ) ∈ R d {{\bf{y}}_i} = f\left( {{{\bf{x}}_i},\left( {{{\bf{x}}_1},{{\bf{x}}_1}} \right), \ldots ,\left( {{{\bf{x}}_n},{{\bf{x}}_n}} \right)} \right) \in {{\mathbb R}^d} yi=f(xi,(x1,x1),…,(xn,xn))∈Rd
n个查询和m个键-值对
Q = t a n h ( W q X ) ∈ R n × d {\bf{Q}} = {\rm{tanh}}\left( {{{\bf{W}}_q}{\bf{X}}} \right) \in {{\mathbb R}^{n \times d}} Q=tanh(WqX)∈Rn×d
K = t a n h ( W k X ) ∈ R m × d {\bf{K}} = {\rm{tanh}}\left( {{{\bf{W}}_k}{\bf{X}}} \right) \in {{\mathbb R}^{m \times d}} K=tanh(WkX)∈Rm×d
V = t a n h ( W v X ) ∈ R m × v {\bf{V}} = {\rm{tanh}}\left( {{{\bf{W}}_v}{\bf{X}}} \right) \in {{\mathbb R}^{m \times v}} V=tanh(WvX)∈Rm×v
自注意力+缩放点积注意力：案例
J. Xu, F. Zhong, and Y. Wang, “Learning multi-agent coordination for enhancing target coverage in directional sensor networks,” in Proc. Neural Information Processing Systems (NeurIPS), Vancouver, BC, Canada, Dec. 2020, pp. 1–16.
https://github.com/XuJing1022/HiT-MAC/blob/main/perception.py
类比多头注意力中
d q = d k = d v = d i n {d_q} = {d_k} = {d_v} = {d_{in}} dq=dk=dv=din
p q = p k = p v = d a t t {p_q} = {p_k} = {p_v} = {d_{att}} pq=pk=pv=datt
x i ∈ R 1 × d i n , X = [ x 1 ⋯ x n m ] ∈ R n m × d i n {{\bf{x}}_i} \in {{\mathbb R}^{1 \times d_{in}}},{\bf{X}} = \left[ {\begin{array}{c} {{{\bf{x}}_1}}\\ \cdots \\ {{{\bf{x}}_{nm}}} \end{array}} \right] \in {{\mathbb R}^{nm \times d_{in}}} xi∈R1×din,X= x1⋯xnm ∈Rnm×din
W ∈ R d a t t × d i n {\bf{W}} \in {{\mathbb R}^{d_{att}\times d_{in}}} W∈Rdatt×din
Q = t a n h ( W q X ⊤ ) ⊤ ∈ R n m × d a t t {\bf{Q}} = {\rm{tanh}}\left( {{{\bf{W}}_q}{\bf{X}}^\top} \right)^\top \in {{\mathbb R}^{nm \times d_{att}}} Q=tanh(WqX⊤)⊤∈Rnm×datt
K = t a n h ( W k X ⊤ ) ⊤ ∈ R n m × d a t t {\bf{K}} = {\rm{tanh}}\left( {{{\bf{W}}_k}{\bf{X}}^\top} \right)^\top \in {{\mathbb R}^{nm \times d_{att}}} K=tanh(WkX⊤)⊤∈Rnm×datt
V = t a n h ( W v X ⊤ ) ⊤ ∈ R n m × d a t t {\bf{V}} = {\rm{tanh}}\left( {{{\bf{W}}_v}{\bf{X}}^\top} \right)^\top \in {{\mathbb R}^{nm \times d_{att}}} V=tanh(WvX⊤)⊤∈Rnm×datt
f ( Q , K , V ) = α ( Q , K ) ⊤ V = s o f t m a x ( 1 d Q K ⊤ ) V ∈ R n m × d a t t f({\bf{Q}},{\bf{K}},{\bf{V}}) = {\boldsymbol{\alpha}} {\left( {{\bf{Q}},{\bf{K}}} \right)^ \top }{\bf{V}} ={\rm{softmax}}\left( {\frac{1}{{\sqrt d }}{\bf{Q}}{{\bf{K}}^ \top }} \right){\bf{V}} \in {{\mathbb R}^{nm \times d_{att}}} f(Q,K,V)=α(Q,K)⊤V=softmax(d1QK⊤)V∈Rnm×datt
class AttentionLayer(torch.nn.Module):def __init__(self, feature_dim, weight_dim, device):super(AttentionLayer, self).__init__()self.in_dim = feature_dimself.device = deviceself.Q = xavier_init(nn.Linear(self.in_dim, weight_dim))self.K = xavier_init(nn.Linear(self.in_dim, weight_dim))self.V = xavier_init(nn.Linear(self.in_dim, weight_dim))self.feature_dim = weight_dimdef forward(self, x):# param x: [num_agent, num_target, in_dim]# return z: [num_agent, num_target, weight_dim]# z = softmax(Q,K)*Vq = torch.tanh(self.Q(x)) # [batch_size, sequence_len, weight_dim]k = torch.tanh(self.K(x)) # [batch_size, sequence_len, weight_dim]v = torch.tanh(self.V(x)) # [batch_size, sequence_len, weight_dim]z = torch.bmm(F.softmax(torch.bmm(q, k.permute(0, 2, 1)), dim=2), v) # [batch_size, sequence_len, weight_dim]global_feature = z.sum(dim=1)return z, global_feature
Transformer
