File size: 7,135 Bytes
315f1bc
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
import math

import torch
import torch.nn.functional as F

"""



An implementation of the parallel scan operation in PyTorch (Blelloch version).

Please see docs/pscan.ipynb for a detailed explanation of what happens here.



"""

def npo2(len):
    """

    Returns the next power of 2 above len

    """

    return 2 ** math.ceil(math.log2(len))

def pad_npo2(X):
    """

    Pads input length dim to the next power of 2



    Args:

        X : (B, L, D, N)



    Returns:

        Y : (B, npo2(L), D, N)

    """

    len_npo2 = npo2(X.size(1))
    pad_tuple = (0, 0, 0, 0, 0, len_npo2 - X.size(1))
    return F.pad(X, pad_tuple, "constant", 0)

class PScan(torch.autograd.Function):
    @staticmethod
    def pscan(A, X):
        # A : (B, D, L, N)
        # X : (B, D, L, N)

        # modifies X in place by doing a parallel scan.
        # more formally, X will be populated by these values :
        # H[t] = A[t] * H[t-1] + X[t] with H[0] = 0
        # which are computed in parallel (2*log2(T) sequential steps (ideally), instead of T sequential steps)

        # only supports L that is a power of two (mainly for a clearer code)
        
        B, D, L, _ = A.size()
        num_steps = int(math.log2(L))

        # up sweep (last 2 steps unfolded)
        Aa = A
        Xa = X
        for _ in range(num_steps-2):
            T = Xa.size(2)
            Aa = Aa.view(B, D, T//2, 2, -1)
            Xa = Xa.view(B, D, T//2, 2, -1)
            
            Xa[:, :, :, 1].add_(Aa[:, :, :, 1].mul(Xa[:, :, :, 0]))
            Aa[:, :, :, 1].mul_(Aa[:, :, :, 0])

            Aa = Aa[:, :, :, 1]
            Xa = Xa[:, :, :, 1]

        # we have only 4, 2 or 1 nodes left
        if Xa.size(2) == 4:
            Xa[:, :, 1].add_(Aa[:, :, 1].mul(Xa[:, :, 0]))
            Aa[:, :, 1].mul_(Aa[:, :, 0])

            Xa[:, :, 3].add_(Aa[:, :, 3].mul(Xa[:, :, 2] + Aa[:, :, 2].mul(Xa[:, :, 1])))
        elif Xa.size(2) == 2:
            Xa[:, :, 1].add_(Aa[:, :, 1].mul(Xa[:, :, 0]))
            return
        else:
            return

        # down sweep (first 2 steps unfolded)
        Aa = A[:, :, 2**(num_steps-2)-1:L:2**(num_steps-2)]
        Xa = X[:, :, 2**(num_steps-2)-1:L:2**(num_steps-2)]
        Xa[:, :, 2].add_(Aa[:, :, 2].mul(Xa[:, :, 1]))
        Aa[:, :, 2].mul_(Aa[:, :, 1])

        for k in range(num_steps-3, -1, -1):
            Aa = A[:, :, 2**k-1:L:2**k]
            Xa = X[:, :, 2**k-1:L:2**k]

            T = Xa.size(2)
            Aa = Aa.view(B, D, T//2, 2, -1)
            Xa = Xa.view(B, D, T//2, 2, -1)

            Xa[:, :, 1:, 0].add_(Aa[:, :, 1:, 0].mul(Xa[:, :, :-1, 1]))
            Aa[:, :, 1:, 0].mul_(Aa[:, :, :-1, 1])

    @staticmethod
    def pscan_rev(A, X):
        # A : (B, D, L, N)
        # X : (B, D, L, N)

        # the same function as above, but in reverse
        # (if you flip the input, call pscan, then flip the output, you get what this function outputs)
        # it is used in the backward pass

        # only supports L that is a power of two (mainly for a clearer code)

        B, D, L, _ = A.size()
        num_steps = int(math.log2(L))

        # up sweep (last 2 steps unfolded)
        Aa = A
        Xa = X
        for _ in range(num_steps-2):
            T = Xa.size(2)
            Aa = Aa.view(B, D, T//2, 2, -1)
            Xa = Xa.view(B, D, T//2, 2, -1)
                    
            Xa[:, :, :, 0].add_(Aa[:, :, :, 0].mul(Xa[:, :, :, 1]))
            Aa[:, :, :, 0].mul_(Aa[:, :, :, 1])

            Aa = Aa[:, :, :, 0]
            Xa = Xa[:, :, :, 0]

        # we have only 4, 2 or 1 nodes left
        if Xa.size(2) == 4:
            Xa[:, :, 2].add_(Aa[:, :, 2].mul(Xa[:, :, 3]))
            Aa[:, :, 2].mul_(Aa[:, :, 3])

            Xa[:, :, 0].add_(Aa[:, :, 0].mul(Xa[:, :, 1].add(Aa[:, :, 1].mul(Xa[:, :, 2]))))
        elif Xa.size(2) == 2:
            Xa[:, :, 0].add_(Aa[:, :, 0].mul(Xa[:, :, 1]))
            return
        else:
            return

        # down sweep (first 2 steps unfolded)
        Aa = A[:, :, 0:L:2**(num_steps-2)]
        Xa = X[:, :, 0:L:2**(num_steps-2)]
        Xa[:, :, 1].add_(Aa[:, :, 1].mul(Xa[:, :, 2]))
        Aa[:, :, 1].mul_(Aa[:, :, 2])

        for k in range(num_steps-3, -1, -1):
            Aa = A[:, :, 0:L:2**k]
            Xa = X[:, :, 0:L:2**k]

            T = Xa.size(2)
            Aa = Aa.view(B, D, T//2, 2, -1)
            Xa = Xa.view(B, D, T//2, 2, -1)

            Xa[:, :, :-1, 1].add_(Aa[:, :, :-1, 1].mul(Xa[:, :, 1:, 0]))
            Aa[:, :, :-1, 1].mul_(Aa[:, :, 1:, 0])

    @staticmethod
    def forward(ctx, A_in, X_in):
        """

        Applies the parallel scan operation, as defined above. Returns a new tensor.

        If you can, privilege sequence lengths that are powers of two.



        Args:

            A_in : (B, L, D, N)

            X_in : (B, L, D, N)



        Returns:

            H : (B, L, D, N)

        """

        L = X_in.size(1)

        # cloning is requiered because of the in-place ops
        if L == npo2(L):
            A = A_in.clone()
            X = X_in.clone()
        else:
            # pad tensors (and clone btw)
            A = pad_npo2(A_in) # (B, npo2(L), D, N)
            X = pad_npo2(X_in) # (B, npo2(L), D, N)
        
        # prepare tensors
        A = A.transpose(2, 1) # (B, D, npo2(L), N)
        X = X.transpose(2, 1) # (B, D, npo2(L), N)

        # parallel scan (modifies X in-place)
        PScan.pscan(A, X)

        ctx.save_for_backward(A_in, X)
        
        # slice [:, :L] (cut if there was padding)
        return X.transpose(2, 1)[:, :L]
    
    @staticmethod
    def backward(ctx, grad_output_in):
        """

        Flows the gradient from the output to the input. Returns two new tensors.



        Args:

            ctx : A_in : (B, L, D, N), X : (B, D, L, N)

            grad_output_in : (B, L, D, N)



        Returns:

            gradA : (B, L, D, N), gradX : (B, L, D, N)

        """

        A_in, X = ctx.saved_tensors

        L = grad_output_in.size(1)

        # cloning is requiered because of the in-place ops
        if L == npo2(L):
            grad_output = grad_output_in.clone()
            # the next padding will clone A_in
        else:
            grad_output = pad_npo2(grad_output_in) # (B, npo2(L), D, N)
            A_in = pad_npo2(A_in) # (B, npo2(L), D, N)

        # prepare tensors
        grad_output = grad_output.transpose(2, 1)
        A_in = A_in.transpose(2, 1) # (B, D, npo2(L), N)
        A = torch.nn.functional.pad(A_in[:, :, 1:], (0, 0, 0, 1)) # (B, D, npo2(L), N) shift 1 to the left (see hand derivation)

        # reverse parallel scan (modifies grad_output in-place)
        PScan.pscan_rev(A, grad_output)

        Q = torch.zeros_like(X)
        Q[:, :, 1:].add_(X[:, :, :-1] * grad_output[:, :, 1:])

        return Q.transpose(2, 1)[:, :L], grad_output.transpose(2, 1)[:, :L]
    
pscan = PScan.apply