Blank Language ModelsTianxiao Shen* Victor Quach* Regina Barzilay Tommi Jaakkola (*: equal contribution)
mailto:[email protected]
Left-to-Right Language Model
They also have ice
They also have ice
cream
…
Generate from scratch
Start with partially specified text
• text editing
• template filling
• text restoration
• …
Blank Language Model (BLM)
Fine-grained control over generation location
Respect preceding and following context
Variable number of missing tokens
Output: They also have ice cream which is really good .Input: They also have which .
BLM - Overview
• Dynamic canvas where “ ” controls where tokens can be placed
• At each step,
1. select a “ ”
2. predict a word w
3. replace that blank with “w”, “ w”, “w ”, or “ w ”
• Stop when there is no “ ”
BLM - Overview
They also have which .
• Dynamic canvas where “ ” controls where tokens can be placed
• At each step,
1. select a “ ”
2. predict a word w
3. replace that blank with “w”, “ w”, “w ”, or “ w ”
• Stop when there is no “ ”
BLM - Overview
really
They also have which .
• Dynamic canvas where “ ” controls where tokens can be placed
• At each step,
1. select a “ ”
2. predict a word w
3. replace that blank with “w”, “ w”, “w ”, or “ w ”
• Stop when there is no “ ”
BLM - Overview
They also have which really .
• Dynamic canvas where “ ” controls where tokens can be placed
• At each step,
1. select a “ ”
2. predict a word w
3. replace that blank with “w”, “ w”, “w ”, or “ w ”
• Stop when there is no “ ”
BLM - Overview
ice
They also have which really .
• Dynamic canvas where “ ” controls where tokens can be placed
• At each step,
1. select a “ ”
2. predict a word w
3. replace that blank with “w”, “ w”, “w ”, or “ w ”
• Stop when there is no “ ”
BLM - Overview
They also have ice which really .
• Dynamic canvas where “ ” controls where tokens can be placed
• At each step,
1. select a “ ”
2. predict a word w
3. replace that blank with “w”, “ w”, “w ”, or “ w ”
• Stop when there is no “ ”
BLM - Overview
is
They also have ice which really .
• Dynamic canvas where “ ” controls where tokens can be placed
• At each step,
1. select a “ ”
2. predict a word w
3. replace that blank with “w”, “ w”, “w ”, or “ w ”
• Stop when there is no “ ”
BLM - Overview
They also have ice which is really .
• Dynamic canvas where “ ” controls where tokens can be placed
• At each step,
1. select a “ ”
2. predict a word w
3. replace that blank with “w”, “ w”, “w ”, or “ w ”
• Stop when there is no “ ”
BLM - Overview
cream
They also have ice which is really .
• Dynamic canvas where “ ” controls where tokens can be placed
• At each step,
1. select a “ ”
2. predict a word w
3. replace that blank with “w”, “ w”, “w ”, or “ w ”
• Stop when there is no “ ”
BLM - Overview
good
They also have ice cream which is really .
• Dynamic canvas where “ ” controls where tokens can be placed
• At each step,
1. select a “ ”
2. predict a word w
3. replace that blank with “w”, “ w”, “w ”, or “ w ”
• Stop when there is no “ ”
BLM - Overview
They also have ice cream which is really good .
• Dynamic canvas where “ ” controls where tokens can be placed
• At each step,
1. select a “ ”
2. predict a word w
3. replace that blank with “w”, “ w”, “w ”, or “ w ”
• Stop when there is no “ ”
BLM - Overview
• Dynamic canvas where “ ” controls where tokens can be placed
• At each step,
1. select a “ ”
2. predict a word w
3. replace that blank with “w”, “ w”, “w ”, or “ w ”
• Stop when there is no “ ”Grammar
• Nonterminal:
• Terminals: w V
• Production rules: ? w ?
(dist. depends on model and context)
∈
→
BLM - Architecture
alsoThey have ____ which ____
Transformer
.
Linear & Softmax1) Choose a blank 2) Predict a word
3) Create new blanks
Linear & Softmax
really
MLP
Fill and repeat
really
really
really____
really
____
________
✓
Trajectory: generating process from an initial “ ” to complete text
BLM - Likelihood
BLM - Likelihood
A sentence with words can be realized by trajectories, each trajectory corresponds to a different word insertion order
x
AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexKQL0FBPGYgHlAsoTZSW8yZnZ2mZkVwxLw7sWDIl79JG/+jZPHQaMFDUVVN91dQSK4Nq775eRWVtfWN/Kbha3tnd294v5BU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWjq6nfukeleSxvzThBP6IDyUPOqLFS/aFXLLlldwbyl3gLUoIFar3iZ7cfszRCaZigWnc8NzF+RpXhTOCk0E01JpSN6AA7lkoaofaz2aETcmKVPgljZUsaMlN/TmQ00nocBbYzomaol72p+J/XSU144WdcJqlByeaLwlQQE5Pp16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv/yXNM/KXqV8Wa+UqteP8zjycATHcAoenEMVbqAGDWCA8AQv8OrcOc/Om/M+b805iwgP4Recj28RXI2M
n
AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexKQL0FBPGYgHlAsoTZSW8yZnZ2mZkVwhLw7sWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGN1O/9YhK81jem3GCfkQHkoecUWOluuwVS27ZnYGsEm9BSrBArVf86vZjlkYoDRNU647nJsbPqDKcCZwUuqnGhLIRHWDHUkkj1H42O3RCzqzSJ2GsbElDZurviYxGWo+jwHZG1Az1sjcV//M6qQmv/IzLJDUo2XxRmApiYjL9mvS5QmbE2BLKFLe3EjakijJjsynYELzll1dJ86LsVcrX9Uqpevs0jyMPJ3AK5+DBJVThDmrQAAYIz/AKb86D8+K8Ox/z1pyziPAY/sD5/AECNI2C
n!
AAAB6XicbVDLSgNBEOyNrxhfUY9eRoPgKexKQL0FBPEYxTwgWcLsZDYZMju7zPQKYQn4AV48KOLVP/Lm3zh5HDSxoKGo6qa7K0ikMOi6305uZXVtfSO/Wdja3tndK+4fNEycasbrLJaxbgXUcCkUr6NAyVuJ5jQKJG8Gw+uJ33zk2ohYPeAo4X5E+0qEglG00r067hZLbtmdgiwTb05KMEetW/zq9GKWRlwhk9SYtucm6GdUo2CSjwud1PCEsiHt87alikbc+Nn00jE5tUqPhLG2pZBM1d8TGY2MGUWB7YwoDsyiNxH/89ophpd+JlSSIldstihMJcGYTN4mPaE5QzmyhDIt7K2EDaimDG04BRuCt/jyMmmcl71K+equUqrePM3iyMMRnMAZeHABVbiFGtSBQQjP8ApvztB5cd6dj1lrzplHeAh/4Hz+AFmKja0=
p(x; ✓) =X
�2Sn
p(x,�; ✓) =X
�2Sn
n�1Y
t=0
p(ax,�t |cx,�t ; ✓)
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
order action, canvas at step t
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
1 2 3 4 5 6 7 8 9 10
3
1
10
6
2
8
4
7
5
9
BLM - Training
log p(x; ✓) = logX
�2Sn
n�1Y
t=0
p(ax,�t |cx,�t ; ✓)
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
intractable
log
1
m
mX
i=1
ai
!>=
1
m
mX
i=1
log ai
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
� log(n!) + 1n!
X
�2Sn
n�1X
t=0
log p(ax,�t |cx,�t ; ✓)
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
BLM - Training
log p(x; ✓) = logX
�2Sn
n�1Y
t=0
p(ax,�t |cx,�t ; ✓) intractable
log
1
m
mX
i=1
ai
!>=
1
m
mX
i=1
log ai
� log(n!) + 1n!
X
�2Sn
n�1X
t=0
log p(ax,�t |cx,�t ; ✓)
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
1. Uniformly sample from
2. Uniformly sample from to
3. Construct canvas
4. Compute estimated loss
cx,�t
AAADbXiclVJbixMxFE6nXtbx1l3xwQuStRRb3JYZWVCRyoIIPq5odxeadkjTzDRskhmSM7JlnCf/oW/+BV/8C6bTqttdF/RA4Mt3bt9JziSTwkIQfKt59UuXr1zduOZfv3Hz1u3G5taBTXPD+IClMjVHE2q5FJoPQIDkR5nhVE0kP5wcv1n4Dz9xY0WqP8I84yNFEy1iwSg4KtqsfWkRmSY4a5+8IjDjQDu4jyuK2FxFBbEiUZQIjT9EusQkM+k0KqAflONCd8PSZdJxcbKzjCsj+MzWrvhXWb9qRCSPoU1iQ1kRloUql11EPyzHCtNIECOSGXTwa6figqhKnQt1FRNe3dp6u4Of/knQ2+UF8ivylPrV8P88QsX5636/1f0toos1YdMU/qtu1GgGvaAyfB6EK9BEK9uPGl/JNGW54hqYpNYOwyCDUUENCCZ56ZPc8oyyY5rwoYOaKm5HRbUtJW45Zorj1LijAVfs6YyCKmvnauIiFYWZPetbkH/zDXOIX4wKobMcuGbLRnEuMaR4sXp4KgxnIOcOUGaE04rZjLovA7egvnuE8OzI58HBs16423v5fre593b1HBvoAXqM2ihEz9Eeeof20QCx2nev4d3z7ns/6nfrD+uPlqFebZVzB61Z/clP4VEWPg==
� log(n!)� n · log p(ax,�t |cx,�t ; ✓)
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
t
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
0
AAADcHiclVJtixMxEE63vpzrW8/7IqiYWgqt2rIrBypSORDBjyfau4OmXdI0uw2XZJdkVq6s+9Uf6Dd/hF/8Babbqtc7D3QgMHnmmZlnkplmUlgIgm81r37p8pWrW9f86zdu3rrd2L5zYNPcMD5kqUzN0ZRaLoXmQxAg+VFmOFVTyQ+nx2+W8cNP3FiR6o+wyPhY0USLWDAKDoq2a1/aRKYJzjonrwjMOdAuHuAKIjZXUUGsSBQlQuMPkS4xyUw6iwoYBOWk0L2wdJl0Upw8XfHKCD6zjSv+VdavGhHJY+iQ2FBWhGWhylUXMQjLicI0EsSIZA5d/NqpuIBVqXNUVzHh1a2jm1385E+CbpYXyK/AU+rXw//zCBXmB357k+K3e7919LAmbJbCf5WOGq2gH1SGzzvh2mmhte1Hja9klrJccQ1MUmtHYZDBuKAGBJO89ElueUbZMU34yLmaKm7HRbUwJW47ZIbj1LijAVfo6YyCKmsXauqYisLcno0twb/FRjnEL8aF0FkOXLNVoziXGFK83D48E4YzkAvnUGaE04rZnLpfA7ejvnuE8OzI552DZ/1wt//y/W5r7+36ObbQPfQIdVCInqM99A7toyFite/ejnffe+D9qN+tP6w3V1Svts7ZQRtWf/wTUQIWuw==
n� 1
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
�
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
Sn
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
one action loss per pass :(
BLM - Training
� log(n!) + 1n!
X
�2Sn
n�1X
t=0
log p(ax,�t |cx,�t ; ✓)
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
depends only on the first elements of
—> combine into one pass loss calculations of trajectories that are the samecx,�t
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
t
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
�
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
t
AAAFqnicrVRbb9MwFM62Bka4dfDIi0dV0Wpr1UyTgKGiSQjEC9K4tB2q28hx3NRa4kTxCVoVLPHT+A288W9wk4616yZWwFKk4+9cvu+cE9mNAy6h1fq5tr5RMm/c3Lxl3b5z99798taDrozShLIOjYIoOXaJZAEXrAMcAnYcJ4yEbsB67smrqb/3hSWSR+ITTGI2CIkv+IhTAhpytja+V3EQ+Siunb7AMGZA6qiNcgjLNHQyLLkfEswF+ugIhXCcRJ6TQbulhplo2EpnkmF2ulvEKQe+0oUrOitr5UQ4YCOo4VFCaGarLFQFC2/bahgi4nCccH8MdfRSq7giKlenQ3VFn+W3mtiuo53zBLGtrpCfg3PqZ81fu4Ucs6q6mFXVBSywqovBVrXxW1EDCUy9CFYkac+3tCT4zz3+PduZ3pDA2HXRa2fOnNE4mX0ASrMsO2DHVupSfO9ATFOmu++vJK/4GwZa5fye/13l/9NyLRU52Wx1Qq+uAYu7O1e1uh6nXGk1W/lBy4Y9MyrG7Bw55R/Yi2gaMgE0IFL27VYMg4wkwGnAlIVTyWJCT4jP+toUJGRykOVPjUJVjXhoFCX6E4BydD4jI6GUk9DVkdNpyIu+KXiZr5/C6Nkg4yJOgQlaEI3SAEGEpu8W8njCKAQTbRCacK0V0THRAwX9ull6CPbFlpeN7l7T3m8+f79fOXzzrRjHpvHIeGzUDNt4ahwab40jo2PQ0pPSu1K31DN3zQ/mZ7NfhK6vzUb40Fg4pvcLKDjk3w==
t+ 1
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
in the first steps but different at the step
BLM - Training depends only on the first elements of
—> combine into one pass loss calculations of trajectories that are the samecx,�t
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
t
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
�
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
t
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
t+ 1
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
in the first steps but different at the step
� log(n!) +n�1X
t=0
1
n!
X
�2Sn
log p(ax,�t |cx,�t ; ✓)
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
= log(n!) + n · EtE�1:tE�t+1E�t+2:n [log p(ax,�t |c
x,�t ; ✓)]
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
= log(n!) + EtE�1:t
2
4 nn� t
X
�t+1
log p(ax,�t |cx,�t ; ✓)
3
5
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
BLM - Training
= log(n!) + EtE�1:t
2
4 nn� t
X
�t+1
log p(ax,�t |cx,�t ; ✓)
3
5
AAAFp3icpVTbbtNAEHXbGIq5pfDIy5ZgKVFJFFeVgKKgSgiEeKGoTRopTqz1Zp2saq8t7xg1MivxY/wEb/wN60tp0rSioStZmj1zOWdmrHUjnwlot3+vrW9U9Dt3N+8Z9x88fPS4uvWkJ8IkJrRLQj+M+y4W1GecdoGBT/tRTHHg+vTEPX2f+U++0ViwkB/DLKLDAE848xjBoCBna+OnafvhBEX1s7c2TCngBuqgHLJFEjipLdgkwDbj6MjhEtlRHI6dFDptOUp505IqE4/Ss5dFnHTgO1m4ovOyRk5k+9SDuu3FmKSWTANZsLCOJUcBwg6zYzaZQgO9UyquicrVqVBVcULzW51vN9DORQLfltfIz8E59WXzN24hxwxTFTNMVcAwF0MNs/lXTxNxm4xDWJGiM9/Qktx/d/j/bOd6AwxT10UfnDmzpHFSax+kYll2wI4l5ZX47j7PUrLND1aSV/wLQ6Vyfsu3V3kbLQtSbiQi5yo3x9XmmrC4ugtRq8txqrV2q50ftGxYpVHTynPoVH/Z45AkAeVAfCzEwGpHMExxDIz4VBp2ImiEySme0IEyOQ6oGKb5OyORqZAx8sJYfRxQjs5npDgQYha4KjKbhrjsy8CrfIMEvNfDlPEoAcpJQeQlPoIQZY8WGrOYEvBnysAkZkorIlOsBgrqaTPUEKzLLS8bvd2Wtdd683WvdvDxRzGOTe2Z9lyra5b2SjvQPmmHWlcjlReVz5WjyrHe0L/oPb1fhK6vlSN8qi0cHf8BXfLkHg==
1. Uniformly sample from to
2. Uniformly sample
3. Construct canvas
4. Compute estimated loss
cx,�t
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
t
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
0
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
n� 1
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
�1:t
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
� log(n!)� nn� t
X
�t+1
log p(ax,�t |cx,�t ; ✓)
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
n/2 action losses per pass :)
BLM - Training
They also have ice cream which is really good .
1 2 3 4 5 6 7 8 9 10x =
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
AAAB63icbVBNSwMxEJ2tX7V+VT16CRbBU8mKoheh6MVjBWsL7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMCxPBjcX42yutrK6tb5Q3K1vbO7t71f2DRxOnmrIWjUWsOyExTHDFWpZbwTqJZkSGgrXD8W3ut5+YNjxWD3aSsECSoeIRp8Tmkrr2cb9aw3U8A1omfkFqUKDZr371BjFNJVOWCmJM18eJDTKiLaeCTSu91LCE0DEZsq6jikhmgmx26xSdOGWAoli7UhbN1N8TGZHGTGToOiWxI7Po5eJ/Xje10VWQcZWklik6XxSlAtkY5Y+jAdeMWjFxhFDN3a2Ijogm1Lp4Ki4Ef/HlZfJ4Vvcv6vj+vNa4KeIowxEcwyn4cAkNuIMmtIDCCJ7hFd486b14797HvLXkFTOH8Afe5w84oI2y
n = 10
1. Uniformly sample from to
2. Uniformly sample
3. Construct canvas
4. Compute estimated loss
cx,�t
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
t
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
0
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
n� 1
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
�1:t
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
� log(n!)� nn� t
X
�t+1
log p(ax,�t |cx,�t ; ✓)
AAAGEXicrVRNb9NAEHVLDMV8pXDksiWySFQSxVUloCioEgJxLIK0leLEWm82zqr22vKOUSOzEr+AC3+FCwcQ4sqNG/+GtZ1Sp2lFQ1nJ1uzbmXlvZlbrRj4T0G7/Wlq+VNEvX1m5aly7fuPmrerq7V0RJjGhXRL6YbzvYkF9xmkXGPh0P4opDlyf7rkHz7Lzvbc0Fizkb2AS0X6APc5GjGBQkLNauW/afuihqH74xIYxBdxAHZRDtkgCJ7UF8wJsM45eO1wiO4rDoZNCpy0HKW9aUkXiQXr4oPCTDrwjM1t0lNbIiWyfjqBuj2JMUkumgSxYWMeSgwBhh9kx88bQQE+VijO8cnXKVWX0aL6r87UGWj8O4GvyDPk5WFI/Lf7cJeSYk1pbIA1TpTRMlcYwYV39ZqMMs/lHWhNxmwxDWJTNo+Xq5rT/vdzF+DoltiPFAYax66LnTslMy21QLPMHqh9SnopvbPEsJLsGvYXkFRejr1SWR35xlf9Py7lU5GTT0XE1uibMzu5Y1T/oaaLSlZvhuDCFU621W+18oXnDmho1bbp2nOpPexiSJKAciI+F6FntCPopjoERn0rDTgSNMDnAHu0pk+OAin6av2gSmQoZolEYq48DytFyRIoDISaBqzyzTouTZxl42lkvgdGjfsp4lADlpCAaJT6CEGXPIxqymBLwJ8rAJGZKKyJjrBoJ6hE1VBOskyXPG7sbLWuz9fjVZm37xfuiHSvaXe2eVtcs7aG2rb3UdrSuRiofKp8qXypf9Y/6Z/2b/r1wXV6atvCONrP0H78Bf9EM+g==
n/2 action losses per pass :)
BLM - Training
t = 5
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
They also have ice cream which is really good .
1 2 3 4 5 6 7 8 9 10x =
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
AAAB63icbVBNSwMxEJ2tX7V+VT16CRbBU8mKoheh6MVjBWsL7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMCxPBjcX42yutrK6tb5Q3K1vbO7t71f2DRxOnmrIWjUWsOyExTHDFWpZbwTqJZkSGgrXD8W3ut5+YNjxWD3aSsECSoeIRp8Tmkrr2cb9aw3U8A1omfkFqUKDZr371BjFNJVOWCmJM18eJDTKiLaeCTSu91LCE0DEZsq6jikhmgmx26xSdOGWAoli7UhbN1N8TGZHGTGToOiWxI7Po5eJ/Xje10VWQcZWklik6XxSlAtkY5Y+jAdeMWjFxhFDN3a2Ijogm1Lp4Ki4Ef/HlZfJ4Vvcv6vj+vNa4KeIowxEcwyn4cAkNuIMmtIDCCJ7hFd486b14797HvLXkFTOH8Afe5w84oI2y
n = 10
1. Uniformly sample from to
2. Uniformly sample
3. Construct canvas
4. Compute estimated loss
cx,�t
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
t
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
0
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
n� 1
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
�1:t
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
� log(n!)� nn� t
X
�t+1
log p(ax,�t |cx,�t ; ✓)
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
n/2 action losses per pass :)
BLM - Training
t = 5
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
They also have ice cream which is really good .
1 2 3 4 5 6 7 8 9 10x =
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
�1:t = (6, 2, 1, 3, 10)
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
AAAB63icbVBNSwMxEJ2tX7V+VT16CRbBU8mKoheh6MVjBWsL7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMCxPBjcX42yutrK6tb5Q3K1vbO7t71f2DRxOnmrIWjUWsOyExTHDFWpZbwTqJZkSGgrXD8W3ut5+YNjxWD3aSsECSoeIRp8Tmkrr2cb9aw3U8A1omfkFqUKDZr371BjFNJVOWCmJM18eJDTKiLaeCTSu91LCE0DEZsq6jikhmgmx26xSdOGWAoli7UhbN1N8TGZHGTGToOiWxI7Po5eJ/Xje10VWQcZWklik6XxSlAtkY5Y+jAdeMWjFxhFDN3a2Ijogm1Lp4Ki4Ef/HlZfJ4Vvcv6vj+vNa4KeIowxEcwyn4cAkNuIMmtIDCCJ7hFd486b14797HvLXkFTOH8Afe5w84oI2y
n = 10
1. Uniformly sample from to
2. Uniformly sample
3. Construct canvas
4. Compute estimated loss
cx,�t
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
t
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
0
AAADcHiclVJtixMxEE63vpzrW8/7IqiYWgqt2rIrBypSORDBjyfau4OmXdI0uw2XZJdkVq6s+9Uf6Dd/hF/8Babbqtc7D3QgMHnmmZlnkplmUlgIgm81r37p8pWrW9f86zdu3rrd2L5zYNPcMD5kqUzN0ZRaLoXmQxAg+VFmOFVTyQ+nx2+W8cNP3FiR6o+wyPhY0USLWDAKDoq2a1/aRKYJzjonrwjMOdAuHuAKIjZXUUGsSBQlQuMPkS4xyUw6iwoYBOWk0L2wdJl0Upw8XfHKCD6zjSv+VdavGhHJY+iQ2FBWhGWhylUXMQjLicI0EsSIZA5d/NqpuIBVqXNUVzHh1a2jm1385E+CbpYXyK/AU+rXw//zCBXmB357k+K3e7919LAmbJbCf5WOGq2gH1SGzzvh2mmhte1Hja9klrJccQ1MUmtHYZDBuKAGBJO89ElueUbZMU34yLmaKm7HRbUwJW47ZIbj1LijAVfo6YyCKmsXauqYisLcno0twb/FRjnEL8aF0FkOXLNVoziXGFK83D48E4YzkAvnUGaE04rZnLpfA7ejvnuE8OzI552DZ/1wt//y/W5r7+36ObbQPfQIdVCInqM99A7toyFite/ejnffe+D9qN+tP6w3V1Svts7ZQRtWf/wTUQIWuw==
n� 1
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
�1:t
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
� log(n!)� nn� t
X
�t+1
log p(ax,�t |cx,�t ; ✓)
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
n/2 action losses per pass :)
BLM - Training
t = 5
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
alsoThey have ____ which ____ .cx,�t
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
They also have ice cream which is really good .
1 2 3 4 5 6 7 8 9 10x =
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
�1:t = (6, 2, 1, 3, 10)
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
(6, 2, 1, 3, 10)
AAAB+XicbVDLSgMxFL1TX7W+Rl26CRahQhlm2lrtrujGZQX7gHYomTRtQzMPkkyhDP0TNy4UceufuPNvTF+g1gP3cjjnXnJzvIgzqWz7y0htbG5t76R3M3v7B4dH5vFJQ4axILROQh6Klocl5SygdcUUp61IUOx7nDa90d3Mb46pkCwMHtUkoq6PBwHrM4KVlrqmmSvnUSGPnDwq6m5fds2sbdlzINsqF68KlRJyVsqKZGGJWtf87PRCEvs0UIRjKduOHSk3wUIxwuk004kljTAZ4QFtaxpgn0o3mV8+RRda6aF+KHQFCs3VnxsJ9qWc+J6e9LEayr/eTPzPa8eqf+MmLIhiRQOyeKgfc6RCNIsB9ZigRPGJJpgIpm9FZIgFJkqHldEhrH15nTQKllOyKg+lbPV2GUcazuAccuDANVThHmpQBwJjeIIXeDUS49l4M94XoyljuXMKv2B8fAMl45Aj
AAAB63icbVBNSwMxEJ2tX7V+VT16CRbBU8mKoheh6MVjBWsL7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMCxPBjcX42yutrK6tb5Q3K1vbO7t71f2DRxOnmrIWjUWsOyExTHDFWpZbwTqJZkSGgrXD8W3ut5+YNjxWD3aSsECSoeIRp8Tmkrr2cb9aw3U8A1omfkFqUKDZr371BjFNJVOWCmJM18eJDTKiLaeCTSu91LCE0DEZsq6jikhmgmx26xSdOGWAoli7UhbN1N8TGZHGTGToOiWxI7Po5eJ/Xje10VWQcZWklik6XxSlAtkY5Y+jAdeMWjFxhFDN3a2Ijogm1Lp4Ki4Ef/HlZfJ4Vvcv6vj+vNa4KeIowxEcwyn4cAkNuIMmtIDCCJ7hFd486b14797HvLXkFTOH8Afe5w84oI2y
n = 10
1. Uniformly sample from to
2. Uniformly sample
3. Construct canvas
4. Compute estimated loss
cx,�t
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
t
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
0
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
n� 1
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
�1:t
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
� log(n!)� nn� t
X
�t+1
log p(ax,�t |cx,�t ; ✓)
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
n/2 action losses per pass :)
BLM - Training
t = 5
AAAGHHicrVTLbtNAFHVLAsU8msKSzZQoUqKSyI6KgKKgCoTEsgjSVopTazwZO6PaY8tzjRqZ+RA2/AobFiDEhgUSf8PYcanTtKKhjGTrzn2dcx8aJ/KZAMP4tbR8pVK9em3lun7j5q3bq7W1O7siTGJC+yT0w3jfwYL6jNM+MPDpfhRTHDg+3XMOX2T2vXc0Fizkb2ES0WGAPc5cRjAolb1W6TYsP/RQ1Dx6asGYAm6hHspVlkgCO7UE8wJsMY7e2FwiK4rDkZ1Cz5AHKW+bUkXig/TowdRP2vCezFzRcVo9B7J86kLTcmNMUlOmgZyisJ4pDwKEbWbFzBtDCz1TLM7xytkpV5XRo/mtyddbaOMkgK/Lc+jnyhL7ovgLl5Dr7NTcAqk3VEq9odLoDdhQv9kovdH+Q62NuEVGISyK5tFydXPc/17uYni9Etox4wDD2HHQS7skpuU2KJR5g+qHlGfqu1s8C8nWYLAQveliDBXL8sgvz/L/cbkQixysGB1Xo2vD7OxOWP0LnzYq7dwMyKUx1Kb3TEOH3kO7Vjc6Rn7QvGAWQl0rzo5d+2GNQpIElAPxsRAD04hgmOIYGPGp1K1E0AiTQ+zRgRI5DqgYpvnjJlFDaUbIDWP1cUC5thyR4kCISeAoz6zp4rQtU55lGyTgPh6mjEcJUE6mQG7iIwhR9lKiEYspAX+iBExiprgiMsaqpaDeU101wTxd8ryw2+2Ym50nrzfr28+Ldqxo97T7WlMztUfatvZK29H6Gql8qHyqfKl8rX6sfq5+q36fui4vFTF3tZlT/fkbAV4PNA==
Transformer
alsoThey have ____ which ____ .cx,�t
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
ice ____ ____ cream
is ____ ____ really ____
____ goodax,�t
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
They also have ice cream which is really good .
1 2 3 4 5 6 7 8 9 10x =
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
�1:t = (6, 2, 1, 3, 10)
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
(6, 2, 1, 3, 10)
AAAB+XicbVDLSgMxFL1TX7W+Rl26CRahQhlm2lrtrujGZQX7gHYomTRtQzMPkkyhDP0TNy4UceufuPNvTF+g1gP3cjjnXnJzvIgzqWz7y0htbG5t76R3M3v7B4dH5vFJQ4axILROQh6Klocl5SygdcUUp61IUOx7nDa90d3Mb46pkCwMHtUkoq6PBwHrM4KVlrqmmSvnUSGPnDwq6m5fds2sbdlzINsqF68KlRJyVsqKZGGJWtf87PRCEvs0UIRjKduOHSk3wUIxwuk004kljTAZ4QFtaxpgn0o3mV8+RRda6aF+KHQFCs3VnxsJ9qWc+J6e9LEayr/eTPzPa8eqf+MmLIhiRQOyeKgfc6RCNIsB9ZigRPGJJpgIpm9FZIgFJkqHldEhrH15nTQKllOyKg+lbPV2GUcazuAccuDANVThHmpQBwJjeIIXeDUS49l4M94XoyljuXMKv2B8fAMl45Aj
AAAB63icbVBNSwMxEJ2tX7V+VT16CRbBU8mKoheh6MVjBWsL7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMCxPBjcX42yutrK6tb5Q3K1vbO7t71f2DRxOnmrIWjUWsOyExTHDFWpZbwTqJZkSGgrXD8W3ut5+YNjxWD3aSsECSoeIRp8Tmkrr2cb9aw3U8A1omfkFqUKDZr371BjFNJVOWCmJM18eJDTKiLaeCTSu91LCE0DEZsq6jikhmgmx26xSdOGWAoli7UhbN1N8TGZHGTGToOiWxI7Po5eJ/Xje10VWQcZWklik6XxSlAtkY5Y+jAdeMWjFxhFDN3a2Ijogm1Lp4Ki4Ef/HlZfJ4Vvcv6vj+vNa4KeIowxEcwyn4cAkNuIMmtIDCCJ7hFd486b14797HvLXkFTOH8Afe5w84oI2y
n = 10
1. Uniformly sample from to
2. Uniformly sample
3. Construct canvas
4. Compute estimated loss
cx,�t
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
t
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
0
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
n� 1
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
�1:t
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
� log(n!)� nn� t
X
�t+1
log p(ax,�t |cx,�t ; ✓)
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
n/2 action losses per pass :)
BLM - Inference
Simple greedy decoding or beam search to fill in the blanks in input
not search for sentence with the maximum marginal likelihood ,
but for sentence and trajectory that have the maximum joint likelihood
p(x; ✓)
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
p(x,�; ✓)
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
x
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
x
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
�
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
Note:
Experiments
Input: They also have which . Output: They also have ice cream which is really good .
Input: τε εγγονον εισαι???????σοφιαι Output: τε εγγονον εισαιου του σοφιαι
Input: The employees were super nice and efficient ! Output: The employees were rude and unprofessional !
Output: They also have ice cream which is really good.
Text infilling
Ancient text restoration
Sentiment transfer
Language modeling
Text Infilling
Dataset: Yahoo Answers (100K documents, max length 200 words)
Test data: randomly mask tokens with ratio , contiguous masked tokens —> “ ”
Metrics:
• Accuracy:
• Fluency:
r
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
BLEU score against original document
perplexity evaluated by a pre-trained left-to-right LM
Text Infilling
Baselines:
• BERT+LM
- use BERT representation of each blank as seed for a left-to-right LM that learns to
generate tokens in the corresponding blank
- at test time, multiple blanks are filled in one after another
Text Infilling
Baselines:
• BERT+LM
• Masked Language Model (MLM) with oracle length
- replace blanks with the target number of ⟨mask⟩ tokens
- fill masks autoregressively by most-confident-first heuristic
Text Infilling
Baselines:
• BERT+LM
• Masked Language Model (MLM) with oracle length
• Insertion Transformer (Stern et al., 2019)
- not allow users to specify where to insert
- force it to generate at valid locations, disable ⟨eos⟩ unless all blanks are filled,
prioritize slots that haven’t been filled yet; otherwise failure rate > 88%
Text Infilling
Baselines:
• BERT+LM
• Masked Language Model (MLM) with oracle length
• Insertion Transformer (Stern et al., 2019)
• Seq2seq-full (Donahue et al., 2020)
- output the full document from input
- may miss existing tokens in input or generate tokens in incorrect locations
• Seq2seq-fill (Donahue et al., 2020)
- output tokens to be placed in blanks, using ‘|’ to indicate separation
- may not generate the correct number of ‘|’
Text Infilling
Baselines:
• BERT+LM
• Masked Language Model (MLM) with oracle length
• Insertion Transformer (Stern et al., 2019)
• Seq2seq-full (Donahue et al., 2020)
- output the full document from input
- may miss existing tokens in input or generate tokens in incorrect locations
• Seq2seq-fill (Donahue et al., 2020)
- output tokens to be placed in blanks, using ‘|’ to indicate separation
- may not generate the correct number of ‘|’
failure rate from 15% to 47%
Text Infilling
BLEU
20
30
40
50
60
70
80
90
Mask ratio10% 20% 30% 40% 50%
BERT+LM MLM (oracle length) InsT BLM
Text Infilling
Perp
lexi
ty
0
10
20
30
40
50
60
Mask ratio10% 20% 30% 40% 50%
BERT+LM MLM (oracle length) InsT BLM
original data
Text Infilling
when time flies , where does it go ? to the center of the earth to be recycled came made into new time .
when time flies , where does it go ? from the center of the earth to be recycled converted made into new time .
when time flies , where does it go ? for the center of the earth has to be recycled and made into new time .
when time flies , where does it go ? for the center of the earth to be recycled and made into new time .
Original
Blanked
BERT+LM
MLM (oracle len)
InsT
BLM
when time flies , where does it go ? to the center of the universe to be recycled and made into new time .
when time flies , does it go ? the center of the to be recycled made into new time .
Mask ratio 10%
Text Infilling
when time flies , where does it go ? to the center of the universe to be recycled and made into new time .
when time , where ? the of universe to recycled made into .
when time is , where to ? i need to find the way of the universe to be recycled and made into a lot .
when time is , where is the universe ? from the creation of the universe to be recycled and made into the universe .
when time was created , where was it ? what was the name of the universe to be recycled and made into space .
when time was created , where did it come from ? it was the first part of the universe to be recycled and made into space .
Original
Blanked
BERT+LM
MLM (oracle len)
InsT
BLM
Mask ratio 50%
Ancient Text Restoration
Dataset (Assael et al., 2019): ancient Greek inscriptions (18M characters/3M words)
• number of characters to recover is assumed to be known
l 2 {0, · · · , t� 1}
AAAGgXicrVRbb9MwFM4uLSPcNnjkxWNEarWkTcrGbgqahEA8DsEuUt1Fjuum1hInih20KVjiX/C/eOPPIJy0Y+m6iY1hKdHx8Tnn+87F9pOQcmHbP2dm5+Zr9XsL9/UHDx89frK49PSAx1mKyT6Owzg98hEnIWVkX1ARkqMkJSjyQ3Lon7wtzg+/kJTTmH0WZwnpRShgdEAxEkrlLc1/N2AYByBpnO5AMSQCNYELShXkWeTlkNMgQpAy8MljEsAkjfteLlxbHufMcqTyRMf5qTmyk574iie24DysXgLBkAxEAw5ShHNH5pEcoVDXkccRQB6FKQ2GogneKBbXWJXslKmKGJBy12DLTbB64cCW5TX0S2WF/Tj5G6dQ6rzc2RZSN1RI3VBhdEOsqt9kEN2w/lCzAIO4H4vbogWkmt0U97+nezs8t4J2zjhCYuj74J1XEfNqGRTK9IGqh5RX6jvbrHApxqB7K3qjwegpltWW353l/+NyIxYl2Lh1TLXOEpO9u2D1L3wsUJm5CZA7Y6hJdx1bjbq7PnkP3MZrs2M65ivTsStXpIBw19rr7Y32ZntLN9QLM456+TrpoRpYmNtm2UtuCsuB0ltcsVt2ucC04IyFFW289rzFH7Af4ywiTOAQcd517ET0cpQKikMidZhxkiB8ggLSVSJDEeG9vHxBJTCUpg8Gcao+JkCprXrkKOL8LPKVZdFZfvmsUF511s3EYLOXU5ZkgjA8AhpkIRAxKJ5j0KcpwSI8UwLCKVVcAR4i1TehHm1dFcG5nPK0cNBpOWutrY9rK7vvv43KsaA9115oDc3RNrRd7YO2p+1reP5X7WXNqrXqc/Vm3a53RqazM+MSPtMmVn3nN0y9KlU=
t� 1� l
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
Length-aware BLM (L-BLM)
• [t] denotes blank of length t
• predict length of the new left blank,
new right blank has length
Baselines (Assael et al., 2019):
• Pythia: character-level seq2seq model to fill in one slot at a time
• Pythia-word: use both character and word representations
Ancient Text Restoration
Cha
ract
er e
rror r
ate
20
25
30
35
40
45
50
55
60
1% 25% 40% 50%
Human Pythia Pythia-word L-BLM
Single-slot Multi-slot
Mask ratio
Sentiment Transfer
Two-step approach:
1. Remove expressions of high polarity
- train a sentiment classifier and mask words with attention weight above average
2. Complete the partial sentence with expressions of the target sentiment
- train two instances of BLM, one for each sentiment
everyone that i spoke with was very helpful and kind .
everyone that i spoke with was rude and unprofessional .
there is definitely not enough room in that part of the venue .
there is always enough parking in that part of the venue .
it is n’t terrible , but it is n’t very good either .
it is n’t fancy , but it is still very good either .
Input
BLM
Input
BLM
Input
BLM
Sentiment Transfer
30
40
50
60
70
80
90
100
Accuracy
MLM (Wu et al. 2019) BLM
10
12
14
16
18
20
22
BLEU
Yelp reviews
Language Modeling
To compute , use Monte-Carlo estimate
• estimated perplexity is likely to be higher than actual perplexity
• as increases, it converges to actual perplexity
p(x; ✓) =X
�2Sn
p(x,�; ✓)
AAAGvXicrVRbb9MwFM7GWka4bfDIi8cUqdOSNSkdu6DAJATicQh2keo2OK6bmjlOFDuwKUTiN8IT/wYn7Vi6bmJjWEpycnx8vu9cfPyYUSFt+9fM7K25Wv32/B397r37Dx4uLD7aF1GaYLKHIxYlhz4ShFFO9iSVjBzGCUG
Top Related