[Zhang+ ACL2014] Kneser-Ney Smoothing on Expected Count [Pickhardt+ ACL2014] A Generalized Language Model as the

Comination of Skipped n-grams and Modified Kneser-Ney Smoothing

2014/7/12 ACL Reading @ PFI

Nakatani Shuyo, Cybozu Labs Inc.

Kneser-Ney Smoothing [Kneser+ 1995]

• Discounting & Interpolation

𝑃 𝑤𝑖 𝑤𝑖−𝑛+1𝑖−1

=max 𝑐 𝑤𝑖−𝑛+1

𝑖 − 𝐷, 0

𝑐 𝑤𝑖−𝑛+1𝑖−1

+𝐷

𝑐 𝑤𝑖−𝑛+1𝑖−1

𝑁1+ 𝑤𝑖−𝑛+1𝑖−1 ∙ 𝑃 𝑤𝑖 𝑤𝑖−𝑛+2

𝑖−1

• where

𝑤𝑚𝑛 = 𝑤𝑚 ⋯𝑤𝑛, 𝑁1+ 𝑤𝑚

𝑛 ⋅ = 𝑤𝑖|𝑐 𝑤𝑚𝑛𝑤𝑖 > 0

Number of Discounting

Modified KN-Smoothing [Chen+ 1999]

𝑃 𝑤𝑖 𝑤𝑖−𝑛+1𝑖−1

=𝑐 𝑤𝑖−𝑛+1

𝑖 − 𝐷 𝑤𝑖−𝑛+1𝑖

𝑐 𝑤𝑖−𝑛+1𝑖−1

+ 𝛾 𝑤𝑖−𝑛+1𝑖−1 𝑃 𝑤𝑖 𝑤𝑖−𝑛+2

𝑖−1

• where 𝐷 𝑐 = 0 if 𝑐 = 0, 𝐷1 if 𝑐 = 1, 𝐷2 if 𝑐 = 2, _ 𝐷3+ if 𝑐 ≥ 3

𝛾 𝑤𝑖−𝑛+1𝑖−1 =

[amount of discounting]

𝑐 𝑤𝑖−𝑛+1𝑖−1

Weighted Discounting (D_n are estimated by leave-1-out CV)

[Zhang+ ACL2014] Kneser-Ney Smoothing on Expected Count

• When each sentence has fractional

weight

– Domain adaptation

– EM-algorithm on word alignment

• Propose KN-smoothing using expected

fractional counts

I’m interested in it!

Model

• 𝒖 means 𝑤𝑖−𝑛+1𝑖−1 , and 𝒖′ means 𝑤𝑖−𝑛+2

𝑖−1

• A sequence 𝒖𝑤 occurs 𝑘 times and each

occurring has probability 𝑝𝑖 (𝑖 = 1,⋯ , 𝑘) as weight,

• then count 𝑐(𝒖𝑤) is distributed according to Poisson Binomial Distribution.

• 𝑝 𝑐 𝑢𝑤 = 𝑟 = 𝑠 𝑘, 𝑟 , where

𝑠 𝑘, 𝑟 =

𝑠 𝑘 − 1, 𝑟 1 − 𝑝𝑘

+ 𝑠 𝑘 − 1, 𝑟 − 1 𝑝𝑘

if 0 ≤ 𝑟 ≤ 𝑘1 if 𝑘 = 𝑟 = 00 otherwise

MLE on this model

• Expectations

– 𝔼 𝑐 𝒖𝑤 = 𝑟 ⋅ 𝑝 𝑐 𝒖𝑤 = 𝑟𝑟

– 𝔼 𝑁𝑟 𝒖 ⋅ = 𝑝 𝑐 𝒖𝑤 = 𝑟𝑤

– 𝔼 𝑁𝑟+ 𝒖 ⋅ = 𝑝 𝑐 𝒖𝑤 ≥ 𝑟𝑤

• Maximize (expected) likelihood

– 𝔼 𝐿 = 𝔼 𝑐 𝒖𝑤 log 𝑝 𝑤 𝒖𝒖𝑤

= 𝔼 𝑐 𝒖𝑤 log 𝑝 𝑤 𝒖𝒖𝑤

– obtain 𝑝MLE 𝑤 𝒖 =𝔼 𝑐 𝒖𝑤

𝔼 𝑐 𝒖⋅

Expected Kneser-Ney

• 𝑐 𝒖𝑤 =

max 0, 𝑐 𝒖𝑤 − 𝐷 + 𝑁1+ 𝒖 ⋅ 𝐷𝑝′(𝑤|𝒖′)

• So, 𝔼 𝑐 𝒖𝑤 = 𝔼 𝑐 𝒖𝑤 − 𝑝 𝑐 𝒖𝑤 > 0 𝐷 +

𝔼 𝑁1+ 𝒖 ⋅ 𝐷𝑝′(𝑤|𝒖′)

– where 𝑝′ 𝑤 𝒖′ = 𝔼 𝑁1+ ⋅𝒖′𝑤

𝔼 𝑁1+ ⋅𝒖′⋅

• then 𝑝 𝑤 𝒖 =𝔼 𝑐 𝒖𝑤

𝔼 𝑐 𝒖⋅

Language model adaptation

• Our corpus consists on

– large general-domain data and

– small specific domain data

• Sentence 𝒘 ‘s weight:

– 𝑝 𝒘 is in − domain =1

1+exp −𝐻 𝒘

– where 𝐻 𝒘 =log 𝑝in 𝒘 −log 𝑝out 𝒘

𝒘,

– 𝑝in:lang. model of in-domain, 𝑝out: out’s one

• Figure 1: On the language model adaptation task, expected KN outperforms all other methods across all sizes of selected subsets. Integral KN is applied to unweighted instances, while fractional WB, fractional KN and expected KN are applied to weighted instances. (via [Zhang+ ACL2014])

from general-domain data

in-domain data - training: 54k - testing: 3k

192

162

156

148

Why isn't there Modified KN as a

baseline?

[Pickhardt+ ACL2014] A Generalized Language Model as the Comination of Skipped n-grams and Modified Kneser-Ney Smoothing

• Higher-order n-grams are very sparse

– Especially remarkable on small data(e.g.

domain specific data!)

• Improve performance for small data

by skipped n-grams and Modified KN-

smoothing

– Perplexity reduces 25.7% for very small

training data of only 736KB text

“Generalized Language Models”

• 𝜕3𝑤1𝑤2𝑤3𝑤4 = 𝑤1𝑤2_𝑤4

– “_” means a word placeholder

𝑃GLM 𝑤𝑖 𝑤𝑖−𝑛+1𝑖−1 =

𝑐 𝑤𝑖−𝑛+1𝑖 − 𝐷 𝑐 𝑤𝑖−𝑛+1

𝑖

𝑐 𝑤𝑖−𝑛+1𝑖−1

+𝛾high 𝑤𝑖−𝑛+1𝑖−1

1

𝑛 − 1𝑃 GLM

𝑛−1

𝑗=1

𝑤𝑖 𝜕𝑗𝑤𝑖−𝑛+1𝑖−1

𝑃 GLM 𝑤𝑖 𝜕𝑗𝑤𝑖−𝑛+1𝑖−1 =

𝑁1+ 𝜕𝑗𝑤𝑖−𝑛𝑖 − 𝐷 𝑐 𝜕𝑗𝑤𝑖−𝑛+1

𝑖

𝑁1+ 𝜕𝑗𝑤𝑖−𝑛+1𝑖−1 ∗

+𝛾mid 𝜕𝑗𝑤𝑖−𝑛+1𝑖−1

1

𝑛 − 2𝑃 GLM 𝑤𝑖 𝜕𝑗𝜕𝑘𝑤𝑖−𝑛+1

𝑖−1

𝑛−1

𝑘=1,𝑘≠𝑗

• The bold arrows correspond to interpolation of models in traditional modified Kneser-Ney smoothing. The lighter arrows illustrate the additional interpolations introduced by our generalized language models. (via [Pickhardt+ ACL2014])

• shrunk training data sets for the English Wikipedia

small domain specific data

Space Complexity

model size = 9.5GB # of entries = 427M

model size = 15GB # of entries = 742M

References

• [Zhang+ ACL2014] Kneser-Ney Smoothing on Expected Count

• [Pickhardt+ ACL2014] A Generalized Language Model as the Comination of Skipped n-grams and Modified Kneser-Ney Smoothing

• [Kneser+ 1995] Improved backing-off for m-gram language modeling

• [Chen+ 1999] An Empirical Study of Smoothing Techniques for Language Modeling