The Zero Problem
LM Smoothing (The EM Algorithm)
PA154 Jazykové modelování (3) Pavel Rychlý
pary@fi.muni.cz
March 9, 2017
Source: Introduction to Natural Language Processing (( Jan Hajic, CS Dept., Johns Hopkins Univ. www.cs.jhu.edu/~hajic
"Raw" n-gram language model estimate:
- necessarily, some zeros
► !many: trigram model -> 2.16 x 1014 parameters, data ~109 words
- which are true 0?
► optimal situation: even the least grequent trigram would be seen several times, in order to distinguish it's probability vs. other trigrams
► optimal situation cannot happen, unfortunately (open question: how many data would we need?)
—> we don't know
- we must eliminate zeros
Two kinds of zeros: p(w|h) = 0, or even p(h) = 0!
PA154 Jazykové modelování (3)
Why do we need Nonzero Probs?
Eliminating the Zero Probabilites: Smoothing
To avoid infinite Cross Entropy:
- happens when an event is found in test data which has not been seen in training data
H(p) = oo: prevents comparing data with > 0 "errors"
To make the system more robust
- low count estimates:
► they typically happen for "detailed" but relatively rare appearances
- high count estimates: reliable but less "detailed"
Get new p'(w) (same Q): almost p(w) but no zeros Discount w for (some) p(w) > 0: new p'(w) < p(w)
Ewediscounted(P(w)-P'(w)) = D
Distribute D to all w; p(w) = 0: new p'(w) > p(w) - possibly also to other w with low p(w) For some w (possibly): p'(w) = p(w) Make sure EwenP'(w) = 1 There are many ways of smoothing
PA154 Jazykové modelování (3)
PA154 Jazykové modelování (3)
Smoothing by Adding 1
■ Simplest but not really usable:
Predicting words w from a vocabulary V, training data T:
p/(w\h) -
c(h, w) +1 c(h) + \V\
► for non-conditional distributions: p'(w) -Problem if \V\ > c(h) (as is often the case; even >> c(h)\) ■ Example:
Training data: <s> what is it what is small?   |T| =
V = {what, is, it, small, ?,<s> ,flying, birds, are, a, bird, .}, |V| = 12 p(it) = .125, p(what) = .25, p(.)=0   p(what is it?) = .252 x .1252 s .001 p(it is flying.) = .125X.25 x 02 = 0 p'fwhat is it?) = .15x.12 s .0002
p'(it) = .1, p'(what) = .15, p'(.) = .05
p'(it is flying.) = .1 x .15 x .052 = .00 004
Adding less than 1
■ Equally simple:
Predicting word w from a vocabulary V, training data T:
pf(w\h) =
c{h, w) + A c(h) + \\V\'
A < 1
for non-conditional distributions: p'(w) = ^",)+A
Example:
\\v\
Training data: <s> what is it what is small?   |T| = 8
V = {what, is, it, small, ?,<s> ,flying, birds, are, a, bird,.}, |V| = 12 p(it) = .125, p(what) = .25, p(.)=0   p(what is it?) = .252 x .1252 ~ .001 p(it is flying.) = .125X.25 x O2 = 0
Use A = .1
p'(it) = .12, p'(what) = .23, p'(what is it?) = .232 x .122 s .0007
p'(.) = .01
p'(it is flying.) = .12x.23 x .012 Si .000003
PA154 Jazykové modelování (3)
PA154 Jazykové modelování (3)
LM Smoothing
Good-Turing
Good-Turing: An Example
Suitable for estimation from large data
- similar idea: discount/boost the relative frequency estimate:
Pr(w) =
(c(iv) + 1) x A/(c(iv) + 1)
7~| x N(c(w))
where W(c) is the count of words with count c (count-of-counts) specifically, for c(w) = 0 (unseen words), pr(w)
- good for small counts (< 5-10, where W(c) is high)
- normalization! (so that we have ^ivP'M = 1)
*N(0)
Remember: P,W ='«XtoT'
Training data: <s> what is it what is small?   |T| =
V = {what, is, it, small, ?,<s> ,flying, birds, are, a, bird, .}, |V| = 12 p(it) = .125, p(what) = .25, p(.)=0   p(what is it?) = .252 x .1252 s .001 p(it is flying.) = .125X.25 x O2 = 0
Raw estimation (W(0) = 6, W(1) = 4, W(2) = 2, W(/) = 0, for / > 2): Pr(it) = (1+1)xN(1+1)/(8xN(1)) = 2x2/(8x4) = .125 Pr(what) = (2+1)xN(2+1)/(8xN(2)) = 3x0/(8x2) = 0:
keep orig. p(what) pr(.) = (0+1)xN(0+1)/(8xN(0)) = 1 x4/(8x6) = .083
Normalize (divide by 1.5 = Y<we\v\ Pr(w)) ancl compute: p'(it) ^ .08, p'(what) ^ .17, p'(.) = .06 p'jwhat is it?) = .172 x .082 = .0002 p'(it is flying.) = .082 x .17 x .062 = .00004
PA154 Jazykové modelování (3)
PA154 Jazykové modelování (3)
Smoothing by Combination: Linear Interpolation
Typical n-gram LM Smoothing
Combine what?
► distribution of various level of detail vs. reliability n-gram models:
► use (n-1)gram, (n-2)gram.....uniform
—> reliability <— detail
Simplest possible combination: - sum of probabilities, normalize:
► p(0|0) = .8,p(1|0) = .2,p(0|1) = 1,p(1|1) = 0, p(0) = .4, p(1) = .6
► p'(0|0) = .6,p'(1|0) = .4,p'(1|0) = .7,p'(1|1) = .3
Weight in less detailed distributions using A = (A0, Ai, A2, A3):
p'^(lV/|lV/_2, = A3P3(W,'|W,_2, +
X2Pz(Wi\Wi-i) + A1p1(iv,-) + A0/| V\
Normalize:
xi > 0, E"=o xi = 1 is sufficient (A0 = 1 - Eti A<)(n = 3) Estimation using MLE:
-fjx the P3,p2,pi and |V| parameters as estimated from the training data
- then find such {A,} which minimizes the cross entropy (maximazes probablity of data): -4t Y!,\=\ '^(PaW1*))
PA154 Jazykové modelování (3)
PA154 Jazykové modelování (3)
Held-out Data
What data to use?
- try training data T: but we will always get A3 = 1
► why? let Pit be an i-gram distribution estimated using r.f. from T)
► minimizing HT(p\) over a vector A, p\ =
A3P3r + A2p2r + A,plr + A0/| V\
- remember HT(p\) = H(p3r) + D(P3t||p'a); P3rfixed -> H(p3r) fixed, best)
-which p\ minimizes HT(p\)? Obviously, a p\ for which D(p3r||p'A) = 0
- ...and that's p3r (because D(p||p) = 0, as we know)
- ...and certainly p\ = p3rifA3 = 1 (maybe in some other cases, too).
- (p'A = 1 x p3r + 0 x p2T + 1 x p, T + 0/|V|)
- thus: do not use the training data for estimation of A!
► must hold out part of the training data (heldout data, H)
► ...call remaining data the (true/raw) training data, T
► the test data S (e.g., for comparison purposes): still different data!
PA154 Jazykové modelování (3)
The Formulas
-1 IHI
Repeat: minimizing — ^j.J, logz{^x{Wi\hi)) over A
ptx(Wi\hi) = f/x(Wi\Wi_2, Wi_i) =
= hP3{Wi\Wi-2, W,-i) + \2P2(Wi\Wi-i) + hP^W-i) + A0^
"Expected counts of lambdas": j = 0..3
PA154 Jazykové modelování (3)
The (Smoothing) EM Algorithm
Remark on Linear Interpolation Smoothing
Start with some A, such that A > 0 for all j e 0..3
Compute "Expected Counts" for eachAy.
Compute new set of Ay, using "Next A" formula.
Start over at step 2, unless a termination condition is met.
Termination condition: convergence of A.
- Simply set an e, and finish if |Ay - Ay„ext| < e for each j (step 3).
Guaranteed to converge: follows from Jensen's inequality, plus a technical proof.
"Bucketed Smoothing":
- use several vectors of A instead of one, based on (the frequency of) history: A(h)
► e.g. for h = (micrograms,per) we will have
A(h) = (.999, .0009, .00009, .00001) (because "cubic" is the only word to follow...)
- actually: not a separate set for each history, but rather a set for "similar" histories ("bucket")
A(b(h)), where b: V2 -> W (in the case of trigrams) b classifies histories according to their reliability (-frequency)
PA154 Jazykové modelování (3)
PA154 Jazykové modelování (3)
Bucketed Smoothing: The Algorithm
First, determine the bucketing function b (use heldoutl):
- decide in advance you want e.g. 1000 buckets
- compute the total frequency of histories in 1 bucket (fmax(b))
- gradually fill your buckets from the most frequent bigrams so that the sum of frequencies does not exceed fmax(i>) (you might end up with slightly more than 1000 buckets)
Divide your heldout data according to buckets
Apply the previous algorithm to each bucket and its data
PA154 Jazykové modelování (3)
Simple Example
■ Raw distribution (unigram only; smooth with uniform):
p(a) = .25, p(b) = .5, p(a) = 1/64 for a e {c.r}, = 0 for the rest: s, t, u, v, w,
x, y, z
■ Heldout data: baby; use one set of A (A^ unigram, A0: uniform)
■ Start with A0 = Aj = .5:
p'x(b) = .5 x .5 + .5/26 = .27 p'A(a) = .5 x .25 + .5/26= .14 p'x(y) = .5x0 + .5/26 = .02
c(Xi) = .5x.5/.27 + .5x.25/.14+ .5x.5/.27+ .5x0/.02 = 2.27 c(A0) = .5x.04/.27 + .5x.04/.14+ .5x.04/.27 + .5x.04/.02 = 1.28 Normalize A1inext = .68, \0,next = -32
Repeat from step 2 (recomputep'a first for efficient computation, then c(A,), ...).
Finish when new lambdas almost equal to the old ones (say, < 0.01 difference).
PA154 Jazykové modelování (3)
Some More Technical Hints
Set V = {all words from training data}.
► You may also consider V = T u H, but it does not make the coding in any way simpler (in fact, harder).
► But: you must never use the test data for your vocabulary
Prepend two "words" in front of all data:
► avoids beginning-of-data problems
► call these index -1 and 0: then the formulas hold exactly When c„(w,h) = 0:
► Assing 0 probability to p„(w|h) where c„_i(h) > 0, but a uniform probablity (1/|V|) to those p„(w|h) where c„_i(h) = 0 (this must be done both when working on the heldout data during EM, as well as when computing cross-entropy on the test data!)
PA154 Jazykové modelování (3)