Examples with importance weights Sometimes some examples are more important. Importance weights pop up in: boosting, differing train/test distributions, active learning, etc. John can reduce everything to importance weighted binary classification.
Principle Having an example with importance weight h should be equivalent to having the example h times in the dataset.
Learning with importance weights
y
Learning with importance weights
wt> x
y
Learning with importance weights
−η(∇`)> x
wt> x
y
Learning with importance weights
−η(∇`)> x
> wt> x wt+1 x
y
Learning with importance weights
−6η(∇`)> x
wt> x
y
Learning with importance weights
−6η(∇`)> x
wt> x
y
> wt+1 x ??
Learning with importance weights
−η(∇`)> x
wt> x
y
> wt+1 x
Learning with importance weights
> wt> x wt+1 x y
Learning with importance weights
s(h)||x||2 > wt> x wt+1 x y
What is s(·)? Losses for linear models `(w > x, y ). ∇w ` = Update must be given by
∂`(p,y ) ∂p x
wt+1 = wt − s(h)x s(h) must satisfy ∂`(p, y ) s(h + ) = s(h) + η ∂p p=(wt −s(h)x)> x ∂`(p, y ) 0 s (h) = η ∂p p=(wt −s(h)x)> x Finally s(0) = 0
Many loss functions Loss
`(p, y )
Squared
(y − p)2
Logistic
log(1 + e −yp )
Exponential
e −yp
Logarithmic
y log yp + (1 − y ) log 1−y 1−p
W (e
p−y x>x hηx > x+yp+e yp
)−hηx > x−e yp for y ∈ {−1, 1} yx > x py −log(hηx > x+e py ) for y ∈ {−1, 1} x > xy q p−1+ (p−1)2 +2hηx > x
if y = 0
q x>x p− p 2 +2hηx > x x>x p−1+ 1 (12hηx > x+8(1−p)3/2 )2/3 4 y =0 x>x p− 1 (12hηx > x+8p 3/2 )2/3 4 y =1 “ ” x>x −y min hη, 1−yp for y ∈ {−1, 1} x>x if y > p −τ min(hη, y −p ) τx>x p−y if y ≤ p (1 − τ ) min(hη, ) (1−τ )x > x
if y = 1
Hellinger
(
√
p−
√ √ √ 2 y ) − ( 1 − p − 1 − y )2
if if
Hinge τ -Quantile
max(0, 1 − yp) if y > p if y ≤ p
τ (y − p) (1 − τ )(p − y )
Update s(h) „ « > 1 − e −hηx x
Robust results for unweighted problems astro - logistic loss
And now something completely different Adaptive, individual learning rates in VW. It’s really GD separately on each coordinate i with ηt,i = r Pt
s=1
1
∂`(ws> xs ,ys ) ∂ws,i
2
Coordinate-wise scaling of the data less of an issue Can state this formally (Duchi, Hazan, and Singer / McMahan and Streeter, COLT 2010)
Some tricks involved Store sum of squared gradients w.r.t wi near wi . float InvSqrt(float x){ float xhalf = 0.5f * x; int i = *(int*)&x; i = 0x5f3759d5 - (i >> 1); x = *(float*)&i; x = x*(1.5f - xhalf*x*x); return x; } Special SSE rsqrt instruction is a little better
Experiments Raw Data ./vw --adaptive -b 24 --compressed -d tmp/spam_train.gz average loss = 0.02878 ./vw -b 24 --compressed -d tmp/spam_train.gz -l 100 average loss = 0.03267 TFIDF scaled data ./vw --adaptive -b 24 --compressed -d tmp/rcv1_train.gz -l 1 average loss = 0.04079 ./vw -b 24 --compressed -d tmp/rcv1_train.gz -l 256 average loss = 0.04465
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