covariate shift correction - alex smolaalex.smola.org/drafts/covariate.pdf · covariate shift...
TRANSCRIPT
Covariate Shift Correction& Propensity Scores
Alex J. Smola
Monday, September 6, 2010
The Problem... aka two problems and one hammer ...
Monday, September 6, 2010
Covariate Shift• Basic setting
• Training data is drawn from• Test data is drawn from
• Examples• Training data from last week, deploy today • Training data for USA market, deploy in UK• Training data for mithril, deploy on axonite • Speech recogntion - adapt to speaker
• No labels on test set but
p(x)p(y|x)q(x)p(y|x)
p(y|x) = q(y|x)
Monday, September 6, 2010
Covariate Shift• Importance Sampler Identity
• Radon Nikodym derivative
(need measure theory to avoid ∞/∞ division)• Reweighted Empirical Risk
Ex∼q(x)[l(x, y, f(x))] = Ex∼p(x)
dq(x)
dp(x)l(x, y, f(x))
β(x) :=dq(x)
dp(x)
minimizef
i
β(xi)l(xi, yi, f(xi)) + λΩ[f ]
Monday, September 6, 2010
Propensity Scores• What if questions in experiments
• Display ad a for user u, what about a’• New feature for advertisers but uneven opt in• Efficacy of medical treatment
(stent vs. drugs for coronary artery problem)• More than 2 choices
• Customized module / page layout• Real-valued dosage level of drug
Monday, September 6, 2010
Propensity Scores• Basic goal - changing the conditioning
• Improvement estimation
This yields improvement when drawing from q. • Doubly robust estimation (variance reduction)
estimate f - we can evaluate the estimate on q
Eq[f(x)] = Ep[β(x)f(x)] −→1
m
m
i=1
β(xi)fi
Eq[f(x)− g(x)] = Ep[β(x)f(x)]−Eq[g(x)] −→1
m
m
i=1
β(xi)fi −1
n
n
i=1
gi
Eq[f(x)] = Eq[f(x)− f(x)] +Eq[f(x)] = Ep[β(x)[f(x)− f(x)]] +Eq[f(x)]
Monday, September 6, 2010
The goal
• Estimate the Radon Nikodym Derivative
based on samples from p and q
β(x) :=dq(x)
dp(x)
Monday, September 6, 2010
Logistic Regression... aka the idiot-proof simple method ...
Monday, September 6, 2010
Logistic Regression 101• Logistic transfer function
• Samples
• Risk minimization
p(y|x, f) = 1
1 + e−yf(x)
Z = (x1, y1), . . . , (xm, ym) where (xi, yi) ∼ p(y, x)
minimizef
m
i=1
log [1 + exp(−yif(xi))] + λΩ[f ]
Monday, September 6, 2010
Logistic to Radon Nikodym• Key idea
Generate artificial distribution from p and q
• Connection to Radon Nikodym
• Efficient optimization in primal spaceStochastic gradient descent in f (VW, Dios)
ρ(x, y) :=1
2δy,1p(x) +
1
2δy,−1q(x)
ρ(y|x) = 1
1 + e−yf(x)=⇒ β(x) =
ρ(−1|x)ρ(1|x) =
1 + ef(x)
1 + e−f(x)= ef(x)
f(x) = φ(x), θMonday, September 6, 2010
Moment Matching Theorem• Maximum Entropy Estimation (primal)
• Maximum a Posteriori Estimation (dual)
here g is the conditional log-partition function• Proof - analogous to Altun&Smola, 2006
via Fenchel Duality Theorem & operators
maximizep∈P
m
i
H(y|xi) subject to
m
i=1
φ(xi, yi)−Ey|xi[φ(xi, y)]
2
≤
minimizeθ
m
i=1
g(θ|xi)− φ(xi, yi), θ+λ
2θ2
Monday, September 6, 2010
Mean Operators... aka Fortet & Mourier 1946 revisited ...
Monday, September 6, 2010
Mean operators• Expectation map
• Empirical average
• Convergence theorem (Altun&Smola, 2006)
f → Ex∼p[f(x)] = Ex∼p[φ(x), θ] = f,Ex∼p[φ(x)] =: f, µ[p]
X → µ[X] :=1
m
m
i=1
φ(xi) hence f, µ[X] = 1
m
m
i=1
f(xi)
Pr µX − µ[p] > + ρ ≤ e−n2R−2
where ρ2 = n−1Ex,x∼p [k(x, x)− k(x, x)]
Monday, September 6, 2010
Mean operators• Key idea
• Have empirical mean operator for p and q• Find reweighted combination from X to X’
• By Cauchy-Schwartz this gives bound
minimizeβ
1
m
m
i=1
βiφ(xi)−1
m
m
i=1
φ(xi)
1
m
m
i=1
βif(xi)−1
m
m
i=1
f(xi)
≤ f
1
m
m
i=1
βiφ(xi)−1
m
m
i=1
φ(xi)
Monday, September 6, 2010
Guarantees• Radon Nikodym derivative is unique solution
when plugging in distributions.• For empirical averages approximation error is
small (upper bound by using RND).
where• We can find it by optimization
Pr
1
m
m
i=1
βiφ(xi)−1
m
m
i=1
φ(xi)
> + ρ
≤ exp
−m2
R2
1
m=
B2
m+
1
m and ρ ≤ R/√m
Monday, September 6, 2010
Optimization template• Constrained problem
• Quadratic penalty: Kernel Mean Matching• L infinity penalty: Bounded Mean Matching• Entropy penalty: Entropy Mean Matching
(Sugiyama, Bickel, Brefeld, Tsuboi, ...)
minimizeβ
Ω[β]
subject to
1
m
m
i=1
βiφ(xi)−1
m
m
i=1
φ(xi)
≤
Monday, September 6, 2010
Optimization Problems... applied duality theory ...
Monday, September 6, 2010
Quadratic Program• Quadratic penalty on RND
(this favors large effective sample size)
looks like a single-class SVM• Bounded range of RND
(this bounds variance in McDiarmid tail)
minimizeα
1
2α[K + λ1]α− αu subject to α1 = 1 and αi ≥ 0.
minimizeα
1
2αKα− αu subject to α1 = 1 and αi ∈ [0,λ]
Monday, September 6, 2010
Quadratic Program• Problem
Optimization problem is cubic in sample size• Solution
Find (ante)-primal problem and solve via SGD
where and via subdifferentials
minimizeθ,b
1
2θ2 + b+
1
2λ
n
i=1
(ui − φ(xi), θ − b)2+
minimizeθ,b
1
2θ2 + b+ λ
n
i=1
(ui − φ(xi), θ − b)+
ui =1
n
n
j=1
k(xi, xj) βi
Monday, September 6, 2010
Convex Program• Minimum KL-Divergence regularization dual
where• Problem
Computing the normalization g is expensive• Solutions
• MCMC sampler for gradient of g• Retain estimate of g (update parts frequently)
minimizeθ
g(θ)− θ, µ+ 1
2λθ2 with g(θ) = log
n
i=1
eφ(xi),θ
β(x) = eφ(x),θ−g(θ)
Monday, September 6, 2010
Conclusions... good/bad news ...
Monday, September 6, 2010
Experimental results
• All methods work well (much better than doing nothing)
• Online optimization is effective• Logistic regression works very well• Logistic regression works very well• Entropy regularization works best
(even though we have theory for the norms)(but not for entropy)
Monday, September 6, 2010