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COMS 4771
This homework is to be done alone. No late homeworks are allowed. To receive credit, a type-
setted copy of the homework pdf must be uploaded to Gradescope by the due date. You must show
your work to receive full credit. Discussing possible approaches for solutions for homework ques-
tions is encouraged on the course discussion board and with your peers, but you must write your
own individual solutions and not share your written work/code. You must cite all resources (includ-
ing online material, books, articles, help taken from/given to specific individuals, etc.) you used to
complete your work.
1 Combining multiple classifiers
The concept of “wisdom-of-the-crowd” posits that collective knowledge of a group as expressed
through their aggregated actions or opinions is superior to the decision of any one individual in the
group. Here we will study a version of the “wisdom-of-the-crowd” for binary classifiers: how can
one combine prediction outputs from multiple possibly low-quality binary classifiers to achieve an
aggregate high-quality final output? Consider the following iterative procedure to combine classifier
results.
Input:
- S – a set of training samples: S = {(x1, y1), . . . , (xm, ym)}, where each yi ∈ {−1,+1}
- T – number of iterations (also, number of classifiers to combine)
- F – a set of (possibly low-quality) classifiers. Each f ∈ F , is of the form f : X → {−1,+1}
Output:
- F – a set of selected classifiers {f1, . . . , fT }, where each fi ∈ F .
- A – a set of combination weights {α1, . . . , αT }
Iterative Combination Procedure:
- Initialize distribution weights D1(i) = 1m [for i = 1, . . . ,m]
- for t = 1, . . . , T do
- // ϵj is weighted error of j-th classifier w.r.t. Dt
- Define ϵj :=
∑m
i=1Dt(i) · 1[yi ̸= fj(xi)] [for each fj ∈ F]
- // select the classifier with the smallest (weighted) error
- ft = argminfj∈F ϵj
- ϵt = minfj∈F ϵj
- // recompute weights w.r.t. performance of ft
- Compute classifier weight αt = 12 ln
(
1−ϵt
ϵt
)
- Compute distribution weight Dt+1(i) = Dt(i) exp(−αtyift(xi))
1
- Normalize distribution weights Dt+1(i) =
Dt+1(i)∑
iDt+1(i)
- endfor
- return weights αt, and classifiers ft for t = 1, . . . , T .