An O(log(n))-Approximation for Decision Trees Brent Heeringa [email protected] (joint work with...
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Transcript of An O(log(n))-Approximation for Decision Trees Brent Heeringa [email protected] (joint work with...
An O(log(n))-Approximation for Decision Trees
Brent Heeringa
[email protected](joint work with Micah Adler)
11 March 2005
Question:• I am thinking of a Williams College CS faculty
member. Which one?
• Rule: Ask YES/NO questions from a finite set Q
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Q1: Is the professor female?
Q1: Is the professor female? YES
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Q1YES NO
Q1: Is the professor female? YES
Q2: Does the professor drive an old Volvo?
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Q1YES NO
Q1: Is the professor female? YES
Q2: Does the professor drive an old Volvo? NO
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Q1YES NO
Q2YES
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Q1: Is the professor female? YES
Q2: Does the professor drive an old Volvo? NO
Q3: Ph.D. from CMU?
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Q1: Is the professor female? YES
Q2: Does the professor drive an old Volvo? NO
Q3: Ph.D. from CMU? YES
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Professor Barbara Lerner
Modeling the Professor Game
• Q = Set of m YES/NO questions• X = Set of possible professors = {Bailey, Bruce, …, Wyman}
= {10…1, 11…1, …, 01…00}
• Goal: Minimize average number of questions– Solution: Build a binary tree– Each professor is a leaf
m m m
profs are bits strings where bit k answers question k
Decision Tree Problem (DT)• Input: A set X=(x1,…,xn) of m-bit binary
strings (called items)
• Solution: A binary tree with n leaves– Each internal node is a bit k
• partitions items into two groups
– Each item is a leaf
• Cost: Total Sum of Leaf Depths
• Optimal Solution: DT with minimum cost
k0 1
Example:11110 10111 11010 01101
30 1
11010
01101
0
1
1
2
0
10111
1
11110
1
2
3 3
Cost: 1 + 2 + 3 + 3 = 9
Example:11110 10111 11010 01101
50 1
11010 01101
0 11
1011111110
2
Cost: 2 + 2 + 2 + 2 = 8
OPTIMAL!0 13
2 2 2
Decision Trees• Decision Trees (DT) model many natural tasks in
– 20 Questions– Medical Diagnosis– Compiler optimizations
• DT is NP-Complete– No known polynomial-time algorithm (intractable)– As hard as Traveling Salesperson, 3SAT, etc.
• How do we deal with intractability?
Approximation AlgorithmsOptimization Problems (minimization)
Goal: minimize some costExamples: Traveling Salesperson, Max-3SAT
-approximation:
C = cost given by approximation algorithmCopt = cost of optimal solution
log(n)-approximation
• X be an instance of an optimization problemoptimal solution cost Copt
Approximation gives a solution w cost at most:
In DT, on input of size n, we know
So log of both sides is:
Outline
• Problem Introduction
• A Greedy Approximation Algorithm for DT
• An Analysis of the Greedy Algorithm– O(log n)-approximation
• Clearing up the historical confusion
A Greedy DT Algorithm
01101 10001 11101 11110 10111 11010
?
IDEA: Always choose bit which most evenly partitions items
0 1
A Greedy DT Algorithm
01101 10001 11101 11110 10111 11010
4
IDEA: Always choose bit which most evenly partitions items
0 1
01101 10001 11101 11110 10111 11010
A Greedy DT Algorithm
4
IDEA: Always choose bit which most evenly partitions items
0 1
11101
11110 10111 11010
1
01101 2
10001
01101 10001 11101 11110 10111 11010
A Greedy DT Algorithm
4
IDEA: Always choose bit which most evenly partitions items
0 1
1110111010
1
01101 2
10001
01101 10001 11101 11110 10111 11010
10111
2
11110
3
A Greedy DT AlgorithmIDEA: Always choose bit which
most evenly partitions items
GREEDY-DT(X)If X=Ø
Return NILElse
Let k be the bit most evenly separating XLet T be a tree nodeT[left] GREEDY-DT({X | X(k)=0})T[right] GREEDY-DT({X | X(k)=1})Return T
Optimal vs. Greedy
a
b c
h
e b
c df g
a
Optimal Tree T* Greedy Tree T
Cost(T)=26Cost(T*)=25
d eh
f g
Outline
• Problem Introduction
• A Greedy Approximation Algorithm for DT
• An Analysis of the Greedy Algorithm– O(log n)-approximation
• Clearing up the historical confusion
Analysis Outline• Accounting Scheme
– Each pair of items {xi, xj} is separated exactly once in any decision tree
• Analyze the cost of greedy tree with respect to the structure of the optimal tree
Theorem: The greedy algorithm has cost at most a factor of O(log n) greater than the optimal tree
Accounting Method
• Divide size of each interior node Sij equally among the pairs of items {xi,xj} split at Sij
xi
Sij
Sij-
Node Sij separates xi from xj
Sij+
xj
|Sij+| ≥ |Sij
-|
Greedy Tree T
# pairs:
size of Sij:
Pair cost (cij):
Accounting Method
pairs of items
Claim:
Proof:
1.
2.
3.
xi
Sij
Sij-
Node Sij separates xi from xj
Sij+
xj
|Sij+| ≥ |Sij
-|
Greedy Tree T
Accounting MethodAccounting Method
01101 10001 11101 11110 10111
40 1
01101 10001 11101 11110 10111
x1 x2 x3 x4 x5
Sij-
Sij+
Cost(x1, x4) = c14 = 2/2 = 1
Sij
Sij =
Accounting Method
01101 10001 11101 11110 10111
40 1
01101 10001 11101 11110 10111
x1 x2 x3 x4 x5
Sij-
Sij+
Cost(x1, x4) = c14 = 2/2 = 1
Cost(x3, x4) = c34 = 2/2 = 1
Sij
Sij =
cij = 1
= 2 x 3 = 6 pairs of items
6 x cij = 6 x 1 = 6 ≥ |Sij| = 5
Bound holds:
Accounting Method
40 1
01101 10001 11101 11110 10111
Sij-
Sij+
Sij
cij = 2/3
= 3 x 3 = 9 pairs of items
9 x cij = 9 x (2/3) = 6 = |Sij| = 6
Bound holds:
Accounting Method
40 1
01101 10001 11101 11110 10111 00111
Sij-
Sij+
Sij
Accounting Method
Bound holds for all nodes Sij in T
Cost C of greedy tree T is the sum of each node size
Accounting Method
Bound holds for all nodes Sij in T
Any ordering of cij is acceptable.
Sum of all cij
Cost C of greedy tree T is the sum of each node size
h
e b
c df g
a
Optimal Tree T* Greedy Tree T
a
b c d eh
f g
f,d f,e g,d g,e h,d h,e
2/4 2/1 2/4 2/1cij:
h
e b
c df g
a
Optimal Tree T* Greedy Tree T
a
b c d eh
f g
f,d f,e g,d g,e h,d h,e
2/4 2/1 2/4 2/1 2/4cij:
h
e b
c df g
a
Optimal Tree T* Greedy Tree T
a
b c d eh
f g
f,d f,e g,d g,e h,d h,e
2/4 2/1 2/4 2/1 2/4 2/1cij:
h
e b
c df g
a
Optimal Tree T* Greedy Tree T
a
b c d eh
f g
f,d f,e g,d g,e h,d h,e
2/4 2/1 2/4 2/1 2/4 2/1cij: + + + + + = 7.5
h
e b
c df g
a
Optimal Tree T* Greedy Tree T
a
b c d eh
f g
f,d f,e g,d g,e h,d h,e
2/4 2/1 2/4 2/1 2/4 2/1+ + + + + = 7.5
7.5 ≥ | | = 5
cij:
h
e b
c df g
a
Optimal Tree T* Greedy Tree T
a
b c d eh
f g
f,d f,e g,d g,e h,d h,e
2/4 2/1 2/4 2/1 2/4 2/1+ + + + + = 7.5
log(n)| | ≥ 7.5 ≥ | | = 5
cij:
Lemma: For any node Zij in T*
• Proof Intuition:– cij help capture trade-offs between a good move locally and a good
move globally– For any {xi,xj} the split at Sij is better than the split at Zij for the
pairs in common to both nodes– Enough wiggle room to show that a greedy split can’t lose too
much ground to an optimal split– For every step taken by the optimal algorithm, the greedy
algorithm takes at most log(n) steps
Reorder Cij according to T*
T*
xi
Zij
Zij-Zij
+
xj
C* = Cost of optimal tree T*
Lemma: For any node Zij in T*
OPTIMAL
Outline
• Problem Introduction
• A Greedy Approximation Algorithm for DT
• An Analysis of the Greedy Algorithm– O(log n)-approximation
• Clearing up the historical confusion
The ConDT Problem:• Input: A set X=(x1,…,xn) of m-bit binary strings (called items)
– Each item xi has a label TRUE or FALSE
• Solution: A binary tree– Each internal node is a bit k– Each leaf is a label– The tree correctly labels each item
• (consistent decision tree)
• Cost: Total number of leaves• Optimal Solution: Consistent decision tree with minimum
number of leaves
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F F F FF F FTT
Each professor has a TRUE / FALSE label
Label answers some “hidden” question
Approximate “hidden” question with YES/NO questions
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F F F FF F FTT
Q1: Is the professor female?
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F F F FF F FTT
Q1: Is the professor female?
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YES NO
FALSE
Q1
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F F F FF F FTT
Q1: Is the professor female?
Q2: Ph.D. in mathematics?
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YES NO
FALSE
Q1
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F F F FF F FTT
Q1: Is the professor female?
Q2: Ph.D. in mathematics?
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YES NO
FALSE
Q1
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NOYES
FALSE TRUE
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F F F FF F FTT
Hidden Question:
Does the professor have a beard?
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YES NO
FALSE
Q1
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NOYES
FALSE TRUE
Clarification
• NP-Complete• No polytime log(n)-approximation
– modulo unlikely complexity results
• Many consider DT and ConDT equivalent• log(n)-approximation proves otherwise
– DT and ConDT are fundamentally different problems
• log(n)-approximation is also first non-trivial upperbound on the aprox. ratio for DT
Mind the Gap!
• Lower bound on approximation for DT:– No PTAS– No -approximation for some constant
• Close the gap between upper bound and lower bound– No log(n)-approximation for some