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Learning to Identify Winning
Coalitions in the PAC Model
A. D. Procaccia & J. S. Rosenschein
Lecture Outline
• Cooperative Games
• Learning:
• PAC model
• VC dimension
• Motivation
• Results
• Closing Remarks
Simple Cooperative Games
• Cooperative n-person game =def (N;v).
N={1,…,n} is the set of players, v:2N→R.
Coalitions
• v(C) is the value of coalition C.
• Simple games: v is binary-valued. C is
winning if v(C)=1, losing if v(C)=0.
• 2N is partitioned into W and L, s.t.
1.  in L.
2. N in W.
3. Superset of winning coalition is winning.
PAC Model
• Sample space X; wish to learn target concept
c:X{0,1} in concept class C.
• Pairs (xi,c(xi)) given, according to a fixed
distribution on X.
• Produce concept but allow mistakes:
• Probability  that learning algorithm fails.
• -approximation of target concept.
• How many samples are needed? Sample
Complexity mC(,).
VC-Dimension
• X = sample space, C contains functions
c:X{0,1}.
• S={x1,…xm}, C(S) =def {(c(x1),...,c(xm)): c in C}
• S is shattered by C iff |C(S)|=2m.
• VC-dim(C) =def size of largest set shattered by
C.
• VC dimension yields upper and lower bounds
on sample complexity of concept class.
VC Dimension: Example
X = R, C={f: a,b s.t. f(x)=1 iff x is in [a,b]}
•
•
•
•
X = sample
space, C
contains
functions
c:X{0,1}.
S={x1,…xm},
C(S)={c(x1),.
..,c(xm): c in
C}
S is shattered
by C if
|C(S)|=2m.
VC-dim(C) =
size of largest
set shattered
by C.
Motivation
• Multiagent community shows interest in
learning, but almost all work is reinforcement
learning.
• Cooperative games are interesting in multiagent
context.
• Real world simple cooperative games settings:
• Parliament.
• Advisers.
Minimum Winning Coalitions
• Simple cooperative games defined by sets of
minimum winning coalitions.
• X = coalitions, C* = sets of minimum winning
coalitions.
{}
{1,2}
{1}
{2}
{3}
{4}
{1,3}
{1,4}
{2,3}
{2,4}
{1,2,3}
{1,2,4}
{1,3,4}
{2,3,4}
{1,2,3,4}
{3,4}
VC-dim(C*)
• Theorem:
 n 

VC  dim( C )  
n
/
2


*
{}
{1,2}
{1}
{2}
{3}
{4}
{1,3}
{1,4}
{2,3}
{2,4}
{1,2,3}
{1,2,4}
{1,3,4}
{2,3,4}
{3,4}
{1,2,3,4}
• F is antichain if A,B in F: AB.
• Sperner’s Theorem: F = antichain of subsets of {1,..,n}. Then
 n 

| F | 
 n / 2 
Restricted Simple Games
• Dictator:
• Single minimum winning coalition with one player.
• VC-dim = logn.
• Junta Coalition:
• Single minimum winning coalition.
• VC-dim = n.
Restricted Simple Games II
• Proper games:
• C is winning  N\C is losing.
• It holds that:
 n 1 

VC  dim  
 (n  1) / 2 
• Elimination of dummies:
• i C s.t. C is winning but C\{i} is losing.
• Same lower bound.
Closing Remarks
• Easy to learn simple games with dictator or
junta coalition; general games are much harder.
Algorithms
included!
• Monotone DNF formulae are equivalent to
minimum winning coalitions.
• Need to find implementation.