1. Learning Submodular Functions
Maria Florina Balcan
LGO, 11/16/2010
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2. Submodular functions V={1,2, …, n}; set-function f : 2V ! R
f(S)+f(T) ¸ f(S Å T) + f(S [ T), 8 S,Tµ V
Decreasing marginal return
f(T [ {x})-f(T)¸ f(S [ {x})-f(S), 8 S,Tµ V, T µ S, x not in T
Examples:
Vector Spaces Let V={v1,,vn}, each vi 2 Fn. For each S µ V, let f(S) = rank(V[S])
Concave Functions Let h : R ! R be concave. For each S µ V, let f(S) = h(|S|)

3. Submodular set functions
Set function f on V is called submodular if
For all S,T µ V: f(S)+f(T) ¸ f(S[T)+f(SÅT)
Equivalent diminishing returns characterization: + +
S [ T S T
SÅT S + x
Large improvement
Submodularity: T + x
Small improvement
For TµS, xS, f(T [ {x}) – f(T) ¸ f(S [ {x}) – f(S)

4. Example: Set cover
Want to cover floorplan with discs
Place sensors in building
Possible locations V
For S µ V: f(S) = “area (# locations) covered by sensors placed at S”
Node predicts
values of positions
with some radius
Formally:
W finite set, collection of n subsets Wi µ W
For S µ V={1,…,n} define f(S) = |i2 S Wi|

5. Set cover is submodular
T={W1,W2} W1 W2 x
f(T[{x})-f(T)
f(S[{x})-f(S) W1 W2 W3 x W4
S = {W1,W2,W3,W4}

6. Submodular functions V={1,2, …, n}; set-function f : 2V ! R
f(S)+f(T) ¸ f(S Å T) + f(S [ T), 8 S,Tµ V
Decreasing marginal return
f(T [ {x})-f(T)· f(S [ {x})-f(S), 8 S,Tµ V, S µ T, x not in T
Examples:
Vector Spaces Let V={v1,,vn}, each vi 2 Fn. For each S µ V, let f(S) = rank(V[S])
Concave Functions Let h : R ! R be concave. For each S µ V, let f(S) = h(|S|)

7. Submodular functions V={1,2, …, n}; set-function f : 2V ! R
f(S)+f(T) ¸ f(S Å T) + f(S [ T), 8 S,Tµ V
Monotone:
f(S) · f(T) , 8 S µ T
Non-negative:
f(S) ¸ 0, 8 S µ V

8. Submodular functions A lot of work on optimization and submodularity.
Can be minimized in polynomial time.
Algorithmic game theory
decreasing marginal utilities.
Substantial interest in the ML community recently.
Tutorials, workshops at ICML, NIPS, etc.
owned by ML.

9. Learnability of Submodular Fns
Important to also understand their learnability.
Previous Work:
Exact learning with value queries
Goemans, Harvey, Iwata, Mirrokni, SODA 2009
[GHIM’09] Model
There is an unknown submodular target function.
Algorithm allowed to (adaptively) pick sets and query the value of the target on those sets.
Can we learn the target with a polynomial number of queries in poly time?
Output a function that approximates the target within a factor of ® on every single subset.

10. Exact learning with value queries
Goemans, Harvey, Iwata, Mirrokni, SODA 2009
Theorem: (General upper bound)
9 an alg. for learning a submodular function with an approx. factor O(n1/2).
Theorem: (General lower bound)
Any alg. for learning a submodular must have an approx. factor of (n1/2).

11. Problems with the GHIM model
- Lower bound fails if our goal is to do well on most of the points.
- Many simple functions that are easy to learn in the PAC model (e.g., conjunctions) are impossible to get exactly from a poly number of queries  
- Well known that value queries are undesirable in some learning applications.
Is there a better model that gets around these problems?

12. Problems with the GHIM model
- Lower bound fails if our goal is to do well on most of the points.
- Many simple functions that are easy to learn in the PAC model (e.g., conjunctions) are impossible to get exactly from a poly number of queries  
- Well known that value queries are undesirable in some learning applications.
Learning submodular fns in a distributional learning setting [BH10]

13. Our model: Passive Supervised Learning
Data Source
Distribution D on {0,1}n
Expert / Oracle
Learning Algorithm
Labeled Examples
(x1,f(x1)),…, (xk,f(xk))
f : {0,1}n  R+
Alg.outputs
g : {0,1}n  R+

14. Our model: Passive Supervised Learning
Distribution D on {0,1}n
Labeled Examples
Learning Algorithm
Expert / Oracle
Data Source
Alg.outputs
f : {0,1}n  R+
g : {0,1}n  R+
(x1,f(x1)),…, (xk,f(xk))
Algorithm sees (x1,f(x1)),…, (xk,f(xk)), xi i.i.d. from D
Algorithm produces “hypothesis” g. (Hopefully g ¼ f)
Prx1,…,xm[ Prx[g(x) · f(x)· ® g(x)]¸ 1-²] ¸ 1-±
“Probably Mostly Approximately Correct”

15. Main results Theorem: (Our general upper bound)
9 an alg. for PMAC-learning the class of non-negative, monotone, submodular fns (w.r.t. an arbitrary distribution) with an approx. factor O(n1/2).
Note:
Much simpler alg. compared to GIHM’09
Theorem: (Our general lower bound)
No algorithm can PMAC learn the class of non-neg., monotone, submodular fns with an approx. factor õ(n1/3).
Note:
The GIHM’09 lower bound fails in our model.
Theorem: (Product distributions)
Matroid rank functions, const. approx.

16. A General Upper Bound Theorem:
9 an alg. for PMAC-learning the class of non-negative, monotone, submodular fns (w.r.t. an arbitrary distribution) with an approx. factor O(n1/2).

17. Subaddtive Fns are Approximately Linear
Let f be non-negative, monotone and subadditive
Claim: f can be approximated to within factor n by a linear function g.
Proof Sketch: Let g(S) = s in S f({s}). Then f(S) · g(S) · n ¢ f(S).
Subadditive: f(S)+f(T) ¸ f(S[ T) S,T µ V
Monotonicity: f(S) · f(T) Sµ T
Non-negativity: f(S) ¸ S µ V

18. Subaddtive Fns are Approximately Linear
f(S) · g(S) · n¢f(S).
n¢f g f V

19. PMAC Learning Subadditive Fns
f non-negative, monotone, subadditive approximated to within factor n by a linear function g,
g (S) =w ¢ Â (S).
Labeled examples ((Â(S), f(S) ), +) and ((Â(S), n¢f(S) ), -) are linearly separable in Rn+1.
Idea: learn a linear separator. Use std sample complex.
Problem: data not i.i.d.
Solution: create a related distribution.
w chi(S) – f(S) >0
w chi(S) – (n+1) f(S)<0
w chi(S) – z f(S) >0
W chi(S) – z(n+1) f(S) <0 ===
Sample S from D; flip a coin.
If heads add ((Â(S), f(S) ), +).
Else add ((Â(S), n¢f(S) ), -).

20. PMAC Learning Subadditive Fns
Algorithm:
Note:
Deal with the set {S:f(S)=0 } separately.
Input: (S1, f(S1)) …, (Sm, f(Sm))
For each Si, flip a coin.
If heads add ((Â(S), f(Si) ), +).
Else add ((Â(S), n¢f(Si) ), -).
Learn a linear separator u=(w,-z) in Rn+1.
Output: g(S)=1/(n+1) w ¢ Â (S).
Theorem: For m = £(n/²), g approximates f to within a factor n on a 1-² fraction of the distribution.

21. PMAC Learning Submodular Fns
Algorithm:
Note:
Deal with the set {S:f(S)=0 } separately.
Input: (S1, f(S1)) …, (Sm, f(Sm))
For each Si, flip a coin.
If heads add ((Â(S), f2(S_i)) ), +).
Else add ((Â(S), n f2(S_i) ), -).
Learn a linear separator u=(w,-z) in Rn+1.
Output: g(S)=1/(n+1)1/2 w ¢ Â (S)
Theorem: For m = £(n/²), g approximates f to within a factor \sqrt{n} on a 1-² fraction of the distribution.
Proof idea: f non-negative, monotone, submodular approximated to within factor \sqrt{n} by a \sqrt{linear function}.
[GHIM, 09]

22. PMAC Learning Submodular Fns
Algorithm:
Note:
Deal with the set {S:f(S)=0 } separately.
Input: (S1, f(S1)) …, (Sm, f(Sm))
For each Si, flip a coin.
If heads add ((Â(S), f2(S_i)) ), +).
Else add ((Â(S), n f2(S_i) ), -).
Learn a linear separator u=(w,-z) in Rn+1.
Output: g(S)=1/(n+1)1/2 w ¢ Â (S)
Much simpler than [GIHM09]. More robust to variations.
the target only needs to be within an ¯ factor of a submodular fnc.
9 a submodular fnc that agrees with target on all but a ´ fraction of the points (on the points it disagrees it can be arbitrarily far).
[the alg is inefficient in this case]

23. A General Lower Bound Theorem: (Our general lower bound)
No algorithm can PMAC learn the class of non-neg., monotone, submodular fns with an approx. factor õ(n1/3).
Plan:
Use the fact that any matroid rank fnc is submodular.
Construct a hard family of matroid rank functions.
High=n1/3 X X
L=nlog log n
Low=log2n X X A1 A2 A3
… … …. …. AL

24. Matroids (V,Ind) is a matroid:
Subsets of independent sets are independent.
If I,J are independent and |I| < |J| then I U {j} is independent for some j in J.
Rank(S)= max{|I|, I 2 Ind, I µ S}, for any S µ V
Examples:
Uniform Matroids
V={1,2, …, n},
Ind={I µ V: |I | · k}
Graphical Matroid:
Elements are the edges of an undirected graph G = (V;E)
Set of edges is independent if it does not contain a cycle

25. Ind={I: |I Å Aj| · uj, for all j }
Partition Matroids
A1, A2, …, Ak µ V={1,2, …, n}, all disjoint; ui · |Ai|-1
Ind={I: |I Å Aj| · uj, for all j }
Then (V, Ind) is a matroid.
If sets Ai are not disjoint, then (V,Ind) might not be a matroid.
E.g., n=5, A1={1,2,3}, A2={3,4,5}, u1=u2=2.
{1,2,4,5} and {2,3,4} both maximal sets in Ind; do not have the same cardinality.

26. Almost partition matroids
k=2, A1, A2 µ V (not necessarily disjoint); ui · |Ai|-1
Ind={I: |I Å Aj| · uj , |I Å (A1 [ A2)| · u1 +u2 - |A1 Å A2|}
Then (V,Ind) is a matroid.

27. Almost partition matroids
More generally
A1, A2, …, Ak µ V={1,2, …, n}, ui · |Ai|-1;
f : 2[k] ! Z
=<0
f(J)= j 2 J uj +|A(J)|-j 2 J|Aj|, 8 J µ [k]
Ind= { I: |I Å A(J)| · f(J), 8 J µ [k] }
Then (V, Ind) is a matroid (if nonempty).
Rewrite f, f(J)=|A(J)|-j 2 J(|Aj| - uj), 8 J µ [k]

28. A generalization of partition matroids
More generally
f : 2[k] ! Z
f(J)=|A(J)|-j 2 J(|Aj| - uj), 8 J µ [k]
Ind= { I: |I Å A(J)| · f(J), 8 J µ [k] }
Then (V, Ind) is a matroid (if nonempty).
Proof technique:
Uncrossing argument
For a set I, define T(I) to be the set of tight constraints
T(I)= {J µ [k], |I Å A(J)|=f(J)}
8 I 2 Ind, J1, J2 2 T(I), then (J1 [ J2 2 T(I)) or (J1 Å J2 =)
Ind is the family of independent sets of a matroid.

29. A generalization of almost partition matroids
f : 2[k] ! Z,
f(J)=|A(J)|-j 2 J(|Aj| -uj), 8 J µ [k]; ui · |Ai|-1
Note: This requires k· n (for k > n, f becomes negative)
But we want k=n^{log log n}.
Do some sort of truncation to allow k>>n.
f(J) is (¹, ¿) good if f(J) ¸ 0 for J µ [k], |J| · ¿ and
f(J) ¸ ¹ for J µ [k], ¿ ·|J| · 2¿ -2
h(J)=f(J) if |J| · ¿ and h(J)=¹, otherwise.
Ind= { I: |I Å A(J)| · h(J), 8 J µ [k] }
Then (V,Ind) is a matroid (if nonempty).

30. A generalization of partition matroids
Let L = nlog log n. Let A1, A2, …, AL be random subsets of V.
(Ai -- include each elem of V indep with prob n-2/3. )
Let ¹=n^{1/3} log2 n, u=log2 n, ¿=n1/3
Each subset J µ {1,2, …, L} induces a matroid s.t. for any i not in J, Ai is indep in this matroid
Rank(Ai), i not in J, is roughly |Ai| (i.e., £(n^{1/3})),
The rank of sets Aj, j in J is u=log2 n.
High=n1/3 X X
L=nlog log n
Low=log2n X X A1 A2 A3
… … …. …. AL

31. Product distributions, Matroid Rank Fns
Talagrand implies:
Let D be a product distribution on V, R=rank(X), X drawn from D. If E[R] ¸ 4000,
If E[R]· 500 log(1/²),
Related Work:
[Chekuri, Vondrak ’09] and [Vondrak ’10] prove a slightly more general result by two different techniques

32. Product distributions, Matroid Rank Fns
Talagrand implies:
Let D be a product distribution on V, R=rank(X), X drawn from D. If E[R] ¸ 4000,
If E[R]· 500 log(1/²),
Algorithm:
Let ¹= i=1m f (xi) / m
Let g be the constant function with value ¹
This achieves approximation factor O(log(1/²)) on a 1-² fraction of points, with high probability.

33. Conclusions and Open Questions
Analyze intrinsic learnability of submodular fns
Our analysis reveals interesting novel extremal and structural properties of submodular fns.
Open questions
Improve (n1/3) lower bound to (n1/2)
Non-monotone submodular functions

34. Other interesting structural properties
Let h : R ! R+ be concave, non-decreasing. For each Sµ V, let f(S) = h(|S|)
Claim: These functions f are submodular, monotone, non-negative. ; V

35. Lots of Theorem: Every submodular function looks like this.
approximately
usually. V ;

36. Theorem: Every submodular function looks like this.
Lots of
approximately
usually. V ;
Theorem
Let f be a non-negative, monotone, submodular, 1-Lipschitz function. For any ²>0, there exists a concave function h : [0,n] ! R s.t. for every k2[0,n], and for a 1-² fraction of SµV with |S|=k, we have:
h(k) · f(S) · O(log2(1/²))¢h(k).
In fact, h(k) is just E[ f(S) ], where S is uniform on sets of size k.
Proof: Based on Talagrand’s Inequality.

37. Conclusions and Open Questions
Analyze intrinsic learnability of submodular fns
Our analysis reveals interesting novel extremal and structural properties of submodular fns.
Open questions
Improve (n1/3) lower bound to (n1/2)
Non-monotone submodular functions
Any algorithm?
Lower bound better than (n1/3)