Submodular Optimization

Transcription

1 Submodular Optimization Nathaniel Grammel N. Grammel Submodular Optimization 1 / 28

2 Submodularity Captures the notion of Diminishing Returns Definition Suppose U is a set. A set function f :2 U! R is submodular if for any S, T U: f (S \ T )+f (S [ T ) apple f (S)+f (T ) N. Grammel Submodular Optimization 2 / 28

3 Submodularity Captures the notion of Diminishing Returns Definition Suppose U is a set. A set function f :2 U! R is submodular if for any S, T U: f (S \ T )+f (S [ T ) apple f (S)+f (T ) How does this represent diminishing returns? N. Grammel Submodular Optimization 2 / 28

4 Submodularity Captures the notion of Diminishing Returns Definition Suppose U is a set. A set function f :2 U! R is submodular if for any S, T U: f (S \ T )+f (S [ T ) apple f (S)+f (T ) How does this represent diminishing returns? Equivalent Definition: Definition A function f :2 U! R is submodular if for any S, T U such that S T,andanyx 2 U \ T : f (T [{x}) f (T ) apple f (S [{x}) f (S) N. Grammel Submodular Optimization 2 / 28

5 Submodularity and Diminishing Returns Imagine choosing items from U one by one At any point, let S be the set of items chosen so far Choosing x as the next item gives an increase in utility of f (S [{x}) f (S) N. Grammel Submodular Optimization 3 / 28

6 Submodularity and Diminishing Returns Imagine choosing items from U one by one At any point, let S be the set of items chosen so far Choosing x as the next item gives an increase in utility of f (S [{x}) f (S) If we choose some other items first to get a set T (notice: S T ), and then choose x, theincreaseinutilityis f (T [{x}) f (T ) apple f (S [{x}) f (S) N. Grammel Submodular Optimization 3 / 28

7 Submodularity and Diminishing Returns Imagine choosing items from U one by one At any point, let S be the set of items chosen so far Choosing x as the next item gives an increase in utility of f (S [{x}) f (S) If we choose some other items first to get a set T (notice: S T ), and then choose x, theincreaseinutilityis f (T [{x}) f (T ) apple f (S [{x}) f (S) Adding x give less utility if we start with more N. Grammel Submodular Optimization 3 / 28

8 Submodularity and Diminishing Returns Imagine choosing items from U one by one At any point, let S be the set of items chosen so far Choosing x as the next item gives an increase in utility of f (S [{x}) f (S) If we choose some other items first to get a set T (notice: S T ), and then choose x, theincreaseinutilityis f (T [{x}) f (T ) apple f (S [{x}) f (S) Adding x give less utility if we start with more This is the concept of diminishing returns N. Grammel Submodular Optimization 3 / 28

9 Monotonicity: Another Useful Property Definition A set function f :2 U! R is monotone if for every S, T U with S T : f (S) apple f (T ) N. Grammel Submodular Optimization 4 / 28

10 Monotonicity: Another Useful Property Definition A set function f :2 U! R is monotone if for every S, T U with S T : f (S) apple f (T ) We are generally interested in monotone submodular functions. Often, these are utility functions. N. Grammel Submodular Optimization 4 / 28

11 AConcreteExample Recall this example utility function from early in the semester: f ({apple, orange}) =5 f ({apple}) =f ({orange}) =3 f ({}) =f (?) =0 Notice it is monotone submodular! N. Grammel Submodular Optimization 5 / 28

12 Why do we care? Diminishing Returns N. Grammel Submodular Optimization 6 / 28

13 Why do we care? Diminishing Returns Closely Related to Convexity and Concavity N. Grammel Submodular Optimization 6 / 28

14 Why do we care? Diminishing Returns Closely Related to Convexity and Concavity Optimization problems are very hard in general (often, NP-hard), especially with set functions. N. Grammel Submodular Optimization 6 / 28

15 Why do we care? Diminishing Returns Closely Related to Convexity and Concavity Optimization problems are very hard in general (often, NP-hard), especially with set functions. But with submodularity, we can make strong guarantees: e cient algorithms for close approximations. N. Grammel Submodular Optimization 6 / 28

16 Why do we care? Diminishing Returns Closely Related to Convexity and Concavity Optimization problems are very hard in general (often, NP-hard), especially with set functions. But with submodularity, we can make strong guarantees: e cient algorithms for close approximations. Many natural functions have these properties N. Grammel Submodular Optimization 6 / 28

17 Why do we care? Diminishing Returns Closely Related to Convexity and Concavity Optimization problems are very hard in general (often, NP-hard), especially with set functions. But with submodularity, we can make strong guarantees: e cient algorithms for close approximations. Many natural functions have these properties Inferring Influence in a Network (Stay tuned!) N. Grammel Submodular Optimization 6 / 28

18 Why do we care? Diminishing Returns Closely Related to Convexity and Concavity Optimization problems are very hard in general (often, NP-hard), especially with set functions. But with submodularity, we can make strong guarantees: e cient algorithms for close approximations. Many natural functions have these properties Inferring Influence in a Network (Stay tuned!) Determining representative sentences in a document N. Grammel Submodular Optimization 6 / 28

19 Why do we care? Diminishing Returns Closely Related to Convexity and Concavity Optimization problems are very hard in general (often, NP-hard), especially with set functions. But with submodularity, we can make strong guarantees: e cient algorithms for close approximations. Many natural functions have these properties Inferring Influence in a Network (Stay tuned!) Determining representative sentences in a document Many applications to image and signal processing. N. Grammel Submodular Optimization 6 / 28

20 Why do we care? Diminishing Returns Closely Related to Convexity and Concavity Optimization problems are very hard in general (often, NP-hard), especially with set functions. But with submodularity, we can make strong guarantees: e cient algorithms for close approximations. Many natural functions have these properties Inferring Influence in a Network (Stay tuned!) Determining representative sentences in a document Many applications to image and signal processing. Sensor Placement N. Grammel Submodular Optimization 6 / 28

21 Why do we care? Diminishing Returns Closely Related to Convexity and Concavity Optimization problems are very hard in general (often, NP-hard), especially with set functions. But with submodularity, we can make strong guarantees: e cient algorithms for close approximations. Many natural functions have these properties Inferring Influence in a Network (Stay tuned!) Determining representative sentences in a document Many applications to image and signal processing. Sensor Placement Graph Cuts N. Grammel Submodular Optimization 6 / 28

22 Why do we care? Diminishing Returns Closely Related to Convexity and Concavity Optimization problems are very hard in general (often, NP-hard), especially with set functions. But with submodularity, we can make strong guarantees: e cient algorithms for close approximations. Many natural functions have these properties Inferring Influence in a Network (Stay tuned!) Determining representative sentences in a document Many applications to image and signal processing. Sensor Placement Graph Cuts And many more! (Check out submodularity.org) N. Grammel Submodular Optimization 6 / 28

23 Submodular Optimization Two main types of submodular optimization: N. Grammel Submodular Optimization 7 / 28

24 Submodular Optimization Two main types of submodular optimization: Maximization N. Grammel Submodular Optimization 7 / 28

25 Submodular Optimization Two main types of submodular optimization: Maximization Want to find S to maximize f (S) subject to some constraints Most simply: Want to find S that maximizes f (S) subjectto S = k for some k (cardinality constraint) More generally: Other types of constraints (e.g. knapsack constraints, matroid constraints) N. Grammel Submodular Optimization 7 / 28

26 Submodular Optimization Two main types of submodular optimization: Maximization Want to find S to maximize f (S) subject to some constraints Most simply: Want to find S that maximizes f (S) subjectto S = k for some k (cardinality constraint) More generally: Other types of constraints (e.g. knapsack constraints, matroid constraints) Cover/Minimization N. Grammel Submodular Optimization 7 / 28

27 Submodular Optimization Two main types of submodular optimization: Maximization Want to find S to maximize f (S) subject to some constraints Most simply: Want to find S that maximizes f (S) subjectto S = k for some k (cardinality constraint) More generally: Other types of constraints (e.g. knapsack constraints, matroid constraints) Cover/Minimization Want to minimize the cost of covering f Most simply: Find S with minimum size that achieves f (S) =f (U) More generally: Find S such that f (S) =f (U) while minimizing cost(s) for some cost function. N. Grammel Submodular Optimization 7 / 28

28 Submodular Maximization: The Simplest Case Suppose we are given a utility function f :2 U! R and a cardinality constraing k Goal: Find S = argmax f (S) S U: S applek This is NP-hard in general! What if f is submodular? Monotone? Nonnegative? We can find good near-optimal solutions! N. Grammel Submodular Optimization 8 / 28

29 Submodular Maximization: A Simple Greedy Algorithm Suppose f :2 U! R is monotone submodular. Assume f is nonnegative. N. Grammel Submodular Optimization 9 / 28

30 Submodular Maximization: A Simple Greedy Algorithm Suppose f :2 U! R is monotone submodular. Assume f is nonnegative. Start with S = {} N. Grammel Submodular Optimization 9 / 28

31 Submodular Maximization: A Simple Greedy Algorithm Suppose f :2 U! R is monotone submodular. Assume f is nonnegative. Start with S = {} At each step, pick the item x 2 U \ S that maximizes f (S [{x}) f (S) (theitemthatmaximizesthegain in utility) and let S = S [{x}. N. Grammel Submodular Optimization 9 / 28

32 Submodular Maximization: A Simple Greedy Algorithm Suppose f :2 U! R is monotone submodular. Assume f is nonnegative. Start with S = {} At each step, pick the item x 2 U \ S that maximizes f (S [{x}) f (S) (theitemthatmaximizesthegain in utility) and let S = S [{x}. Continue until S = k N. Grammel Submodular Optimization 9 / 28

33 Submodular Maximization: A Simple Greedy Algorithm Suppose f :2 U! R is monotone submodular. Assume f is nonnegative. Start with S = {} At each step, pick the item x 2 U \ S that maximizes f (S [{x}) f (S) (theitemthatmaximizesthegain in utility) and let S = S [{x}. Continue until S = k Theorem (Nemhauser et al. 1978) Let S be the k-element set constructed as above, and let S be the set that maximizes f over all sets of size at most k. Then: f (S) (1 1/e)f (S ) N. Grammel Submodular Optimization 9 / 28

34 Submodular Maximization: A Simple Greedy Algorithm Suppose f :2 U! R is monotone submodular. Assume f is nonnegative. Start with S = {} At each step, pick the item x 2 U \ S that maximizes f (S [{x}) f (S) (theitemthatmaximizesthegain in utility) and let S = S [{x}. Continue until S = k Theorem (Nemhauser et al. 1978) Let S be the k-element set constructed as above, and let S be the set that maximizes f over all sets of size at most k. Then: f (S) (1 1/e)f (S ) Thus, the set S provides a (1 1/e)-approximation, and the construction gives a polynomial-time approximation algorithm. N. Grammel Submodular Optimization 9 / 28

35 Greedy Algorithm: Proof For convenience, let S(x) =f (S [{x}) f (S). Then submodularity states that for S T and x 2 U \ T,wehave T (x) apple S (x). Let S i U be the subset of i elements chosen greedily: S i = S i 1 [{arg max x2u S i 1 (x)} with S 0 = {}. Proof that f (S) (1 1/e)f (S ). Due to monotonicity, S = k. LetS = {e 1, e 2,...,e k }.Further, also due to monotonicity, for any i < k: f (S ) apple f (S [ S i ) (1) N. Grammel Submodular Optimization 10 / 28

36 Greedy Algorithm: Proof Proof that f (S) (1 1/e)f (S ). We also have the following equality f (S [ S i )=f (S i )+ kx j=1 S i [{e 1,...,e j 1 }(e j ) (1) since the terms S i [{e 1,...,e j 1 }(e j )=f (S i [{e 1,...,e j }) f (S i [{e 1,...,e j 1 }) are telescoping so the sum is equal to f (S i [{e 1,...,e j }) f (S i [ {}) =f (S i [ S ) f (S i ). N. Grammel Submodular Optimization 10 / 28

37 Greedy Algorithm: Proof Proof that f (S) (1 1/e)f (S ). Due to submodularity, we have f (S i )+ kx j=1 S i [{e 1,...,e j 1 }(e j ) apple f (S i )+ kx j=1 S i (e j ) (1) The greedy rule states that f (S i+1 ) f (S i ) S i (x) foranyx. Thus: f (S i )+ kx j=1 kx S i (e j ) apple f (S i )+ (f (S i+1 ) f (S i )) j=1 apple f (S i )+k(f (S i+1 ) f (S I )) (2) where the second inequality holds since S = k. N. Grammel Submodular Optimization 10 / 28

38 Greedy Algorithm: Proof Proof that f (S) (1 1/e)f (S ). Putting it all together: f (S ) f (S i ) apple k(f (S i+1 ) f (S i )) (1) Let i = f (S ) f (S i ). Then, we can rearrange to get 1 i apple k( i i+1 ), or i+1 apple i 1 k which yields k apple k k N. Grammel Submodular Optimization 10 / 28

39 Greedy Algorithm: Proof Proof that f (S) (1 1/e)f (S ). Since f is nonnegative, 0 = f (S ) f ({}) apple f (S ). A famous inequality states: 1 x apple e x for all x 2 R. Thisyields k apple 1 1 k k 0 apple 1 Since k = f (S ) f (S k ), we get 1 k f (S ) apple e k/k f (S ) k k = f (S ) f (S k ) apple e 1 f (S ) (1) f (S ) e 1 f (S ) apple f (S k ) (2) f (S )(1 1/e) apple f (S k ) (3) N. Grammel Submodular Optimization 10 / 28

40 (1 1/e) isthebestwecanget Theorem (Nemhauser and Wolsey 1978) Any algorithm that evaluate f on at most a polynomial number of inputs cannot do better than a (1 1/e)-approximation of the optimal solution. N. Grammel Submodular Optimization 11 / 28

41 Speedup with Lazy Evaluations (Minoux 1978) Evaluating S(x) foreveryx at each iteration may be costly Store for each item x avalue (x), representing an upper bound on S(x). Store the items sorted in order of. At each step, pick the element x at the front of the list (i.e. with maximum (x)) Lazy Evaluation: Evaluate S only for element x, andupdate (x) S(x). If after update, (x) (x 0 ) for all other x 0,thenx is still the best choice for the greedy algorithm! We ve avoided re-evaluating for all the other elements! N. Grammel Submodular Optimization 12 / 28

42 Submodular Cover Suppose instead we want to find S so that f (S )=f (U) while minimizing S. Again, suppose f is monotone and submodular. But: this time let f :2 U! N. Suppose we apply the same greedy rule until our constructed set S has f (S) =f (U). Do we have bounds on S? Yes! Theorem (Wolsey 1982) S apple(1 + ln ) S where =max x2u f ({x}) is the maximum possible increase in utility. N. Grammel Submodular Optimization 13 / 28

43 What about non-uniform costs? Up until now we have focused on cardinality: either constrained to S applek (maximization), or trying to minimize S (cover) N. Grammel Submodular Optimization 14 / 28

44 What about non-uniform costs? Up until now we have focused on cardinality: either constrained to S applek (maximization), or trying to minimize S (cover) What if we instead have a cost function c(x) forallx 2 U and want to maximize f (S) subjectto X c(x) apple B x2s for some budget B. This is called a Knapsack Constraint. N. Grammel Submodular Optimization 14 / 28

45 What about non-uniform costs? Up until now we have focused on cardinality: either constrained to S applek (maximization), or trying to minimize S (cover) What if we instead have a cost function c(x) forallx 2 U and want to maximize f (S) subjectto X c(x) apple B x2s for some budget B. This is called a Knapsack Constraint. Or minimize P x2s c(x) suchthats covers f (i.e. f (S) =f (U))? N. Grammel Submodular Optimization 14 / 28

46 Results for non-uniform costs Standard Greedy Algorithm could be arbitrarily bad: Doesn t consider costs at all! N. Grammel Submodular Optimization 15 / 28

47 Results for non-uniform costs Standard Greedy Algorithm could be arbitrarily bad: Doesn t consider costs at all! Modification: At each step, choose x that maximizes best bang for the buck S (x) c(x) N. Grammel Submodular Optimization 15 / 28

48 Results for non-uniform costs Standard Greedy Algorithm could be arbitrarily bad: Doesn t consider costs at all! Modification: At each step, choose x that maximizes best bang for the buck S (x) c(x) Can get similar results to uniform-cost case, with some slight modifications N. Grammel Submodular Optimization 15 / 28

49 Results for non-uniform costs Standard Greedy Algorithm could be arbitrarily bad: Doesn t consider costs at all! Modification: At each step, choose x that maximizes best bang for the buck S (x) c(x) Can get similar results to uniform-cost case, with some slight modifications Cover: Wolsey (1982) generalizes the result of uniform-cost case N. Grammel Submodular Optimization 15 / 28

50 Results for non-uniform costs Standard Greedy Algorithm could be arbitrarily bad: Doesn t consider costs at all! Modification: At each step, choose x that maximizes best bang for the buck S (x) c(x) Can get similar results to uniform-cost case, with some slight modifications Cover: Wolsey (1982) generalizes the result of uniform-cost case For maximization: A bit trickier. This greedy rule doesn t su ce! But some simple modifications can yield similar approximations to uniform-cost case. N. Grammel Submodular Optimization 15 / 28