9. Submodular function optimization

Transcription

1 Submodular function maximization 9-9. Submodular function optimization Submodular function maximization Greedy algorithm for monotone case Influence maximization Greedy algorithm for non-monotone case

2 Combinatorial optimization problems Knapsack problem given n items each with cost c i and value v i, and a budget B, the goal of the knapsack problem is to find a subset of items S such that the value is maximized while cost does not exceed the budget, i.e. maximize subject to v T x c T x B x i 2 f0; g; 8i 2 [n] MaxCover problem consider a set of n elements [n] = f; : : : ; ng, and m subsets T ; : : : ; T m [n], and the goal of MaxCover problem is to find K subsets whose union has the largest cardinality, i.e. maximize T z subject to z j X i:j 2T i x i T x K z j ; x i 2 f0; g; 8j 2 [n] 8i 2 [m]

3 Maximizing submodular functions consider a set of elements [n] = f; : : : ; ng and a real-valued function over a subset of the elements f : 2 n! R X [n] 7! f (X ) a function f is monotone if for all S T, f (S) f (T ) the marginal contribution of a function f of an element i to a set S is defined as f S (i) = f (S [ fig) f (S) a function f is submodular if for any i 2 [n] and any S T we have f S (i) f T (i) and we are interested in the following optimization problem for a submordular objective function f maximize subject to f (S) S 2 F for a subset S [n] that satisfy a certain constraint represented by the feasible set F

4 Submodular function maximization 9-4 equivalently, a function is submodular if and only if for all S ; T [n], f (S) + f (T ) f (S [ T ) + f (S \ T ) knapsack problem is a submodular maximization with f (S) = P i2s v i and F = fs [n] : c(s) Bg where c(s) = P i2s c i, and f is submodular and monotone MaxCover problem is a submodular maximization with f (S) = j [ i2s T i j and F = fs [n] : jsj K g, and f is submodular and monotone many interesting problems can be formulated as submordular function maximization problem

5 Greedy algorithm with approximation guarantee first, consider a special case of monotone and submodular f, and the constraint is cardinality constraint with F = fs [n] : jsj K g Greedy algorithm. set S = ; 2. while jsj K add to S an element i that maximizes f (S [ fig) 3. end while. surprisingly, this greedy algorithm gives a good approximate solution as proved by Fisher, Nemhauser and Wolsey in 978 theorem. (approximation) If f monotone, submodular and f (;) = 0, then the greedy algorithm returns a solution that achieves f (S) OPT e where OPT = max S:jS jk f (S) theorem. (converse result by Feige 998) it is NP-hard to get a ( e ) + approximation for any > 0 for MaxCover problem (proof is beyond the scope of this lecture)

6 proof of the approximation guarantee. let S i be the set after i iterations of the greedy algorithm, then by monotonicity f (S ) f (S i [ S ), where S is the optimal set by submodularity, f (S ) f (S i [ S ) f (S i ) + (f (S i [ fx g) f (S i )) + (f (S i [ fx ; x 2 g) f (S i [ fx g)) + f (S i ) + (f (S i [ fx g) f (S i )) + f Si (x 2 ) + + f Si (x k ) f (S i ) + K (f (S i+ ) f (S i )) for S = fx ; : : : ; x k g, and this gives f (S i+) f (S i) OPT f (Si) K now we use induction to prove that f (S i) i OPT K it is trivially true for i = 0 and we write f (S i+ ) f (S i ) + K OPT f (S i ) = K K = f (S i ) + K OPT K K i+ OPT i OPT + K OPT [by induction hypothesis] this implies f (S) ( ( =K ) K )OPT ( =e)opt Submodular function maximization 9-6

7 Submodular function maximization 9-7 data-dependent bound suppose S is the output of the greedy algorithm and S is the optimal solution let i = f S (i) and suppose i s are sorted such that 2 3 : : : then similar argument shows that f (S ) f (S [ S ) f (S) k gain is significant if f (S)

8 next, we consider a more general case where f is still monotone and submodular, but the constraint is the budget constraint with F = fs [n] : C (S) Bg where C (S) = P i2s c i for some elementwise cost c i Cost-benefit greedy algorithm. set S = ; 2. while there exists an i 2 [n] such that C (S [ fig) B add to S an element i that maximizes f (S [ fig) f (S) subject to C (S [ fig) B 3. end while. c i performance of the above algorithm can be arbitrarily bad, for example if B = and f (f; 2g) = f () + f (2) with item : f () = 2"; c = " item 2 : f (2) = ; c 2 = Cost-benefit greedy algorithm chooses S = fg with F (S) = 2", whereas the optimal solution is F (f2g) =

9 Submodular function maximization 9-9 theorem. [Leskovec et al KDD 07] let S CB be the solution of Cost-benefit greedy algorithm and let S UC be the output of the greedy algorithm treating the costs as equal, then maxff (S CB ); f (S UC )g 2 e OPT

10 Examples of submodular function maximization Influence maximization motivation: given a target customers in a social network, how can we choose a set S of early adopters and market them in order to generate a cascade of adoptions? define influence function f (S) as the expected number of influenced nodes at the end of the process starting with S in a finite network G(V ; E) (to be formally defined below) maximize subject to f (S) c(s) B theorem. [Kempe et al. 03] under general cascade model, influence maximization is NP-hard ro approximate to a factor of n " for any " > 0

11 Submodular function maximization 9- independent cascade model for each edge (i ; j ) there is a probability p ij such that if i is activated then it has one chance to activate j with probability p ij theorem. [Kempe et al 03] f is submodular under independent cascade model proof. consider an equivalent scenario where a biased coin is flipped for each edge in the beginning to determine whether the edge is active or not, then a node i is active if there exists a path of active edges from the node i to set S for a given instance of the active edges, the function of influence is submodular, since for all S T, the (deterministic) function f (S [ fig) f (S) is the number of nodes that are reachable from i using active edges that is not already reachable from S and if T includes S, this number only gets smallet for T, i.e. f (S [ fig) f (S) f (T [ fig) f (T ) since the function is submodular for each instance of the random trial, it is submodular in expectation as well, from the linearity of expectation f (S) = fz (S)P(Z ) PZ

12 linear threshold model for each edge (i ; j ) there is a nonnegative weight w ij such that Pi w ij each node i has threshold i uniformly a trandom in [0; ] such that i becomes active if X w ij j activei theorem. [Kempe et al 03] f is submodular under linear threshold model

13 Submodular function maximization 9-3 feature selection given random variables Y ; X ; : : : ; X n, want to predict Y from subset X A = fx i ; : : : ; X ik g Y : patient is sick X : patient has fever X 2 : patient has rash X 3 : patient is male how do we find k most informative features? f (S) = I (Y ; X S ) = H (Y ) H (Y jx S ) f monotonic and submodular if X i s are conditionally independent given Y [Krause, Guestrin UAI 05]

14 Maximizing non-monotone but submodular function f Now we slightly relax the previous assumptions and suppose that f is submodular, but not necessarily monotone, and there is no constraint on f this problem is NP-hard in general, if f can take negative values, hard to even approximate, i.e. O(n ) approximation is NP-hard however, when f is non-negative, there is an efficient greedy algorithm: Non-negative greedy algorithm. set S = fig where i maximizes f (fig) 2. while there exists an i 2 [n] n S such that f (S [ fig) f (S) add to S an element i return to while there exists an i 2 S such that f (S n fig) f (S) remove i from S return to 2. and achieves good approximation guarantee theorem. [Feige 07] for non-negative and submodular f, Non-negative greedy algorithm achieves 3 -approximation. Submodular function maximization 9-4

15 proof. let S be the optimal solution and S be the output of the greedy algorithm, then f (S ) f (S ) + f (;) + f ([n]) [by non-negativity] this proves that f (S \ S) + f (S \ S c ) + f ([n]) [by sub-modularity] f (S \ S) + f (S [ S) + f (S c ) [by submodularity] f (S) + f (S) + f (S c ) [by the following lemma] maxfs ; S c g 3 OPT lemma. for any subset T of S, i.e. T S or a superset T of S, i.e. S T, we have f (S) f (T ) proof. consider any sequence of increasing sets ; = T 0 T T k = S T k+ T n = [n] where jt i+ n T ij = and a i t i+ n T i for i < k, f (T i+) f (T i) f (S) f (S n fa ig) [by submodularity] 0 [by greedy search]

16 Submodular function maximization 9-6 for i > k, f (T i) f (T i ) f (S [ fa ig) f (S) [by submodularity] 0 [by greedy search]

17 further reading submodular function maximization with matroid constraints greedy algorithm achieves ( =e) approximation robust optimization max S min i f i(s) for a class of submoduular functions f i s [Krause et al 07] does not admit any approximation unless P=NP minimizing submodular function

18 Minimizing submodular functions example: factoring distributions given random variables X ; : : : ; X n, partition the variables into two sets S and [n] n S as independent as possible minimize subject to f (S) c(s) B where f (S) = I (X S ; X [n]ns ) = H (X [n]ns ) H (X [n]ns jx S ) f is submodular and symmetric where a sub modular function is symmetries if and only if for all S [n] f (S) = f (S c ) another example of a symmetric submodular set function is graph cut

19 Submodular function maximization 9-9 remark. a symmetric submodular function is (trivially) minimized at f (;) = f ([n]) proof. f (S) = 2 (f (S) + f (S c )) (f (;) + f ([n])) = f (;) 2 we are interested in the following unconstrained minimization problem: minimize ;S [n] f (S) theorem. [Queyranne 98] for symmetric submodular f, there is an algorithm with runtime O(n 3 )for solving the unconstrained minimization problem.