A survey of submodular functions maximization Yao Zhang 03/19/2015
Example Deploy sensors in the water distribution network to detect contamination F(S): the performance of the detection when a set S of places is selected 2
Definition of submodular functions finite ground set V={1,2,...n} set function f(s): Marginal gain: Submodular: diminishing return 3
Definition of submodular functions Modular function: Supermodular 4
Examples of submodular function Deploy sensors in the water distribution network to detect contamination 5 From Krause s survey
Examples of submodular function From Krause s survey Weighted coverage functions Entropy Mutual information Cut capacity functions Influence function (Kempe 03) f(s): the expected number of infected nodes when nodes in S are infected at the start Propagation model Linear threshold Independent Cascade 6
Properties of submodular function Linear combination if set functions F 1,...,F m are submodular functions, and a 1,...,a m >0 then Concavity is submodular 7
Submodularity optimization Except for the submodularity, we assume: 1. Monotonically non-decreasing 2. 8
Summary of submodularity optimization For more details, see http://submodularity.org/ 9 Focus on this part
Maximization of submodular functions Problem: Simplest constraint cardinality constraints for a given k, we require that Greedy algorithm 10
11 Greedy Algorithm
Matroid constraints Greedy algorithm is an ½-approximation algorithm 12
Knapsack constraint Knapsack constraint: Greedy Algorithms: Analysis: Scb: the solution provided by the cost-benefit greedy algorithm Suc: the solution returned by returned by the uniform cost 13 [Leskovec et al. KDD 2007]
Speeding up the greedy algorithm Lazy evaluation [Leskovec et al. KDD 2007] First iteration as usual Keep an order list of marginal gain Δi from the previous iteration Re-evaluate the marginal gain only for top element i if Δi stays on top, user it, otherwise re-sort Sorted list in the descending order: t, s at i-th iteration In the (i+1)-th iteration: 14
Fast algorithms Summary: Randomized Greedy w is the threshold 15 Badanidiyuru, Ashwinkumar, and Jan Vondrák. "Fast algorithms for maximizing submodular functions." SODA2014.
Lazier Than Lazy Greedy Random Sampling Cardinality Constraint Analysis: (1-1/e-ε) approximation 16 Mirzasoleiman, Baharan, Ashwinkumar Badanidiyuru, Amin Karbasi, Jan Vondrák, and Andreas Krause. "Lazier Than Lazy Greedy." AAAI 2015.
Complex constraints Submodular maximization using the multilinear extension Submodular optimization over graphs the set S forms a path, or a tree on G of weight at most B Robust submodular optimization Consider adversaries (Game theory) Nonmonotone submodular functions E.g., a monotone submodular function f, and a modular cost function c We want to max. non-monotone function 17 All have good approximations using Greedy based algorithm (See Krause s survey for details)
Online maximization of submodular functions The objective may not be known in advance Objectives functions {f 1,,f T } drawn from some distribution At each round, select certain element Two settings: no-regret setting the choices in any round are not constrained by what one did in previous rounds and the goal is to perform well on average competitive setting a sequence of irrevocable decisions Previous round choices may affect the decision of the current round 18
Adaptive submodularity We wish to adaptively select a set, observing and taking into account feedback after selecting any particular element. E.g., when placing the next sensor, adaptively taking into account measurements provided by the sensors selected so far Active learning Daniel Golovin and Andreas Krause,. Adaptive Submodularity: Theory and Applications in Active Learning and Stochastic Optimization. Journal of Artificial Intelligence Research (JAIR), 2011. 19
Recent Progress
Recent Progress Submodular Welfare problem Submodular function over integer lattice Distributed Submodular Maximization Streaming Submodular Maximization Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints 21
Submodular Welfare problem Problem: Algorithm: Continuous Greedy Algorithm provides (1-1/e-o(1))- appriximation 22 Jan Vondrák. "Optimal approximation for the submodular welfare problem in the value oracle model." STOC 2008.
Submodular Welfare problem 23 Jan Vondrák. "Optimal approximation for the submodular welfare problem in the value oracle model." STOC 2008.
Submodular function over integer lattice Integer lattice: vector Submodular function satisfies: Greedy algorithm provides (1-1/e)-approx. for the cardinality constraint (ICML 2014) Recent paper: Consider cardinality, matriod and knapsack constraint 24 1. Soma, Tasuku, and Yuichi Yoshida. "Maximizing Submodular Functions with the Diminishing Return Property over the Integer Lattice." arxiv preprint arxiv:1503.01218 (2015) 2. Soma, Tasuku, Naonori Kakimura, Kazuhiro Inaba, and Ken-ichi Kawarabayashi. "Optimal budget allocation: Theoretical guarantee and efficient algorithm." ICML 2014.
Distributed Submodular Maximization I Greedy Algorithm for the submodular function 25 Mirzasoleiman, Baharan, Amin Karbasi, Rik Sarkar, and Andreas Krause. "Distributed submodular maximization: Identifying representative elements in massive data." NIPS 2013.
Distributed Submodular Maximization II Summary of Greedy Algorithm in MapReduce SPAA 2013 best paper Kumar, Ravi, Benjamin Moseley, Sergei Vassilvitskii, and Andrea Vattani. "Fast greedy algorithms in mapreduce and streaming." SPAA 2013 26
Streaming Submodular Maximization Assume elements set V is ordered any streaming algorithm must process V in the given order At each iteration t the algorithm maintains a memory of subset of elements Mt points; and must be ready to output a candidate feasible solution St When a new point arrives, the algorithm may select to remember it, and discard previous elements Badanidiyuru, Ashwinkumar, Baharan Mirzasoleiman, Amin Karbasi, and Andreas Krause. "Streaming submodular maximization: Massive data summarization on the fly." KDD 2014. 27
Streaming Submodular Maximization Simple example our memory can only store one element Algorithm: Badanidiyuru, Ashwinkumar, Baharan Mirzasoleiman, Amin Karbasi, and Andreas Krause. "Streaming submodular maximization: Massive data summarization on the fly." KDD 2014. 28
Submodular knapsack constraint Two types of problem Submodular Cost Submodular Cover (SCSC) Submodular Cost Submodular Knapsack (SCSK) Both f and g are submodular function 29 Iyer, Rishabh K., and Jeff A. Bilmes. "Submodular optimization with submodular cover and submodular knapsack constraints." NIPS 2013.
30 References 1. http://submodularity.org/ 2. Krause, Andreas, and Daniel Golovin. "Submodular function maximization." Tractability: Practical Approaches to Hard Problems 3 (2012): 19. 3. Mirzasoleiman, Baharan, Ashwinkumar Badanidiyuru, Amin Karbasi, Jan Vondrák, and Andreas Krause. "Lazier Than Lazy Greedy." AAAI 2015. 4. Badanidiyuru, Ashwinkumar, and Jan Vondrák. "Fast algorithms for maximizing submodular functions." SODA2014. 5. Jan Vondrák. "Optimal approximation for the submodular welfare problem in the value oracle model." STOC 2008. 6. Soma, Tasuku, and Yuichi Yoshida. "Maximizing Submodular Functions with the Diminishing Return Property over the Integer Lattice." arxiv preprint arxiv:1503.01218 (2015) 7. Soma, Tasuku, Naonori Kakimura, Kazuhiro Inaba, and Ken-ichi Kawarabayashi. "Optimal budget allocation: Theoretical guarantee and efficient algorithm." ICML 2014.
References 8. Mirzasoleiman, Baharan, Amin Karbasi, Rik Sarkar, and Andreas Krause. "Distributed submodular maximization: Identifying representative elements in massive data." NIPS 2013. 9. Kumar, Ravi, Benjamin Moseley, Sergei Vassilvitskii, and Andrea Vattani. "Fast greedy algorithms in mapreduce and streaming." SPAA 2013. 10. Badanidiyuru, Ashwinkumar, Baharan Mirzasoleiman, Amin Karbasi, and Andreas Krause. "Streaming submodular maximization: Massive data summarization on the fly." KDD 2014. 11. Iyer, Rishabh K., and Jeff A. Bilmes. "Submodular optimization with submodular cover and submodular knapsack constraints." NIPS 2013. 31