Optimization of submodular functions
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1 Optimization of submodular functions
2 Law Of Diminishing Returns According to the Law of Diminishing Returns (also known as Diminishing Returns Phenomenon), the value or enjoyment we get from something starts to decrease after a certain point*. Let s say we go to an amusement park and ride our favorite roller coaster five times in one day. The first time is exhilarating. The second and third times are also exciting. But after the fourth or fifth ride, we start to feel sick and bored - we ve had enough*. You are going to your favorite restaurant, you want to eat as much as possible. Your enjoyment for the first dish is stellar, for the second is great, for the third is good, for the fourth OK, for the fifth you cannot eat anymore and you are actually not enjoying it that much.. The law of diminishing returns also applies to performance. Although it's important to study for an exam or practice for a game, there is a limit to how much time and energy we can invest and still expect to see an improvement. If we practice too much we ll start to feel burned out and may even start performing poorly*. Submodular functions model formally the concept of diminishing returns * From
3 Submodular functions definition (1) Let s consider a set V, and a function ff: 2 VV RR The function assigns to each subset S V a value f(s) You can imagine f(s) as the function that determines the perceived utility eating a set of dishes S from the menu V We define the discrete derivative of f(), given a subset S V and an element e V as ee SS = ff SS + ee ff(ss) Here we use the often adopted notation where S + e means S {e}
4 Submodular functions definition (2) A function is submodular if for every A B V and e V \ B, it holds that ee AA ee BB An equivalent definitions is, for every A, B V ff AA BB + ff AA BB ff AA + ff(bb)
5 There are many examples of submodular functions Linear (modular) functions ff SS = ee SS ff(ee) Coverage functions. Given a universe set Ω = {u 1,u 2,,u n } and a set E = {E 1,E 2,,E m }, s.t. E 2 Ω Consider a set S EE, the function ff SS = EEii SS EE ii is submodular. Weighted coverage functions Entropy of a set of random variables. Given a set of random variables Ω = {X 1,X 2,,X n }, for any S Ω, H(S) is a submodular function Matroid rank functions..
6 Some problems you may have seen Consider these two problems max ff SS SS V } E.g., Max matching, Max Cut, Max Coverage miiii ff SS SS V } Set Cover, Vertex Cover, Minimum Spanning Tree, Min Cut These problems can all be modeled as optimizations of submodular functions
7 Concave or Convex? Concave functions are continuous f: R R is concave if f (x) is non increasing in x Submodular functions are the discrete counterpart f: {0,1} n R is submodular if i= 1,..,n, the discrete partial derivative i f(x) = f(x + e i ) f(x) is non increasing in x Submodularity has however some similarities with convex functions too ( easy to minimize)
8 Optimization of submodular functions Min and Max problems have different complexities Theorem: (Grotschel-Lovasz-Schrijver, 1981) There is an algorithm that computes the minimum of any submodular function in poly (n), (using value queries f(s) =?) Maximization of submodular function is instead usually an NP-Hard, but we can do something nice here too
9 Minimization of a submodular function: the Lovasz extension The approach is similar to the relaxation of a ILP problem to an LP problem We are given a submodular function f: {0,1} n R We define a convex function f L : [0,1] n R Since f L is convex we can minimize it efficiently A minimizer for f L can be converted to a minimizer for f
10 Minimization of a submodular function: the Lovasz extension We define f L : {0,1} n R as follows, for z [0,1] n ff LL zz = EE[ff ii zz ii λ}] where λ is a uniformly distributed random variable in [0,1] Why is f L an extension of f? Let s continue on the black board.. We can prove that f L is convex if and only if f is submodular We can then use standard optimization techniques for convex functions (e.g. gradient descendent, ellipsoid, etc.)
11 Only good news? No The Lovaz extension works well if the problem has no additional constraints However, problem like the following make the minimization hard: miiii ff SS SS kk} miiii ff TT TT iiii aa ssssssssssssssss tttttttt oooo GG} miiii ff PP PP iiii aa sssssssssssssss ppppppp bbbbbbbbbbbbbb ss tt}
12 Maximization of submodular functions These problems are common, intuitive and well represent human behavior Go to an amusement park and ride our favorite roller coaster Picking items in a restaurant menu Maximize your performance Maximize wellbeing In these contexts usually it is assumed that the submodular function is monotonic increasing For each X Y, f(x) f(y) Coverage problems fall in this category too (the more I pick the more I cover) These problems are often NP-Hard, but
13 Greedy maximization We can design greedy algorithm that give us approximation bounds Usually we get (1 1/e) OPT Let s see how the algorithm looks like..
14 Greedy maximization modular budget We can design greedy algorithm that give us approximation bounds Linear non-negative cost-function Same bound (1 1/e) OPT with an algorithm complexity of O(N 5 ) Greedy algorithm gives (1 1/e)/2 Let s see how the algorithm looks like.. [1] Maxim Sviridenko, A note on maximizing a submodular set function subject to a knapsack constraint, Elsevier Operational Research, 2002 [2] Andreas Krause Carlos Guestrin, A Note on the Budgeted Maximization of Submodular Functions
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