Overview Introduce the notion of curvature, to provide better connections between theory and practice. Study the role of curvature in:I I Study the role of curvature in: Approximating submodular functions everywhere Learning Submodular functions Constrained Minimization of submodular functions. I Provide improved curvature-dependent worst case approximation guarantees and matching hardness results Curvature of a Submodular function Define three variants of curvature of a monotone submodular functionas:I Proposition: κˆf (S) ≤ κf (S) ≤ κf .I I Captures the linearity of a submodular function. I A more gradual characterization of the hardness of various problems. I Investigated for submodular maximization (Conforti & Cornuejols, 1984).

Approximating Submodular functions Everywhere Problem:Given a submodular function f in form of a value oracle, find an approximation ˆf (within polynomial time and space), such that ˆf (X ) ≤ f (X ) ≤ α1(n)ˆf (X ), ∀X ⊆ V for a polynomial α1(n). We provide a blackbox technique to transform bounds into curvature dependent ones.I κ I Main technique: Approximate the curve-normalized version f as ˆf κ, such that ˆf κ(X ) ≤ f κ(X ) ≤ α(n)ˆf κ(X ).

Ellipsoidal Approximation:I I The Ellipsoidal Approximation algorithm of Goemans et al, provides a function p√ of the form w f (X ) with an approximation factor of α1(n) = O( n log n)). I Corollary: There exists a function of the form, Lower bound: Given a submodular function f with curvature κf , there does not exist any polynomial-time algorithm that approximates f within a factor of n1/2− , for any  > 0. 1+(n1/2−−1)(1−κ )fI Modular Upper Bound:I Pmˆ (X ) =I A simplest approximation (and upper bound) is f j∈X f (j). j∈X I Lemma: Given a monotone submodular function f , it holds that, This bound is tight for the class of modularapproximations.I Pka I Corollary: The class of functions, f (X ) =λ[w(X)],λ≥0,satisfiesiiii=1 Pf (X ) ≤ j∈X f (j) ≤ |X |1−af (X ). Learning Submodular Functions Problem: Given i.i.d training samples {(Xi , f (Xi )}m i=1 from a distribution D, learn an approximation ˆf (X ) that is, with probability 1 − δ, within a multiplicative factor of α2(n) from f . Balcan & Harvey proposeanalgorithmwhichPMAClearnsanysubmodular√ function upto a factor of n + 1. We improve this bound to a curvature dependent one. Lemma: Let f be a monotone submodular function for which we know an upper bound on its curvature κf and the singleton weights f (j) for all j ∈ V √ . There is n+1.an poly-time algorithm which PMAC-learns f within a factor of 1+(√ n+1−1)(1−κ) We also provide an algorithm which does not need the singleton weights. Lemma: If f is a monotone submodular function with known curvature (or a known upper bound) κˆf (X ), ∀X ⊆ V , then for every , δ > 0 there is an algorithm |X |which PMAC learns f (X ) within a factor of 1 + 1+(|X |−1)(1−κˆf (X )) . Pka I Corollary: The class of functions f (X ) =λ[w(X)], λi ≥ 0, can be learnt toii=1 i a factor of |X |1−a. I Lower bound: Given a class of submodular functions with curvature κf , there does not exist a polynomial-time algorithm that is guaranteed to PMAC-learn f 1/3−0 n0within a factor of 1+(n1/3−,forany>0.0−1)(1−κ ) Constrained Submodular Minimization Problem: Minimize a submodular function f over a family C of feasible sets, i.e., minX ∈C f (X ). C could be constraints of the form cardinality (knapsack) constraints, cuts, paths, matchings, trees etc. Main framework is to choose a surrogate function ˆf , and optimize it instead of f .I I Ellipsoidal Approximation based (EA): I Use the curvature based Ellipsoidal Approximation as the surrogate function. Lemma: For a submodular function with curvature κf < 1, algorithm EA will b that satisfiesreturn a solution X Modular Upper bound based: Use the simple modular upper bound as a surrogate. P bI Lemma: Let X ∈ C be the solution for minimizing j∈X f (j) over C. Then Pka I Corollary: The class of functions, f (X ) =λ[w(X)], λi ≥ 0, can beii=1 i minimized upto a factor of |X ∗|1−a.

Effect of Curvature: Polynomial change in the bounds!I I Experiments: ¯ I Define a function fR (X ) = κ min{|X ∩ R| + β, |X |, α} + (1 − κ)|X |. 1/2+ I Choose α = nand β = n2, and C = {X : |X | ≥ α}.

Acknowledgements Based upon work supported by National Science Foundation Grant No. IIS-1162606, and by a Google, a Microsoft, and an Intel research award. This was also funded in part by Office of Naval Research under grant no. N00014-11-1-0688, NSF CISE Expeditions award CCF-1139158, DARPA XData Award FA8750-12-2-0331, and gifts from Amazon Web Services, Google, SAP, Blue Goji, Cisco, Clearstory Data, Cloudera, Ericsson, Facebook, General Electric, Hortonworks, Intel, Microsoft, NetApp, Oracle, Samsung, Splunk, VMware and Yahoo!