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f6cc031 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 | <Poster Width="1003" Height="1337">
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<Text>Overview</Text>
<Text>Introduce the notion of curvature, to provide better connections between theory</Text>
<Text>and practice.</Text>
<Text>Study the role of curvature in:I</Text>
<Text>I Study the role of curvature in:</Text>
<Text>Approximating submodular functions everywhere</Text>
<Text>Learning Submodular functions</Text>
<Text>Constrained Minimization of submodular functions.</Text>
<Text>I Provide improved curvature-dependent worst case approximation guarantees</Text>
<Text>and matching hardness results</Text>
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<Text>Curvature of a Submodular function</Text>
<Text>Define three variants of curvature of a monotone submodular functionas:I</Text>
<Text>Proposition: κˆf (S) ≤ κf (S) ≤ κf .I</Text>
<Text>I Captures the linearity of a submodular function.</Text>
<Text>I A more gradual characterization of the hardness of</Text>
<Text>various problems.</Text>
<Text>I Investigated for submodular maximization</Text>
<Text>(Conforti & Cornuejols, 1984).</Text>
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<Text>Approximating Submodular functions Everywhere</Text>
<Text>Problem:Given a submodular function f in form of a value oracle,</Text>
<Text>find an approximation ˆf (within polynomial time and space), such that</Text>
<Text>ˆf (X ) ≤ f (X ) ≤ α1(n)ˆf (X ), ∀X ⊆ V for a polynomial α1(n).</Text>
<Text>We provide a blackbox technique to transform bounds into curvature dependent</Text>
<Text>ones.I</Text>
<Text>κ</Text>
<Text>I Main technique: Approximate the curve-normalized version f as ˆf κ, such that</Text>
<Text>ˆf κ(X ) ≤ f κ(X ) ≤ α(n)ˆf κ(X ).</Text>
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<Text>Ellipsoidal Approximation:I</Text>
<Text>I The Ellipsoidal Approximation algorithm of Goemans et al, provides a function</Text>
<Text>p√</Text>
<Text>of the form w f (X ) with an approximation factor of α1(n) = O( n log n)).</Text>
<Text>I Corollary: There exists a function of the form,</Text>
<Text>Lower bound: Given a submodular function f with curvature κf , there does not</Text>
<Text>exist any polynomial-time algorithm that approximates f within a factor of</Text>
<Text>n1/2−</Text>
<Text>, for any > 0.</Text>
<Text>1+(n1/2−−1)(1−κ )fI</Text>
<Text>Modular Upper Bound:I</Text>
<Text>Pmˆ (X ) =I A simplest approximation (and upper bound) is f</Text>
<Text>j∈X f (j).</Text>
<Text>j∈X </Text>
<Text>I Lemma: Given a monotone submodular function f , it holds that,</Text>
<Text>This bound is tight for the class of modularapproximations.I</Text>
<Text>Pka</Text>
<Text>I Corollary: The class of functions, f (X ) =λ[w(X)],λ≥0,satisfiesiiii=1</Text>
<Text>Pf (X ) ≤ </Text>
<Text>j∈X f (j) ≤ |X |1−af (X ).</Text>
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<Text>Learning Submodular Functions</Text>
<Text>Problem: Given i.i.d training samples {(Xi , f (Xi )}m</Text>
<Text>i=1 from a distribution D, learn</Text>
<Text>an approximation ˆf (X ) that is, with probability 1 − δ, within a multiplicative factor of</Text>
<Text>α2(n) from f .</Text>
<Text>Balcan & Harvey proposeanalgorithmwhichPMAClearnsanysubmodular√</Text>
<Text>function upto a factor of n + 1.</Text>
<Text>We improve this bound to a curvature dependent one.</Text>
<Text>Lemma: Let f be a monotone submodular function for which we know an upper</Text>
<Text>bound on its curvature κf and the singleton weights f (j) for all j ∈ V</Text>
<Text>√ . There is</Text>
<Text>n+1.an poly-time algorithm which PMAC-learns f within a factor of </Text>
<Text>1+(√</Text>
<Text>n+1−1)(1−κ)</Text>
<Text>We also provide an algorithm which does not need the singleton weights.</Text>
<Text>Lemma: If f is a monotone submodular function with known curvature (or a</Text>
<Text>known upper bound) κˆf (X ), ∀X ⊆ V , then for every , δ > 0 there is an algorithm</Text>
<Text>|X |which PMAC learns f (X ) within a factor of 1 + </Text>
<Text>1+(|X |−1)(1−κˆf (X )) .</Text>
<Text>Pka</Text>
<Text>I Corollary: The class of functions f (X ) =λ[w(X)], λi ≥ 0, can be learnt toii=1 i</Text>
<Text>a factor of |X |1−a.</Text>
<Text>I Lower bound: Given a class of submodular functions with curvature κf , there</Text>
<Text>does not exist a polynomial-time algorithm that is guaranteed to PMAC-learn f</Text>
<Text>1/3−0</Text>
<Text>n0within a factor of </Text>
<Text>1+(n1/3−,forany>0.0−1)(1−κ )</Text>
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<Text>Constrained Submodular Minimization</Text>
<Text>Problem: Minimize a submodular function f over a family C of feasible sets, i.e.,</Text>
<Text>minX ∈C f (X ). C could be constraints of the form cardinality (knapsack) constraints,</Text>
<Text>cuts, paths, matchings, trees etc.</Text>
<Text>Main framework is to choose a surrogate function ˆf , and optimize it instead of f .I</Text>
<Text>I Ellipsoidal Approximation based (EA):</Text>
<Text>I Use the curvature based Ellipsoidal Approximation as the surrogate function.</Text>
<Text>Lemma: For a submodular function with curvature κf < 1, algorithm EA will</Text>
<Text>b that satisfiesreturn a solution X</Text>
<Text>Modular Upper bound based:</Text>
<Text>Use the simple modular upper bound as a surrogate.</Text>
<Text>P</Text>
<Text>bI Lemma: Let X ∈ C be the solution for minimizing</Text>
<Text>j∈X f (j) over C. Then</Text>
<Text>Pka</Text>
<Text>I Corollary: The class of functions, f (X ) =λ[w(X)], λi ≥ 0, can beii=1 i</Text>
<Text>minimized upto a factor of |X ∗|1−a.</Text>
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<Text>Effect of Curvature: Polynomial change in the bounds!I</Text>
<Text>I Experiments:</Text>
<Text>¯</Text>
<Text>I Define a function fR (X ) = κ min{|X ∩ R| + β, |X |, α} + (1 − κ)|X |.</Text>
<Text>1/2+</Text>
<Text>I Choose α = nand β = n2, and C = {X : |X | ≥ α}.</Text>
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<Text>Acknowledgements</Text>
<Text>Based upon work supported by National Science Foundation Grant No. IIS-1162606,</Text>
<Text>and by a Google, a Microsoft, and an Intel research award. This was also funded in part</Text>
<Text>by Office of Naval Research under grant no. N00014-11-1-0688, NSF CISE Expeditions award</Text>
<Text>CCF-1139158, DARPA XData Award FA8750-12-2-0331, and gifts from Amazon Web Services,</Text>
<Text>Google, SAP, Blue Goji, Cisco, Clearstory Data, Cloudera, Ericsson, Facebook, General</Text>
<Text>Electric, Hortonworks, Intel, Microsoft, NetApp, Oracle, Samsung, Splunk, VMware and Yahoo!</Text>
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