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Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas [article]

Vitaly Feldman, Jan Vondrak
2015 arXiv   pre-print
We investigate the approximability of several classes of real-valued functions by functions of a small number of variables ( juntas).  ...  Our main results are tight bounds on the number of variables required to approximate a function f:{0,1}^n → [0,1] within ℓ_2-error ϵ over the uniform distribution: 1.  ...  Acknowledgements We would like to thank Seshadhri Comandur and Pravesh Kothari for useful discussion. We also thank the anonymous FOCS and SICOMP referees for their comments and useful suggestions.  ... 
arXiv:1307.3301v3 fatcat:tfwzel6yajhk7jznjs2gbacvim

Optimal bounds on approximation of submodular and XOS functions by juntas

Vitaly Feldman, Jan Vondrak
2014 2014 Information Theory and Applications Workshop (ITA)  
It relies crucially on our approximation by junta result. As follows from the lower bounds in [1] both of these algorithms are close to optimal.  ...  We investigate the approximability of several classes of real-valued functions by functions of a small number of variables (juntas).  ...  ACKNOWLEDGEMENTS We would like to thank Seshadhri Comandur, Pravesh Kothari and the anonymous FOCS referees for their comments and useful suggestions.  ... 
doi:10.1109/ita.2014.6804263 dblp:conf/ita/FeldmanV14 fatcat:gxjyiofvrfdyxdgf6dfyrkdsaa

Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas

Vitaly Feldman, Jan Vondrak
2013 2013 IEEE 54th Annual Symposium on Foundations of Computer Science  
It relies crucially on our approximation by junta result. As follows from the lower bounds in [1] both of these algorithms are close to optimal.  ...  We investigate the approximability of several classes of real-valued functions by functions of a small number of variables (juntas).  ...  ACKNOWLEDGEMENTS We would like to thank Seshadhri Comandur, Pravesh Kothari and the anonymous FOCS referees for their comments and useful suggestions.  ... 
doi:10.1109/focs.2013.32 dblp:conf/focs/FeldmanV13 fatcat:wpq2nientjb2he6em6psgfwwz4

Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas

Vitaly Feldman, Jan Vondrák
2016 SIAM journal on computing (Print)  
It relies crucially on our approximation by junta result. As follows from the lower bounds in [1] both of these algorithms are close to optimal.  ...  We investigate the approximability of several classes of real-valued functions by functions of a small number of variables (juntas).  ...  ACKNOWLEDGEMENTS We would like to thank Seshadhri Comandur, Pravesh Kothari and the anonymous FOCS referees for their comments and useful suggestions.  ... 
doi:10.1137/140958207 fatcat:id7g5y2pfbcwjij6u2yhtishp4

Structure and learning of valuation functions

Vitaly Feldman, Jan Vondrák
2014 ACM SIGecom Exchanges  
We discuss structural results and learning algorithms for submodular and fractionally subadditive valuation functions.  ...  While learning these valuation functions over general distributions turns out to be hard, we present compact approximate representations and efficient learning algorithms for such functions over the uniform  ...  XOS functions are 1-selfbounding and submodular functions are 2-self-bounding. However, subadditive and a-self-bounding functions are incomparable.  ... 
doi:10.1145/2692359.2692371 fatcat:4njjqvdmsrcqlmmt7ydo5s4fee

Tight Bounds on Low-Degree Spectral Concentration of Submodular and XOS Functions

Vitaly Feldman, Jan Vondrak
2015 2015 IEEE 56th Annual Symposium on Foundations of Computer Science  
This improves on previous approaches that all showed an upper bound of O(1/ 2 ) for submodular [CKKL12, FKV13, FV13] and XOS [FV13] functions.  ...  Our techniques reveal new structural properties of submodular and XOS functions and the upper bounds lead to nearly optimal PAC learning algorithms for these classes of functions. 1 Here and below we normalize  ...  Acknowledgements We would like to thank Pravesh Kothari for useful discussions and his help with the proof of Lemma 6.1.  ... 
doi:10.1109/focs.2015.61 dblp:conf/focs/FeldmanV15 fatcat:er54355hsfhkvdb34nf2vddr7a

Tight Bounds on ℓ_1 Approximation and Learning of Self-Bounding Functions [article]

Vitaly Feldman, Pravesh Kothari, Jan Vondrák
2019 arXiv   pre-print
Self-bounding functions include such well-known classes of functions as submodular and fractionally-subadditive (XOS) functions.  ...  Our main result is a nearly tight ℓ_1-approximation of self-bounding functions by low-degree juntas.  ...  Feldman and Vondrák [FV16] studied approximation of submodular, XOS and self-bounding functions by juntas.  ... 
arXiv:1404.4702v3 fatcat:wnmchrdiubeujertmzg3zz6u5m

Tight Bounds on Low-degree Spectral Concentration of Submodular and XOS functions [article]

Vitaly Feldman, Jan Vondrak
2015 arXiv   pre-print
This improves on previous approaches that all showed an upper bound of O(1/ϵ^2) for submodular and XOS functions. The best previous lower bound was Ω(1/ϵ^2/3) for monotone submodular functions.  ...  Our techniques reveal new structural properties of submodular and XOS functions and the upper bounds lead to nearly optimal PAC learning algorithms for these classes of functions.  ...  Acknowledgements We would like to thank Pravesh Kothari for useful discussions and his help with the proof of Lemma 6.1.  ... 
arXiv:1504.03391v2 fatcat:y5rwgeraafh67lzpdmgag4esci

Tight Bounds on ℓ 1 Approximation and Learning of Self-Bounding Functions

Vitaly Feldman, Pravesh Kothari, Jan Vondrák
2017 International Conference on Algorithmic Learning Theory  
Self-bounding functions include such wellknown classes of functions as submodular and fractionally-subadditive (XOS) functions.  ...  Our bounds rely on the analysis of noise stability of self-bounding functions together with a stronger connection between noise stability and 1 approximation by low-degree polynomials.  ...  The main tool used in this work is the notion of noise stability. Feldman and Vondrák (2016) studied approximation of submodular, XOS and self-bounding functions by juntas.  ... 
dblp:conf/alt/FeldmanKV17 fatcat:veyihsja3vcxnmyxvhju5o7tdy

Testing Submodularity and Other Properties of Valuation Functions

Eric Blais, Abhinav Bommireddi, Marc Herbstritt
2017 Innovations in Theoretical Computer Science  
coverage functions, and self-bounding functions.  ...  This result, combined with a recent junta theorem of Feldman and Vondrák ( 2016 ), yields the constant-query testability of submodularity.  ...  The authors wish to thank the anonymous referees for valuable feedback, Amit Levi for insightful discussions and Karl Wimmer for pointing out [31] to us.  ... 
doi:10.4230/lipics.itcs.2017.33 dblp:conf/innovations/BlaisB17 fatcat:o7vrglavaff2rkad2uoo2l3bz4

Testing submodularity and other properties of valuation functions [article]

Eric Blais, Abhinav Bommireddi
2016 arXiv   pre-print
coverage functions, and self-bounding functions.  ...  This result, combined with a recent junta theorem of Feldman and Vondrak (2016), yields the constant-query testability of submodularity.  ...  through the lenses of learning theory [2, 3, 15] , optimization [13, 14] , approximation [16, 17] , and sketching [1] .  ... 
arXiv:1611.07879v1 fatcat:3p3yvqz3jfcwxife35jdxvysqu

Constrained signaling for welfare and revenue maximization

Shaddin Dughmi, Nicole Immorlica, Aaron Roth
2013 ACM SIGecom Exchanges  
Each candidate signaling scheme induces an incompleteinformation game among the buyers, and the goal of the mechanism designer is to choose the signaling scheme that optimizes either welfare or revenue  ...  For example, the auctioneer may be able to communicate only a bounded length message for each good (equivalently, he may be constrained to using only a fixed number of signals in total).  ...  4.1.3 we show how to express optimization over these mappings as submodular function maximization subject to a matroid 9 constraint, which admits an e/(e − 1) approximation algorithm by the result of  ... 
doi:10.1145/2509013.2509022 fatcat:auj2ved2bvf6pgfwli6llrhyxi

Approximate F_2-Sketching of Valuation Functions

Grigory Yaroslavtsev, Samson Zhou, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization  
α-Lipschitz submodular and matroid rank functions.  ...  Using known connections with dynamic streaming algorithms, both upper and lower bounds on dimension obtained in our work extend to the space complexity of algorithms evaluating f (x) under long sequences  ...  of classes for which we already have an Ω(n) lower bound (e.g. submodular, subadditive, etc.) or because such a lower bound follows trivially (e.g. for OXS/XOS since for XS-functions a lower bound of  ... 
doi:10.4230/lipics.approx-random.2019.69 dblp:conf/approx/YaroslavtsevZ19 fatcat:dqv5xlu2krfrhoolutor6q7vi4

Representation, Approximation and Learning of Submodular Functions Using Low-rank Decision Trees [article]

Vitaly Feldman and Pravesh Kothari and Jan Vondrak
2013 arXiv   pre-print
Our result is proved by constructing an approximation of a submodular function by a DT of rank 4/ϵ^2 and a proof that any rank-r DT can be ϵ-approximated by a DT of depth 5/2(r+(1/ϵ)).  ...  We study the complexity of approximate representation and learning of submodular functions over the uniform distribution on the Boolean hypercube {0,1}^n.  ...  There exists an algorithm AEFT, that given an integer d, θ > 0 and δ ∈ (0, 1], access to value queries of any f : {0, 1} n → [−1, 1], with probability at least 1 − δ, returns a function h represented by  ... 
arXiv:1304.0730v1 fatcat:utrdrx5nrrffzpidqpho5vjzme

Learning Coverage Functions and Private Release of Marginals [article]

Vitaly Feldman, Pravesh Kothari
2014 arXiv   pre-print
We study the problem of approximating and learning coverage functions.  ...  They are a natural subclass of submodular functions and arise in a number of applications.  ...  Acknowledgements We are grateful to Jan Vondrak for helpful advice and numerous discussions about this work.  ... 
arXiv:1304.2079v3 fatcat:lor5pjhisfd4vo534nxcwml2im
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