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Learning Valuation Functions
 25TH ANNUAL CONFERENCE ON LEARNING THEORY
, 2012
"... A core element of microeconomics and game theory is that consumers have valuation functions over bundles of goods and that these valuations functions drive their purchases. A common assumption is that these functions are subadditive meaning that the value given to a bundle is at most the sum of valu ..."
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Cited by 17 (2 self)
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A core element of microeconomics and game theory is that consumers have valuation functions over bundles of goods and that these valuations functions drive their purchases. A common assumption is that these functions are subadditive meaning that the value given to a bundle is at most the sum of values on the individual items. In this paper, we provide nearly tight guarantees on the efficient learnability of subadditive valuations. We also provide nearly tight bounds for the subclass of XOS (fractionally subadditive) valuations, also widely used in the literature. We additionally leverage the structure of valuations in a number of interesting subclasses and obtain algorithms with stronger learning guarantees.
Optimal bounds on approximation of submodular and xos functions by juntas
 CoRR
"... Abstract—We investigate the approximability of several classes of realvalued 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 `2error over the uniform distribu ..."
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Cited by 14 (5 self)
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Abstract—We investigate the approximability of several classes of realvalued 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 `2error over the uniform distribution: • If f is submodular, then it is close to a function of
Representation, approximation and learning of submodular functions using lowrank decision trees
 In Proceedings of the Conference on Learning Theory (COLT
, 2013
"... We study the complexity of approximate representation and learning of submodular functions over the uniform distribution on the Boolean hypercube {0, 1}n. Our main result is the following structural theorem: any submodular function is close in `2 to a realvalued decision tree (DT) of depth O(1/2) ..."
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Cited by 13 (8 self)
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We study the complexity of approximate representation and learning of submodular functions over the uniform distribution on the Boolean hypercube {0, 1}n. Our main result is the following structural theorem: any submodular function is close in `2 to a realvalued decision tree (DT) of depth O(1/2). This immediately implies that any submodular function is close to a function of at most 2O(1/ 2) variables and has a spectral `1 norm of 2O(1/ 2). It also implies the closest previous result that states that submodular functions can be approximated by polynomials of degree O(1/2) (Cheraghchi et al., 2012). Our result is proved by constructing an approximation of a submodular function by a DT of rank 4/2 and a proof that any rankr DT can be approximated by a DT of depth 52 (r + log(1/)). We show that these structural results can be exploited to give an attributeefficient PAC learning algorithm for submodular functions running in time Õ(n2) · 2O(1/4). The best previous algorithm for the problem requires nO(1/ 2) time and examples (Cheraghchi et al., 2012) but works also in the agnostic setting. In addition, we give improved learning algorithms for a number of related settings. We also prove that our PAC and agnostic learning algorithms are essentially optimal via two lower bounds: (1) an informationtheoretic lower bound of 2Ω(1/ 2/3) on the complexity of learning monotone submodular functions in any reasonable model (including learning with value queries); (2) computational lower bound of nΩ(1/ 2/3) based on a reduction to learning of sparse parities with noise, widelybelieved to be intractable. These are the first lower bounds for learning of submodular functions over the uniform distribution.
Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions
 IN NIPS
, 2013
"... We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAClike setting [28]), and constrained minimization of submodular functions. We show that the complexity of all three problems depe ..."
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Cited by 9 (6 self)
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We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAClike setting [28]), and constrained minimization of submodular functions. We show that the complexity of all three problems depends on the “curvature” of the submodular function, and provide lower and upper bounds that refine and improve previous results [2, 6, 8, 27]. Our proof techniques are fairly generic. We either use a blackbox transformation of the function (for approximation and learning), or a transformation of algorithms to use an appropriate surrogate function (for minimization). Curiously, curvature has been known to influence approximations for submodular maximization [3, 29], but its effect on minimization, approximation and learning has hitherto been open. We complete this picture, and also support our theoretical claims by empirical results.
Learning pseudoBoolean kDNF and submodular functions
, 2013
"... We prove that any submodular function f: {0, 1} n → {0, 1,..., k} can be represented as a pseudoBoolean 2kDNF formula. PseudoBoolean DNFs are a natural generalization of DNF representation for functions with integer range. Each term in such a formula has an associated integral constant. We show t ..."
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Cited by 6 (2 self)
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We prove that any submodular function f: {0, 1} n → {0, 1,..., k} can be represented as a pseudoBoolean 2kDNF formula. PseudoBoolean DNFs are a natural generalization of DNF representation for functions with integer range. Each term in such a formula has an associated integral constant. We show that an analog of H˚astad’s switching lemma holds for pseudoBoolean kDNFs if all constants associated with the terms of the formula are bounded. This allows us to generalize Mansour’s PAClearning algorithm for kDNFs to pseudoBoolean kDNFs, and hence gives a PAClearning algorithm with membership queries under the uniform distribution for submodular functions of the form f: {0, 1} n → {0, 1,..., k}. Our algorithm runs in time polynomial in n, k O(k log k/ɛ) and log(1/δ) and works even in the agnostic setting. The line of previous work on learning submodular functions [Balcan, Harvey (STOC ’11), Gupta, Hardt, Roth, Ullman; (STOC ’11), Cheraghchi, Klivans, Kothari, Lee
Submodular Functions: Learnability, Structure, and Optimization
, 2012
"... Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. They have been used in many areas, including combinatorial optimization, machine learning, and economics. In this work we study submodular functions from a learning theoret ..."
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Cited by 6 (0 self)
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Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. They have been used in many areas, including combinatorial optimization, machine learning, and economics. In this work we study submodular functions from a learning theoretic angle. We provide algorithms for learning submodular functions, as well as lower bounds on their learnability. In doing so, we uncover several novel structural results revealing ways in which submodular functions can be both surprisingly structured and surprisingly unstructured. We provide several concrete implications of our work in other domains including algorithmic game theory and combinatorial optimization. At a technical level, this research combines ideas from many areas, including learning theory (distributional learning and PACstyle analyses), combinatorics and optimization (matroids and submodular functions), and pseudorandomness (lossless expander graphs).
Testing coverage functions
, 2012
"... A coverage function f over a ground set [m] is associated with a universe U of weighted elements and m sets A1,..., Am ⊆ U, and for any T ⊆ [m], f(T) is defined as the total weight of the elements in the union ∪j∈T Aj. Coverage functions are an important special case of submodular functions, and ar ..."
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Cited by 5 (0 self)
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A coverage function f over a ground set [m] is associated with a universe U of weighted elements and m sets A1,..., Am ⊆ U, and for any T ⊆ [m], f(T) is defined as the total weight of the elements in the union ∪j∈T Aj. Coverage functions are an important special case of submodular functions, and arise in many applications, for instance as a class of utility functions of agents in combinatorial auctions. Set functions such as coverage functions often lack succinct representations, and in algorithmic applications, an access to a value oracle is assumed. In this paper, we ask whether one can test if a given oracle is that of a coverage function or not. We demonstrate an algorithm which makes O(mU) queries to an oracle of a coverage function and completely reconstructs it. This gives a polytime tester for succinct coverage functions for which U  is polynomially bounded in m. In contrast, we demonstrate a set function which is “far ” from coverage, but requires 2 ˜ Θ(m) queries to distinguish it from the class of coverage functions.
Influence function learning in information diffusion networks
, 2014
"... Can we learn the influence of a set of people in a social network from cascades of information diffusion? This question is often addressed by a twostage approach: first learn a diffusion model, and then calculate the influence based on the learned model. Thus, the success of this approach relies ..."
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Cited by 5 (1 self)
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Can we learn the influence of a set of people in a social network from cascades of information diffusion? This question is often addressed by a twostage approach: first learn a diffusion model, and then calculate the influence based on the learned model. Thus, the success of this approach relies heavily on the correctness of the diffusion model which is hard to verify for real world data. In this paper, we exploit the insight that the influence functions in many diffusion models are coverage functions, and propose a novel parameterization of such functions using a convex combination of random basis functions. Moreover, we propose an efficient maximum likelihood based algorithm to learn such functions directly from cascade data, and hence bypass the need to specify a particular diffusion model in advance. We provide both theoretical and empirical analysis for our approach, showing that the proposed approach can provably learn the influence function with low sample complexity, be robust to the unknown diffusion models, and significantly outperform existing approaches in both synthetic and real world data. 1.
Learning coverage functions and private release of marginals
 In COLT
, 2014
"... We study the problem of approximating and learning coverage functions. A function c: 2[n] → R+ is a coverage function, if there exists a universe U with nonnegative weights w(u) for each u ∈ U and subsets A1, A2,..., An of U such that c(S) = u∈∪i∈SAi w(u). Alternatively, coverage functions can be ..."
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Cited by 3 (1 self)
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We study the problem of approximating and learning coverage functions. A function c: 2[n] → R+ is a coverage function, if there exists a universe U with nonnegative weights w(u) for each u ∈ U and subsets A1, A2,..., An of U such that c(S) = u∈∪i∈SAi w(u). Alternatively, coverage functions can be described as nonnegative linear combinations of monotone disjunctions. They are a natural subclass of submodular functions and arise in a number of applications. We give an algorithm that for any γ, δ> 0, given random and uniform examples of an unknown coverage function c, finds a function h that approximates c within factor 1 + γ on all but δfraction of the points in time poly(n, 1/γ, 1/δ). This is the first fullypolynomial algorithm for learning an interesting class of functions in the demanding PMAC model of Balcan and Harvey [2012]. Our algorithms are based on several new structural properties of coverage functions. Using the results in [Feldman and Kothari, 2014], we also show that coverage functions are learnable agnostically with excess `1error over all product and symmetric distributions in time nlog(1/). In contrast, we show that, without assumptions on the distribution, learning coverage functions is at least as hard as learning polynomialsize disjoint DNF formulas, a class of functions for which the best known algorithm runs in time 2Õ(n 1/3) [Klivans and Servedio, 2004]. As an application of our learning results, we give simple differentiallyprivate algorithms for releasing monotone conjunction counting queries with low average error. In particular, for any k ≤ n, we obtain private release of kway marginals with average error α ̄ in time nO(log(1/ᾱ)). 1