On the hardness of learning intersections of two halfspaces

Khot, Subhash; Saket, Rishi

doi:10.1016/j.jcss.2010.06.010

Cited by 23 publications

(20 citation statements)

References 35 publications

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“…A key ingredient in our reductions is the smooth version of Label Cover that enables us to devise a more sophisticated decoding procedure that can be combined with the dictatorship test. It is pertinent to note that a couple of the (few) previous results using smooth versions of Label Cover, on hardness of learning intersection of halfspaces [Khot and Saket 2011] and monomials [Feldman et al 2009], have also used analysis based on versions of the Central Limit Theorem. Our results imply that for many geometric (and possibly other combinatorial optimization) problems, using Unique Games Conjecture is not necessary and the Smooth Label Cover (which is NPhard) suffices in its place.…”

Section: Discussion Of Our Resultsmentioning

confidence: 99%

Bypassing UGC from Some Optimal Geometric Inapproximability Results

Guruswami

Raghavendra

Saket

et al. 2016

ACM Trans. Algorithms

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Conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance nevertheless seems critical in these proofs. In this work, we bypass the need for UGC assumption in inapproximability results for two geometric problems, obtaining a tight NP-hardness result in each case. The first problem, known as L p Subspace Approximation, is a generalization of the classic least squares regression problem. Here, the input consists of a set of points X = {a 1 ,. .. , a m } ⊆ R n and a parameter k (possibly depending on n). The goal is to find a subspace H of R n of dimension k that minimizes the p norm of the Euclidean distances to the points in X. For p = 2, k = n− 1, this reduces to the least squares regression problem, while for p = ∞, k = 0 it reduces to the problem of finding a ball of minimum radius enclosing all the points. We show that for any fixed p ∈ (2, ∞), and for k = n− 1, it is NP-hard to approximate this problem to within a factor of γ p − for constant > 0, where γ p is the pth norm of a standard Gaussian random variable. This matches the γ p approximation algorithm obtained by Deshpande, Tulsiani, and Vishnoi who also showed the same hardness result under the UGC. The second problem we study is the related L p Quadratic Grothendieck Maximization Problem, considered by Kindler, Naor, and Schechtman. Here, the input is a multilinear quadratic form n i, j=1 a ij x i x j and the goal is to maximize the quadratic form over the p unit ball, namely, all x with n i=1 |x i | p 1. The problem is polynomial time solvable for p = 2. We show that for any constant p ∈ (2, ∞), it is NP-hard to approximate the quadratic form to within a factor of γ 2 p − for any > 0. The same hardness factor was shown under the UGC by Kindler et al. We also obtain a γ 2 p-approximation algorithm for the problem using the convex relaxation of the problem defined by Kindler et al. A γ 2 p approximation algorithm has also been independently obtained by Naor and Schechtman. These are the first approximation thresholds, proven under P = NP, that involve the Gaussian random variable in a fundamental way. Note that the problem statements themselves do not explicitly involve the Gaussian distribution.

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Section: Discussion Of Our Resultsmentioning

confidence: 99%

Bypassing UGC from Some Optimal Geometric Inapproximability Results

Guruswami

Raghavendra

Saket

et al. 2016

ACM Trans. Algorithms

Self Cite

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“…[33]). It is known that properly learning intersection of even 2 halfspaces is hard [32]. For improper learning, Klivans and Sherstov [35] have shown that learning an intersection of polynomially many half spaces is hard, under a certain cryptographic assumption regarding the shortest vector problem.…”

Section: Learning Intersection Of Halfspacesmentioning

confidence: 99%

From average case complexity to improper learning complexity

Daniely

Linial

Shalev‐Shwartz

2014

Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing

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The basic problem in the PAC model of computational learning theory is to determine which hypothesis classes are efficiently learnable. There is presently a dearth of results showing hardness of learning problems. Moreover, the existing lower bounds fall short of the best known algorithms.The biggest challenge in proving complexity results is to establish hardness of improper learning (a.k.a. representation independent learning). The difficulty in proving lower bounds for improper learning is that the standard reductions from NP-hard problems do not seem to apply in this context. There is essentially only one known approach to proving lower bounds on improper learning. It was initiated in [21] and relies on cryptographic assumptions.We introduce a new technique for proving hardness of improper learning, based on reductions from problems that are hard on average. We put forward a (fairly strong) generalization of Feige's assumption [13] about the complexity of refuting random constraint satisfaction problems. Combining this assumption with our new technique yields far reaching implications. In particular,• Learning DNF's is hard.• Agnostically learning halfspaces with a constant approximation ratio is hard.• Learning an intersection of ω(1) halfspaces is hard.

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“…An instance of LCPP is called δ-smooth if any two labels i = i of v ∈ V map to different labels of w ∈ W with probability at least 1 − δ over the choice of neighbors w of v. The smoothness property was introduced in [Kho02] and has been used for several hardness of approximation reductions [FGRW09,GRSW10,KS11]. The hardness factor achieved by the the reduction from LCPP to MWSPP is bounded by 1/δ and 1/s where δ is the smoothness parameter and s is the soundness of the LCPP instance.…”

Section: Main Results and Overviewmentioning

confidence: 99%