Boolean function analysis meets stochastic optimization: An approximation scheme for stochastic knapsack

De, Anindya

doi:10.1137/1.9781611975031.84

Cited by 5 publications

(4 citation statements)

References 42 publications

(111 reference statements)

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“…We will apply an anti-concentration inequality to the random variable D i f (ρ) = f ↾ρ (i) (see Lemma 13 in [11]): for any real-valued random variable X with variance at least σ 2 and central fourth moment at most t 4 σ 4 ,…”

Section: Proof Of Claim 37 Formentioning

confidence: 99%

Junta Correlation is Testable

Mossel

Neeman

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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The problem of tolerant junta testing is a natural and challenging problem which asks if the property of a function having some specified correlation with a k-Junta is testable. In this paper we give an affirmative answer to this question: We show that given distance parameters 1 2 > c u > c ℓ ≥ 0, there is a tester which given oracle access to f : {−1, 1} n → {−1, 1}, with query complexity 2 k · poly(k, 1/|c u − c ℓ |) and distinguishes between the following cases:

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Section: Proof Of Claim 37 Formentioning

confidence: 99%

Junta Correlation is Testable

Mossel

Neeman

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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“…Goyal and Ravi [17] proposed a PTAS by solving the reformulated two-dimensional BKP and rounding its solution for the CKP with independent normally distributed weights. Recently, De [13] proposed fully polynomial-time approximation schemes (FPTAS) for the CKP with Bernoulli distributed weights. A PTAS for the CKP with random weights whose variation and kurtosis are bounded was also designed by De [13].…”

Section: Introductionmentioning

confidence: 99%

“…Recently, De [13] proposed fully polynomial-time approximation schemes (FPTAS) for the CKP with Bernoulli distributed weights. A PTAS for the CKP with random weights whose variation and kurtosis are bounded was also designed by De [13].…”

Section: Introductionmentioning

confidence: 99%

Robust solutions for stochastic and distributionally robust chance-constrained binary knapsack problems

Ryu¹

2021

Preprint

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We consider chance-constrained binary knapsack problems, where the weights of items are independent random variables with the means and standard deviations known. The chance constraint can be reformulated as a second-order cone constraint under some assumptions for the probability distribution of the weights. The problem becomes a second-order cone-constrained binary knapsack problem, which is equivalent to a robust binary knapsack problem with an ellipsoidal uncertainty set.We demonstrate that optimal solutions to robust binary knapsack problems with inner and outer polyhedral approximations of the ellipsoidal uncertainty set can provide both upper and lower bounds on the optimal value of the second-order cone-constrained binary knapsack problem, and they can be obtained by solving ordinary binary knapsack problems repeatedly. Moreover, we prove that the solution providing the upper bound converges to the optimal solution to the secondorder cone-constrained binary knapsack problem as the approximation of the uncertainty set becomes more accurate. Based on this, a pseudo-polynomial time algorithm is obtained under the assumption that the coefficients of the problem are integer-valued. We also propose an exact algorithm, which iteratively improves the accuracy of the approximation until an exact optimal solution is obtained. The exact algorithm also runs in pseudo-polynomial time. Computational results are presented to show the quality of the upper and lower bounds and efficiency of the exact algorithm compared to CPLEX and a previous study.

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“…Indeed, the technical workhorses of all these results are various central limit theorems which crucially exploit the simple structure of these collective support sets. (These central limit theorems have since found applications in other settings, such as the design of algorithms for approximating equilibrium [DDKT16,DKT15,DKS16c,CDS17] as well as stochastic optimization [De18]. )…”

Section: Introductionmentioning

confidence: 99%

Learning Sums of Independent Random Variables with Sparse Collective Support

Long

Servedio

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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We study the learnability of sums of independent integer random variables given a bound on the size of the union of their supports. For A ⊂ Z + , a sum of independent random variables with collective support A (called an A-sum in this paper) is a distribution S = X 1 + · · · + X N where the X i 's are mutually independent (but not necessarily identically distributed) integer random variables with ∪ i supp(X i ) ⊆ A.We give two main algorithmic results for learning such distributions:1. For the case |A| = 3, we give an algorithm for learning A-sums to accuracy ε that uses poly(1/ε) samples and runs in time poly(1/ε), independent of N and of the elements of A.2. For an arbitrary constant k ≥ 4, if A = {a 1 , ..., a k } with 0 ≤ a 1 < ... < a k , we give an algorithm that uses poly(1/ε) · log log a k samples (independent of N ) and runs in time poly(1/ε, log a k ).We prove an essentially matching lower bound: if |A| = 4, then any algorithm must use Ω(log log a 4 ) samples even for learning to constant accuracy. We also give similar-in-spirit (but quantitatively very different) algorithmic results, and essentially matching lower bounds, for the case in which A is not known to the learner. Our learning algorithms employ new limit theorems which may be of independent interest. Our lower bounds rely on equidistribution type results from number theory. Our algorithms and lower bounds together settle the question of how the sample complexity of learning sums of independent integer random variables scales with the elements in the union of their supports, both in the known-support and unknownsupport settings. Finally, all our algorithms easily extend to the "semi-agnostic" learning model, in which training data is generated from a distribution that is only cε-close to some A-sum for a constant c > 0.Secondary algorithmic results: Learning with unknown support. We also give algorithms for a more challenging unknown-support variant of the learning problem. In this variant the values a 1 , . . . , a k are not provided to the learning algorithm, but instead only an upper bound a max ≥ a k is given. Interestingly, it turns out that the unknown-support problem is significantly different from the known-support problem: as explained below, in the unknown-support variant the dependence on a max kicks in at a smaller value of k 1 Here and throughout we assume a unit-cost model for arithmetic operations +, ×, ÷. 4 than in the known-support variant, and this dependence is exponentially more severe than in the knownsupport variant.Using well-known results from hypothesis selection, it is straightforward to show that upper bounds for the known-support case yield upper bounds in the unknown-support case, essentially at the cost of an additional additive O(k log a max )/ε 2 term in the sample complexity. This immediately yields the following:Theorem 3 (Learning with unknown support of size k). For any k ≥ 3, there is an algorithm and a positive constant c with the following properties: The algorithm is given N , the value k, an accuracy pa...

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Boolean function analysis meets stochastic optimization: An approximation scheme for stochastic knapsack

Cited by 5 publications

References 42 publications

Junta Correlation is Testable

Junta Correlation is Testable

Robust solutions for stochastic and distributionally robust chance-constrained binary knapsack problems

Learning Sums of Independent Random Variables with Sparse Collective Support

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