Upper Bounds in Classical Discrepancy Theory

Chen, William; Skriganov, M. M.

doi:10.1007/978-3-319-04696-9_1

Cited by 6 publications

(5 citation statements)

References 27 publications

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“…For envy-freeness, we consider a similar additive approximation, EF-r, where an agent's utility must not increase by more than r when swapping places with another agent. We make a connection to discrepancy theory (Chen et al 2014) to show that a EF-O( n k • log k) partition always exists, and it can be computed efficiently. We conjecture that a EF-2 partition may always exists for any k.…”

Section: Our Resultsmentioning

confidence: 99%

“…Instead, our focus is on providing worst-case guarantees on the necessary violation of envy-freeness, as is commonly done in the literature on fair resource allocation (Lipton et al 2004;Caragiannis et al 2019;Aziz et al 2019). We make a connection to discrepancy theory (Chen et al 2014) to establish an O( √ n) bound. In discrepancy theory, the goal is to distribute each agent's friends as evenly as possible between the parts, so that not only does an agent not have many more friends in another part than her own part, she also does not have many more friends in her own part than in any other part.…”

Section: Related Workmentioning

confidence: 99%

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Partitioning Friends Fairly

Micha

Nikolov

et al. 2023

AAAI

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We consider the problem of partitioning n agents in an undirected social network into k almost equal in size (differing by at most one) groups, where the utility of an agent for a group is the number of her neighbors in the group. The core and envy-freeness are two compelling axiomatic fairness guarantees in such settings. The former demands that there be no coalition of agents such that each agent in the coalition has more utility for that coalition than for her own group, while the latter demands that no agent envy another agent for the group they are in. We provide (often tight) approximations to both fairness guarantees, and many of our positive results are obtained via efficient algorithms.

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Section: Our Resultsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Partitioning Friends Fairly

Micha

Nikolov

et al. 2023

AAAI

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show abstract

“…Consider a number of points N R from a sequence {θ i }, for i = 1, .., N , in an n-dimensional rectangle R centred upon an origin 0, whose sides are parallel to the coordinate axis, which is a subset of I n : R ⊂ I n , where R is attached with a measure. A sequence has low discrepancy if the proportion of points in the sequence falling into an arbitrary set R is close to the measure of R. LDS satisfies the upper bound condition [33]:…”

Section: Quasi Monte-carlo Sampling Methodsmentioning

confidence: 99%

“…A sequence has low discrepancy if the proportion of points in the sequence falling into an arbitrary set R is close to the measure of R . LDS satisfies the upper bound condition [33]: where D N is the sample discrepancy and k ( n ) is a particular constant depending on the sequence and size of input paraeter space. LDS is designed to place sample points as uniformly as possible mathematically, within a hypercube, instead of the statistical approach adopted in LH.…”

Section: Methodsmentioning

confidence: 99%

Convergence, Sampling and Total Order Estimator Effects on Parameter Orthogonality in Global Sensitivity Analysis

Saxton,

Xu,

Schenkel

et al. 2024

Preprint

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Dynamical system models typically involve numerous input parameters whose ``effects'' and orthogonality need to be quantified through sensitivity analysis, to identify inputs contributing the greatest uncertainty. Whilst prior art has compared total-order estimators’ role in recovering “true” effects, assessing their ability to recover robust parameter orthogonality for use in identifiability metrics has not been investigated. In this paper, we perform: (i) an assessment using a different class of numerical models representing the cardiovascular system, (ii) a wider evaluation of sampling methodologies and their interactions with estimators, (iii) an investigation of the consequences of permuting estimators and sampling methodologies on input parameter orthogonality, (iv) a study of sample convergence through resampling, and (v) an assessment of whether positive outcomes are sustained when model input dimensionality increases. Our results indicate that Jansen or Janon estimators display efficient convergence with minimum uncertainty when coupled with Sobol and the lattice rule sampling methods, making them prime choices for calculating parameter orthogonality and influence. This study reveals that global sensitivity analysis is convergence driven. Unconverged indices are subject to error and therefore the true influence or orthogonality of the input parameters are not recovered. This investigation importantly clarifies the interactions of the estimator and the sampling methodology by reducing the associated ambiguities, defining novel practices for modelling in the life sciences.

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“…A sequence has low discrepancy if the proportion of points in the sequence falling into an arbitrary set R is close to the measure of R . LDS satisfies the upper bound condition [ 33 ]: where D N is the sample discrepancy and k ( n ) is a particular constant depending on the sequence and size of input paraeter space. LDS is designed to place sample points as uniformly as possible mathematically, within a hypercube, instead of the statistical approach adopted in LH.…”

Section: Methodsmentioning

confidence: 99%

Convergence, sampling and total order estimator effects on parameter orthogonality in global sensitivity analysis

Saxton,

Xu,

Schenkel

et al. 2024

PLoS Comput Biol

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Dynamical system models typically involve numerous input parameters whose “effects” and orthogonality need to be quantified through sensitivity analysis, to identify inputs contributing the greatest uncertainty. Whilst prior art has compared total-order estimators’ role in recovering “true” effects, assessing their ability to recover robust parameter orthogonality for use in identifiability metrics has not been investigated. In this paper, we perform: (i) an assessment using a different class of numerical models representing the cardiovascular system, (ii) a wider evaluation of sampling methodologies and their interactions with estimators, (iii) an investigation of the consequences of permuting estimators and sampling methodologies on input parameter orthogonality, (iv) a study of sample convergence through resampling, and (v) an assessment of whether positive outcomes are sustained when model input dimensionality increases. Our results indicate that Jansen or Janon estimators display efficient convergence with minimum uncertainty when coupled with Sobol and the lattice rule sampling methods, making them prime choices for calculating parameter orthogonality and influence. This study reveals that global sensitivity analysis is convergence driven. Unconverged indices are subject to error and therefore the true influence or orthogonality of the input parameters are not recovered. This investigation importantly clarifies the interactions of the estimator and the sampling methodology by reducing the associated ambiguities, defining novel practices for modelling in the life sciences.

show abstract

Upper Bounds in Classical Discrepancy Theory

Cited by 6 publications

References 27 publications

Partitioning Friends Fairly

Partitioning Friends Fairly

Convergence, Sampling and Total Order Estimator Effects on Parameter Orthogonality in Global Sensitivity Analysis

Convergence, sampling and total order estimator effects on parameter orthogonality in global sensitivity analysis

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