arXiv subjects

Inbal Livni-Navon

Publications and source records attributed to Inbal Livni-Navon.

3 recordsLinked to original sources

arXiv2023

Generative Models of Huge Objects

This work initiates the systematic study of explicit distributions that are indistinguishable from a single exponential-size combinatorial object. In this we extend the work of Goldreich, Goldwasser and Nussboim (SICOMP 2010) that focused on the implementation of huge objects that are indistinguishable from the uniform distribution, satisfying some global properties (which they coined truthfulness). Indistinguishability from a single object is motivated by the study of generative models in learning theory and regularity lemmas in graph theory. Problems that are well understood in the setting of pseudorandomness present significant challenges and at times are impossible when considering generative models of huge objects. We demonstrate the versatility of this study by providing a learning algorithm for huge indistinguishable objects in several natural settings including: dense functions and graphs with a truthfulness requirement on the number of ones in the function or edges in the graphs, and a version of the weak regularity lemma for sparse graphs that satisfy some global properties. These and other results generalize basic pseudorandom objects as well as notions introduced in algorithmic fairness. The results rely on notions and techniques from a variety of areas including learning theory, complexity theory, cryptography, and game theory.

cs.CC↗

arXiv2023

Omnipredictors for Constrained Optimization

The notion of omnipredictors (Gopalan, Kalai, Reingold, Sharan and Wieder ITCS 2021), suggested a new paradigm for loss minimization. Rather than learning a predictor based on a known loss function, omnipredictors can easily be post-processed to minimize any one of a rich family of loss functions compared with the loss of hypotheses in a class $\mathcal C$. It has been shown that such omnipredictors exist and are implied (for all convex and Lipschitz loss functions) by the notion of multicalibration from the algorithmic fairness literature. In this paper, we introduce omnipredictors for constrained optimization and study their complexity and implications. The notion that we introduce allows the learner to be unaware of the loss function that will be later assigned as well as the constraints that will be later imposed, as long as the subpopulations that are used to define these constraints are known. We show how to obtain omnipredictors for constrained optimization problems, relying on appropriate variants of multicalibration. We also investigate the implications of this notion when the constraints used are so-called group fairness notions.

cs.LG↗

arXiv2020

On Hardness of Approximation of Parameterized Set Cover and Label Cover: Threshold Graphs from Error Correcting Codes

In the $(k,h)$-SetCover problem, we are given a collection $\mathcal{S}$ of sets over a universe $U$, and the goal is to distinguish between the case that $\mathcal{S}$ contains $k$ sets which cover $U$, from the case that at least $h$ sets in $\mathcal{S}$ are needed to cover $U$. Lin (ICALP'19) recently showed a gap creating reduction from the $(k,k+1)$-SetCover problem on universe of size $O_k(\log |\mathcal{S}|)$ to the $\left(k,\sqrt[k]{\frac{\log|\mathcal{S}|}{\log\log |\mathcal{S}|}}\cdot k\right)$-SetCover problem on universe of size $|\mathcal{S}|$. In this paper, we prove a more scalable version of his result: given any error correcting code $C$ over alphabet $[q]$, rate $ρ$, and relative distance $δ$, we use $C$ to create a reduction from the $(k,k+1)$-SetCover problem on universe $U$ to the $\left(k,\sqrt[2k]{\frac{2}{1-δ}}\right)$-SetCover problem on universe of size $\frac{\log|\mathcal{S}|}ρ\cdot|U|^{q^k}$. Lin established his result by composing the input SetCover instance (that has no gap) with a special threshold graph constructed from extremal combinatorial object called universal sets, resulting in a final SetCover instance with gap. Our reduction follows along the exact same lines, except that we generate the threshold graphs specified by Lin simply using the basic properties of the error correcting code $C$. We use the same threshold graphs mentioned above to prove inapproximability results, under W[1]$\neq$FPT and ETH, for the $k$-MaxCover problem introduced by Chalermsook et al. (SICOMP'20). Our inapproximaiblity results match the bounds obtained by Karthik et al. (JACM'19), although their proof framework is very different, and involves generalization of the distributed PCP framework. Prior to this work, it was not clear how to adopt the proof strategy of Lin to prove inapproximability results for $k$-MaxCover.

cs.CC↗