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1. One-Way Functions and a Conditional Variant of MKTP

   https://doi.org/10.4230/LIPIcs.FSTTCS.2021.7  

   Allender, Eric; Cheraghchi, Mahdi; Myrisiotis, Dimitrios; Tirumala, Harsha; Volkovich, Ilya (November 2021, Leibniz international proceedings in informatics)

   Bojanczy, Mikolaj; Chekuri, Chandra (Ed.)

   One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some natural NP-complete problem. In this paper, we make progress on this question by studying a conditional variant of the Minimum KT-complexity Problem (MKTP), which we call McKTP, as follows. 1) First, we prove that if McKTP is average-case hard on a polynomial fraction of its instances, then there exist OWFs. 2) Then, we observe that McKTP is NP-complete under polynomial-time randomized reductions. 3) Finally, we prove that the existence of OWFs implies the nontrivial average-case hardness of McKTP. Thus the existence of OWFs is inextricably linked to the average-case hardness of this NP-complete problem. In fact, building on recently-announced results of Ren and Santhanam [Rahul Ilango et al., 2021], we show that McKTP is hard-on-average if and only if there are logspace-computable OWFs. 

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2. DAPT: A package enabling distributed automated parameter testing

   https://doi.org/10.46471/gigabyte.22  

   Duggan, Ben; Metzcar, John; Macklin, Paul (June 2021, Gigabyte)

   null (Ed.)

   Modern agent-based models (ABM) and other simulation models require evaluation and testing of many different parameters. Managing that testing for large scale parameter sweeps (grid searches), as well as storing simulation data, requires multiple, potentially customizable steps that may vary across simulations. Furthermore, parameter testing, processing, and analysis are slowed if simulation and processing jobs cannot be shared across teammates or computational resources. While high-performance computing (HPC) has become increasingly available, models can often be tested faster with the use of multiple computers and HPC resources. To address these issues, we created the Distributed Automated Parameter Testing (DAPT) Python package. By hosting parameters in an online (and often free) “database”, multiple individuals can run parameter sets simultaneously in a distributed fashion, enabling ad hoc crowdsourcing of computational power. Combining this with a flexible, scriptable tool set, teams can evaluate models and assess their underlying hypotheses quickly. Here, we describe DAPT and provide an example demonstrating its use. 

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3. Lower bounds for testing graphical models: colorings and antiferromagnetic Ising models

   Bezakova, Ivona; Blanca, Antonio; Chen, Zongchen; Stefankovic, Daniel; Vigoda, Eric (July 2019, Proceedings of Machine Learning Research)

   We study the identity testing problem in the context of spin systems or undirected graphical models, where it takes the following form: given the parameter specification of the model M and a sampling oracle for the distribution \mu_{M^*} of an unknown model M^*, can we efficiently determine if the two models M and M^* are the same? We consider identity testing for both soft-constraint and hard-constraint systems. In particular, we prove hardness results in two prototypical cases, the Ising model and proper colorings, and explore whether identity testing is any easier than structure learning. For the ferromagnetic (attractive) Ising model, Daskalasis et al. (2018) presented a polynomial time algorithm for identity testing. We prove hardness results in the antiferromagnetic (repulsive) setting in the same regime of parameters where structure learning is known to require a super-polynomial number of samples. In particular, for n-vertex graphs of maximum degree d, we prove that if |\beta| d = \omega(\log n) (where \beta is the inverse temperature parameter), then there is no identity testing algorithm for the antiferromagnetic Ising model that runs in polynomial time unless RP = NP. We also establish computational lower bounds for a broader set of parameters under the (randomized) exponential time hypothesis. In our proofs, we use random graphs as gadgets; this is inspired by similar constructions in seminal works on the hardness of approximate counting. In the hard-constraint setting, we present hardness results for identity testing for proper colorings. Our results are based on the presumed hardness of #BIS, the problem of (approximately) counting independent sets in bipartite graphs. In particular, we prove that identity testing for colorings is hard in the same range of parameters where structure learning is known to be hard, which in turn matches the parameter regime for NP-hardness of the corresponding decision problem. 

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4. Distributed Corruption Detection in Networks

   https://doi.org/10.4086/toc.2020.v016a001  

   Alon, Noga; Mossel, Elchanan; Pemantle, Robin (January 2020, Theory of Computing)

   We consider the problem of distributed corruption detection in networks. In this model each node of a directed graph is either truthful or corrupt. Each node reports the type (truthful or corrupt) of each of its outneighbors. If it is truthful, it reports the truth, whereas if it is corrupt, it reports adversarially. This model, first considered by Preparata, Metze and Chien in 1967, motivated by the desire to identify the faulty components of a digital system by having the other components checking them, became known as the PMC model. The main known results for this model characterize networks in which all corrupt (that is, faulty) nodes can be identified, when there is a known upper bound on their number. We are interested in networks in which a large fraction of the nodes can be classified. It is known that in the PMC model, in order to identify all corrupt nodes when their number is t, all in-degrees have to be at least t. In contrast, we show that in d regular-graphs with strong expansion properties, a 1 - O(1/d) fraction of the corrupt nodes, and a 1 - O(1/d) fraction of the truthful nodes can be identified, whenever there is a majority of truthful nodes. We also observe that if the graph is very far from being a good expander, namely, if the deletion of a small set of nodes splits the graph into small components, then no corruption detection is possible even if most of the nodes are truthful. Finally we discuss the algorithmic aspects and the computational hardness of the problem. 

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5. Adversarially Robust Learning Could Leverage Computational Hardness

   Garg, S; Jha, S; Mahloujifar, S; Mahmoody, M (January 2020, International Conference on Algorithmic Learning Theory)

   Over recent years, devising classification algorithms that are robust to adversarial perturbations hasemerged as a challenging problem. In particular, deep neural nets (DNNs) seem to be susceptible tosmall imperceptible changes over test instances. However, the line of work inprovablerobustness,so far, has been focused oninformation theoreticrobustness, ruling out even theexistenceof anyadversarial examples. In this work, we study whether there is a hope to benefit fromalgorithmicnature of an attacker that searches for adversarial examples, and ask whether there isanylearning taskfor which it is possible to design classifiers that are only robust againstpolynomial-timeadversaries.Indeed, numerous cryptographic tasks (e.g. encryption of long messages) can only be secure againstcomputationally bounded adversaries, and are indeedimpossiblefor computationally unboundedattackers. Thus, it is natural to ask if the same strategy could help robust learning.We show that computational limitation of attackers can indeed be useful in robust learning bydemonstrating the possibility of a classifier for some learning task for which computational andinformation theoretic adversaries of bounded perturbations have very different power. Namely, whilecomputationally unbounded adversaries can attack successfully and find adversarial examples withsmall perturbation, polynomial time adversaries are unable to do so unless they can break standardcryptographic hardness assumptions. Our results, therefore, indicate that perhaps a similar approachto cryptography (relying on computational hardness) holds promise for achieving computationallyrobust machine learning. On the reverse directions, we also show that the existence of such learningtask in which computational robustness beats information theoretic robustness requires computationalhardness by implying (average-case) hardness o fNP. 

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