9 research outputs found
A random hierarchical lattice: the series-parallel graph and its properties
We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability and respectively. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at
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A Lower Bound for Adaptively-Secure Collective Coin-Flipping Protocols
In 1985, Ben-Or and Linial (Advances in Computing Research \u2789) introduced the collective coin-flipping problem, where n parties communicate via a single broadcast channel and wish to generate a common random bit in the presence of adaptive Byzantine corruptions. In this model, the adversary can decide to corrupt a party in the course of the protocol as a function of the messages seen so far. They showed that the majority protocol, in which each player sends a random bit and the output is the majority value, tolerates O(sqrt n) adaptive corruptions. They conjectured that this is optimal for such adversaries.
We prove that the majority protocol is optimal (up to a poly-logarithmic factor) among all protocols in which each party sends a single, possibly long, message.
Previously, such a lower bound was known for protocols in which parties are allowed to send only a single bit (Lichtenstein, Linial, and Saks, Combinatorica \u2789), or for symmetric protocols (Goldwasser, Kalai, and Park, ICALP \u2715)
Bandits with many optimal arms
We consider a stochastic bandit problem with a possibly infinite number of
arms. We write for the proportion of optimal arms and for the
minimal mean-gap between optimal and sub-optimal arms. We characterize the
optimal learning rates both in the cumulative regret setting, and in the
best-arm identification setting in terms of the problem parameters (the
budget), and . For the objective of minimizing the cumulative
regret, we provide a lower bound of order and a
UCB-style algorithm with matching upper bound up to a factor of
. Our algorithm needs to calibrate its parameters, and we
prove that this knowledge is necessary, since adapting to in this setting
is impossible. For best-arm identification we also provide a lower bound of
order on the probability of outputting a
sub-optimal arm where is an absolute constant. We also provide an
elimination algorithm with an upper bound matching the lower bound up to a
factor of order in the exponential, and that does not need or
as parameter. Our results apply directly to the three related problems
of competing against the -th best arm, identifying an good arm,
and finding an arm with mean larger than a quantile of a known order.Comment: Substantial rewrite and added experiments. Accepted for NeurIPS 202
Optimally-secure Coin-tossing against a Byzantine Adversary
In their seminal work, Ben-Or and Linial (1985) introduced the full information model for collective coin-tossing protocols involving processors with unbounded computational power using a common broadcast channel for all their communications. The design and analysis of coin-tossing protocols in the full information model have close connections to diverse fields like extremal graph theory, randomness extraction, cryptographic protocol design, game theory, distributed protocols, and learning theory. Several works have focused on studying the asymptotically best attacks and optimal coin-tossing protocols in various adversarial settings. While one knows the characterization of the exact or asymptotically optimal protocols in some adversarial settings, for most adversarial settings, the optimal protocol characterization remains open. For the cases where the asymptotically optimal constructions are known, the exact constants or poly-logarithmic multiplicative factors involved are not entirely well-understood.
In this work, we study -processor coin-tossing protocols where every processor broadcasts an arbitrary-length message once. Note that, in this setting, which processor speaks and its message distribution may depend on the messages broadcast so far. An adaptive Byzantine adversary, based on the messages broadcast so far, can corrupt processor. A bias- coin-tossing protocol outputs 1 with probability ; 0 with probability . For a coin-tossing protocol, its insecurity is the maximum change in the output distribution (in the statistical distance) that an adversarial strategy can cause. Our objective is to identify optimal bias- coin-tossing protocols with minimum insecurity, for every .
Lichtenstein, Linial, and Saks (1989) studied bias- coin-tossing protocols in this adversarial model under the highly restrictive constraint that each party broadcasts an independent and uniformly random bit. The underlying message space is a well-behaved product space, and can only be integer multiples of , which is a discrete problem. The case where every processor broadcasts only an independent random bit admits simplifications, for example, the collective coin-tossing protocol must be monotone. Surprisingly, for this class of coin-tossing protocols, the objective of reducing an adversary’s ability to increase the expected output is equivalent to reducing an adversary’s ability to decrease the expected output. Building on these observations, Lichtenstein, Linial, and Saks proved that the threshold coin-tossing protocols are optimal for all and .
In a sequence of works, Goldwasser, Kalai, and Park (2015), Kalai, Komargodski, and Raz (2018), and (independent of our work) Haitner and Karidi-Heller (2020) prove that k=\mathcal{O}\left(\sqrt n\cdot \polylog{n}\right) corruptions suffice to fix the output of any bias-X coin-tossing protocol. These results consider parties who send arbitrary-length messages, and each processor has multiple turns to reveal its entire message. However, optimal protocols robust to a large number of corruptions do not have any apriori relation to the optimal protocol robust to corruption. Furthermore, to make an informed choice of employing a coin-tossing protocol in practice, for a fixed target tolerance of insecurity, one needs a precise characterization of the minimum insecurity achieved by these coin-tossing protocols.
We rely on an inductive approach to constructing coin-tossing protocols to study a proxy potential function measuring the susceptibility of any bias- coin-tossing protocol to attacks in our adversarial model. Our technique is inherently constructive and yields protocols that minimize the potential function. It happens to be the case that threshold protocols minimize the potential function. We demonstrate that the insecurity of these threshold protocols is 2-approximate of the optimal protocol in our adversarial model. For any other that threshold protocols cannot realize, we prove that an appropriate (convex) combination of the threshold protocols is a 4-approximation of the optimal protocol
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Explicit two-source extractors and more
In this thesis we study the problem of extracting almost truly random bits from imperfect sources of randomness. This is motivated by the wide use of randomness in computer science, and the fact that most accessible sources of randomness generate correlated bits, and at best contain some amount of entropy. We follow Chor and Goldreich [CG88] and Zuckerman [Z90], and model weak sources using min-entropy, where an (n,k)-source X is a distribution on n bits and takes any string x with probability at most 2^-k. It is known that it is impossible to extract random bits from a single (n,k)-source, and Chor and Goldreich [CG88] raised the question of extracting randomness from two such independent (n,k)-sources. Existentially, such 2-source randomness extractors exist for min-entropy k >=log n + O(1), but the best known construction prior to work in this thesis requires min-entropy k >=0.499 n [B2]. One of the main contributions of this thesis is an explicit 2-source extractor for min-entropy log^C n, for some constant C. Other results in this thesis include improved ways of extracting random bits from various other sources of randomness, as well as stronger notions of randomness extraction. Our results have applications in privacy amplification [BBR88,Mau92,BBCM95], which is a classical problem in information cryptography, and give protocols that achieve almost optimal parameters. Other applications include explicit constructions of non-malleable codes, which is a relaxation of the notion of error-detection codes and have applications in tamper-resilient cryptography [DPW10].Computer Science