arXiv subjects

Jan Hązła

Publications and source records attributed to Jan Hązła.

21 records · Page 2Linked to original sources

arXiv2018

Reasoning in Bayesian Opinion Exchange Networks Is PSPACE-Hard

We study the Bayesian model of opinion exchange of fully rational agents arranged on a network. In this model, the agents receive private signals that are indicative of an unkown state of the world. Then, they repeatedly announce the state of the world they consider most likely to their neighbors, at the same time updating their beliefs based on their neighbors' announcements. This model is extensively studied in economics since the work of Aumann (1976) and Geanakoplos and Polemarchakis (1982). It is known that the agents eventually agree with high probability on any network. It is often argued that the computations needed by agents in this model are difficult, but prior to our results there was no rigorous work showing this hardness. We show that it is PSPACE-hard for the agents to compute their actions in this model. Furthermore, we show that it is equally difficult even to approximate an agent's posterior: It is PSPACE-hard to distinguish between the posterior being almost entirely concentrated on one state of the world or another.

cs.CC↗

arXiv2016

Forbidden Subgraph Bounds for Parallel Repetition and the Density Hales-Jewett Theorem

We study a special kind of bounds (so called forbidden subgraph bounds, cf. Feige, Verbitsky '02) for parallel repetition of multi-prover games. First, we show that forbidden subgraph upper bounds for $r \ge 3$ provers imply the same bounds for the density Hales-Jewett theorem for alphabet of size $r$. As a consequence, this yields a new family of games with slow decrease in the parallel repetition value. Second, we introduce a new technique for proving exponential forbidden subgraph upper bounds and explore its power and limitations. In particular, we obtain exponential upper bounds for two-prover games with question graphs of treewidth at most two and show that our method cannot give exponential bounds for all two-prover graphs.

cs.DM↗

arXiv2015

Upper Tail Estimates with Combinatorial Proofs

We study generalisations of a simple, combinatorial proof of a Chernoff bound similar to the one by Impagliazzo and Kabanets (RANDOM, 2010). In particular, we prove a randomized version of the hitting property of expander random walks and apply it to obtain a concentration bound for expander random walks which is essentially optimal for small deviations and a large number of steps. At the same time, we present a simpler proof that still yields a "right" bound settling a question asked by Impagliazzo and Kabanets. Next, we obtain a simple upper tail bound for polynomials with input variables in $[0, 1]$ which are not necessarily independent, but obey a certain condition inspired by Impagliazzo and Kabanets. The resulting bound is used by Holenstein and Sinha (FOCS, 2012) in the proof of a lower bound for the number of calls in a black-box construction of a pseudorandom generator from a one-way function. We then show that the same technique yields the upper tail bound for the number of copies of a fixed graph in an Erdős-Rényi random graph, matching the one given by Janson, Oleszkiewicz and Ruciński (Israel J. Math, 2002).

cs.DM↗