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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

We present a Gaussian random walk in a polytope that starts at a point inside and continues until it gets absorbed at a vertex. Our main result is that the probability distribution induced on the
vertices by this random walk has strong negative dependence properties for matroid polytopes. Such distributions are highly sought after in randomized algorithms as they imply concentration
properties. Our random walk is simple to implement, computationally efficient and can be viewed as an algorithm to round the starting point in an unbiased manner. The proof relies on a simple
inductive argument that synthesizes the combinatorial structure of matroid polytopes with the geometric structure of multivariate Gaussian distributions. Our result not only implies a long
line of past results in a unified and transparent manner, but also implies new results about constructing negatively associated distributions for all matroids.

Yuval Peres, Mohit Singh, and Nisheeth K. Vishnoi. Random Walks in Polytopes and Negative Dependence. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 50:1-50:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{peres_et_al:LIPIcs.ITCS.2017.50, author = {Peres, Yuval and Singh, Mohit and Vishnoi, Nisheeth K.}, title = {{Random Walks in Polytopes and Negative Dependence}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {50:1--50:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.50}, URN = {urn:nbn:de:0030-drops-81464}, doi = {10.4230/LIPIcs.ITCS.2017.50}, annote = {Keywords: Random walks, Matroid, Polytope, Brownian motion, Negative dependence} }

Document

**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

For a rumor spreading protocol, the spread time is defined as the first time that everyone learns the rumor. We compare the synchronous push&pull rumor spreading protocol with its asynchronous variant, and show that for any n-vertex graph and any starting vertex, the ratio between their expected spread times is bounded by O(n^{1/3} log^{2/3} n). This improves the O(sqrt n) upper bound of Giakkoupis, Nazari, and Woelfel (in Proceedings of ACM Symposium on Principles of Distributed Computing, 2016). Our bound is tight up to a factor of O(log n), as illustrated by the string of diamonds graph.

Omer Angel, Abbas Mehrabian, and Yuval Peres. The String of Diamonds Is Tight for Rumor Spreading. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 26:1-26:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{angel_et_al:LIPIcs.APPROX-RANDOM.2017.26, author = {Angel, Omer and Mehrabian, Abbas and Peres, Yuval}, title = {{The String of Diamonds Is Tight for Rumor Spreading}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {26:1--26:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.26}, URN = {urn:nbn:de:0030-drops-75754}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.26}, annote = {Keywords: randomized rumor spreading, push\&pull protocol, asynchronous time model, string of diamonds} }

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**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

We consider the random walk on the hypercube which moves by picking an ordered pair (i,j) of distinct coordinates uniformly at random and adding the bit at location i to the bit at location j, modulo 2. We show that this Markov chain has cutoff at time (3/2)n*log(n) with window of size n, solving a question posed by Chung and Graham (1997).

Anna Ben-Hamou and Yuval Peres. Cutoff for a Stratified Random Walk on the Hypercube. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 29:1-29:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{benhamou_et_al:LIPIcs.APPROX-RANDOM.2017.29, author = {Ben-Hamou, Anna and Peres, Yuval}, title = {{Cutoff for a Stratified Random Walk on the Hypercube}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {29:1--29:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.29}, URN = {urn:nbn:de:0030-drops-75787}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.29}, annote = {Keywords: Mixing times, cutoff, hypercube} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We study the fundamental problem of prediction with expert advice and develop regret lower bounds for a large family of algorithms for this problem. We develop simple adversarial primitives, that lend themselves to various combinations leading to sharp lower bounds for many algorithmic families. We use these primitives to show that the classic Multiplicative Weights Algorithm (MWA) has a regret of (T*ln(k)/2)^{0.5} (where T is the time horizon and k is the number of experts), there by completely closing the gap between upper and lower bounds. We further show a regret lower bound of (2/3)* (T*ln(k)/2)^{0.5} for a much more general family of algorithms than MWA, where the learning rate can be arbitrarily varied over time, or even picked from arbitrary distributions over time. We also use our primitives to construct adversaries in the geometric horizon setting for MWA to precisely characterize the regret at 0.391/(\delta)^{0.5} for the case of 2 experts and a lower bound of (1/2)*(ln(k)/(2*\delta))^{0.5}, for the case of arbitrary number of experts k (here \delta is the probability that the game ends in any given round).

Nick Gravin, Yuval Peres, and Balasubramanian Sivan. Tight Lower Bounds for Multiplicative Weights Algorithmic Families. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 48:1-48:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gravin_et_al:LIPIcs.ICALP.2017.48, author = {Gravin, Nick and Peres, Yuval and Sivan, Balasubramanian}, title = {{Tight Lower Bounds for Multiplicative Weights Algorithmic Families}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {48:1--48:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.48}, URN = {urn:nbn:de:0030-drops-74997}, doi = {10.4230/LIPIcs.ICALP.2017.48}, annote = {Keywords: Multiplicative Weights, Lower Bounds, Adversarial Primitives} }

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