Colloquium Details

New York Area Theory Day

Speaker: IBM/NYU/Columbia

Location: Warren Weaver Hall 109

Date: May 10, 2013, 9:30 a.m.

Host: Courant Institute of Mathematical Sciences


The New York Area Theory Day is a semi-annual conference, aimed to bring together people in the New York Metropolitan area for one day of interaction and discussion. The meeting is free and open to everyone;

in particular, students are encouraged to attend.

9:30--10:00: Coffee and bagels

10:00--10:55: Prof. Tim Roughgarden (Stanford University) Extension Theorems for the Price of Anarchy

10:55--11:05: Short break

11:05--12:00: Dr. Allison Lewko (Microsoft Research New England) On the Complexity of Asynchronous Agreement Against Powerful Adversaries

12:00--02:00: Lunch break (lunch not provided)

02:00--02:55: Prof. Avi Wigderson (Institute for Advanced Study) Grey Boxes, Population Recovery and Partial Identification

02:55--03:15: Coffee break

03:15--04:10: Dr. Ankur Moitra (Institute for Advanced Study) An Information Complexity Approach to Extended Formulations

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Xi Chen: Yevgeniy Dodis: Tal Rabin:

======== Abstracts ========

Extension Theorems for the Price of Anarchy Prof. Tim Roughgarden (Stanford University)

Abstract: The price of anarchy is a measure of the inefficiency of selfish behavior that has been successfully analyzed in many applications, including network routing, resource allocation, network formation, and even models of basketball. It is defined as the worst-case ratio between the welfare of a Nash equilibrium and that of an optimal solution.

Pure-strategy Nash equilibria --- where each player deterministically picks a single action --- are often easier to analyze than their more general cousins like mixed-strategy Nash equilibria (where players can randomize) and Bayes-Nash equilibria (where players are uncertain about what game they're playing in). For this reason, the price of anarchy is often analyzed, at least initially, only for a game's pure-strategy Nash equilibria. But as much as he or she might want to, the conscientious researcher cannot stop there --- such equilibria might not exist, might be hard to compute, and are relevant only when all players' preferences are common knowledge. Can we obtain price of anarchy bounds for more general equilibrium concepts without working any harder than we already do to analyze pure-strategy Nash equilibria? Ideal would be an "extension theorem" that could be used in the following ``black-box'' way: (1) prove a bound on the price of anarchy of pure-strategy Nash equilibria of a game; (2) invoke the extension theorem to conclude immediately that the exact same approximation bound applies to some more general equilibrium concept. Such an extension theorem would dodge the obvious deterrents from generalizing price of anarchy bounds beyond pure Nash equilibria --- no extra work, and no loss in the approximation guarantee. In this talk, I'll describe two such extension theorems: one for the possible outcomes generated by the repeated play of no-regret learners, and one for games of incomplete information.


On the Complexity of Asynchronous Agreement Against Powerful Adversaries Dr. Allison Lewko (Microsoft Research New England)

Abstract: In this talk, we will present new techniques for proving lower bounds on the running time of randomized algorithms for distributed consensus in the presence of mobile failures. We begin with the observation that the early randomized algorithms for consensus presented by Bracha and Ben-Or succeed against even stronger adversaries than they were initially designed to resist. We then prove that the exponential expected running time of these algorithms is an inherent consequence of that stronger resilience. This furthers our understanding of the kind of failures that can (or cannot) be corrected by polynomial time randomized algorithms. This is joint work with Mark Lewko.


Grey Boxes, Population Recovery and Partial Identification Prof. Avi Wigderson (Institute for Advanced Study)

Abstract: The talk, like the title, will have three parts.

In the first, I will describe a new learning model we call "restriction access". This model aims at generalizing prevailing "black-box" access to functions when trying to learn the "device" (e.g. circuit, decision tree, polynomial ...) which computes them. We propose a "grey-box" access that allows certain partial views of the device, obtained from random restrictions. The PAC-learning analog of our model is able to efficiently learn decision trees and DNFs, which are currently beyond reach in the standard "black-box" version of PAC-learning.

To analyze some of these learning algorithms, we study several natural "population recovery" problems in which an unknown distribution over an unknown population of vectors needs to be recovered from partial or noisy samples. Such problems naturally arise in many other contexts in learning, clustering, statistics, data mining and database privacy, where loss and error may be introduced by nature, inaccurate measurements, or on purpose. We give fairly efficient algorithms to recover the data under fairly general assumptions, when loss and noise are close to the information theoretic limit (namely, nearly completely obliterate the original data).

The third part introduces a new combinatorial structure we call a partial identification (PID) graph, which underlies our algorithms. While standard IDs are subsets of features (vector coordinates) that uniquely identify an individual in a population, partial IDs allow ambiguity (and "imposters"), and thus can be shorter. PID graphs capture this imposter-structure. PID graphs yield strategies for dimension reductions of recovery problems, and the re-assembly of this local pieces of statistical information to a global one. The combinatorial heart of this work is proving that every set of vectors admits partial IDs with "cheap" PID graphs (and hence efficient recovery). We further show how to find such near-optimal PIDs efficiently.

Based on joint works with Zeev Dvir, Anup Rao and Amir Yehudayoff


An Information Complexity Approach to Extended Formulations Dr. Ankur Moitra (Institute for Advanced Study)

Often it is possible to express a given polytope $P$ as the projection of a higher dimensional polytope $Q$ and in so doing to drastically reduce the number of facets needed. We call $Q$ an extended formulation for $P$. In fact, small extended formulations are known for a number of polytopes that arise in combinatorial optimization and yet there are many polytopes that we do not expect to have a small extended formulation -- e.g. polytopes that capture the solutions to hard optimization problems. Can we prove unconditional results that these polytopes do not have small extended formulations?

Recently, there has been a revival of interest in this question. A breakthrough result of Fiorini et al (STOC 2012) established that the TSP polytope has no subexponential sized extended formulation. Subsequently Braun et al (FOCS 2012) proved that the clique polytope has no subexponential $O(n^{1/2 - \epsilon})$-approximate extended formulation either. Here we give a new approach to these lower bounds based on information complexity -- roughly a too-good-to-be-true extended formulation would allow us to sample from a distribution using too few bits of entropy, and this is the means by which we prove our lower bounds. We use this framework to prove that the clique polytope has no subexponential $O(n^{1 - \epsilon})$-approximate extended formulation, thus giving an optimal and unconditional lower bound that matches Hastad's celebrated hardness of approximation results for clique.

This is based on joint work with Mark Braverman.


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