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Omer Tamuz

Publications and source records attributed to Omer Tamuz.

At least 73 records · Page 4Linked to original sources
arXiv2015

Symbolic dynamics on amenable groups: the entropy of generic shifts

Let $G$ be a finitely generated amenable group. We study the space of shifts on $G$ over a given finite alphabet $A$. We show that the zero entropy shifts are generic in this space, and that more generally the shifts of entropy $c$ are generic in the space of shifts with entropy at least $c$. The same is shown to hold for the space of transitive shifts and for the space of weakly mixing shifts. As applications of this result, we show that for every entropy value $c \in [0,\log |A|]$ there is a weakly mixing subshift of $A^G$ with entropy $c$. We also show that the set of strongly irreducible shifts does not form a $G_δ$ in the space of shifts, and that all non-trivial, strongly irreducible shifts are non-isolated points in this space.

math.DS↗
arXiv2014

Property (T) and the Furstenberg Entropy of Nonsingular Actions

We establish a new characterization of property (T) in terms of the Furstenberg entropy of nonsingular actions. Given any generating measure $μ$ on a countable group $G$, A. Nevo showed that a necessary condition for $G$ to have property (T) is that the Furstenberg $μ$-entropy values of the ergodic, properly nonsingular $G$-actions are bounded away from zero. We show that this is also a sufficient condition.

math.GR↗
arXiv2014

Convergence, unanimity and disagreement in majority dynamics on unimodular graphs and random graphs

In majority dynamics, agents located at the vertices of an undirected simple graph update their binary opinions synchronously by adopting those of the majority of their neighbors. On infinite unimodular transitive graphs (e.g., Cayley graphs), when initial opinions are chosen from a distribution that is invariant with respect to the graph automorphism group, we show that the opinion of each agent almost surely either converges, or else eventually oscillates with period two; this is known to hold for finite graphs, but not for all infinite graphs. On Erdős-Rényi random graphs with degrees $Ω(\sqrt{n})$, we show that when initial opinions are chosen i.i.d. then agents all converge to the initial majority opinion, with constant probability. Conversely, on random 4-regular finite graphs, we show that with high probability different agents converge to different opinions.

math.PR↗
arXiv2014

Scenery Reconstruction on Finite Abelian Groups

We consider the question of when a random walk on a finite abelian group with a given step distribution can be used to reconstruct a binary labeling of the elements of the group, up to a shift. Matzinger and Lember (2006) give a sufficient condition for reconstructibility on cycles. While, as we show, this condition is not in general necessary, our main result is that it is necessary when the length of the cycle is prime and larger than 5, and the step distribution has only rational probabilities. We extend this result to other abelian groups.

math.PR↗
arXiv2014

Majority Dynamics and the Retention of Information

We consider a group of agents connected by a social network who participate in majority dynamics: each agent starts with an opinion in {-1,+1} and repeatedly updates it to match the opinion of the majority of its neighbors. We assume that one of {-1,+1} is the "correct" opinion S, and consider a setting in which the initial opinions are independent conditioned on S, and biased towards it. They hence contain enough information to reconstruct S with high probability. We ask whether it is still possible to reconstruct S from the agents' opinions after many rounds of updates. While this is not the case in general, we show that indeed, for a large family of bounded degree graphs, information on S is retained by the process of majority dynamics. Our proof technique yields novel combinatorial results on majority dynamics on both finite and infinite graphs, with applications to zero temperature Ising models.

math.PR↗
arXiv2013

Testing Booleanity and the Uncertainty Principle

Let f:{-1,1}^n -> R be a real function on the hypercube, given by its discrete Fourier expansion, or, equivalently, represented as a multilinear polynomial. We say that it is Boolean if its image is in {-1,1}. We show that every function on the hypercube with a sparse Fourier expansion must either be Boolean or far from Boolean. In particular, we show that a multilinear polynomial with at most k terms must either be Boolean, or output values different than -1 or 1 for a fraction of at least 2/(k+2)^2 of its domain. It follows that given oracle access to f, together with the guarantee that its representation as a multilinear polynomial has at most k terms, one can test Booleanity using O(k^2) queries. We show an Ω(k) queries lower bound for this problem. Our proof crucially uses Hirschman's entropic version of Heisenberg's uncertainty principle.

cs.DM↗
arXiv2013

A lower bound on seller revenue in single buyer monopoly auctions

We consider a monopoly seller who optimally auctions a single object to a single potential buyer, with a known distribution of valuations. We show that a tight lower bound on the seller's expected revenue is $1/e$ times the geometric expectation of the buyer's valuation, and that this bound is uniquely achieved for the equal revenue distribution. We show also that when the valuation's expectation and geometric expectation are close, then the seller's expected revenue is close to the expected valuation.

cs.GT↗
arXiv2013

Making Consensus Tractable

We study a model of consensus decision making, in which a finite group of Bayesian agents has to choose between one of two courses of action. Each member of the group has a private and independent signal at his or her disposal, giving some indication as to which action is optimal. To come to a common decision, the participants perform repeated rounds of voting. In each round, each agent casts a vote in favor of one of the two courses of action, reflecting his or her current belief, and observes the votes of the rest. We provide an efficient algorithm for the calculation the agents have to perform, and show that consensus is always reached and that the probability of reaching a wrong decision decays exponentially with the number of agents.

math.ST↗
arXiv2012

From Agreement to Asymptotic Learning

We consider a group of Bayesian agents who are each given an independent signal about an unknown state of the world, and proceed to communicate with each other. We study the question of asymptotic learning: do agents learn the state of the world with probability that approaches one as the number of agents tends to infinity? We show that under general conditions asymptotic learning follows from agreement on posterior actions or posterior beliefs, regardless of the communication dynamics. In particular, we prove that asymptotic learning holds for the Gale-Kariv model on undirected networks and non-atomic private beliefs.

math.ST↗
arXiv2012

Lower Bounds on Revenue of Approximately Optimal Auctions

We obtain revenue guarantees for the simple pricing mechanism of a single posted price, in terms of a natural parameter of the distribution of buyers' valuations. Our revenue guarantee applies to the single item n buyers setting, with values drawn from an arbitrary joint distribution. Specifically, we show that a single price drawn from the distribution of the maximum valuation Vmax = max {V_1, V_2, ...,V_n} achieves a revenue of at least a 1/e fraction of the geometric expecation of Vmax. This generic bound is a measure of how revenue improves/degrades as a function of the concentration/spread of Vmax. We further show that in absence of buyers' valuation distributions, recruiting an additional set of identical bidders will yield a similar guarantee on revenue. Finally, our bound also gives a measure of the extent to which one can simultaneously approximate welfare and revenue in terms of the concentration/spread of Vmax.

cs.GT↗
arXiv2012

Textual Features for Programming by Example

In Programming by Example, a system attempts to infer a program from input and output examples, generally by searching for a composition of certain base functions. Performing a naive brute force search is infeasible for even mildly involved tasks. We note that the examples themselves often present clues as to which functions to compose, and how to rank the resulting programs. In text processing, which is our domain of interest, clues arise from simple textual features: for example, if parts of the input and output strings are permutations of one another, this suggests that sorting may be useful. We describe a system that learns the reliability of such clues, allowing for faster search and a principled ranking over programs. Experiments on a prototype of this system show that this learning scheme facilitates efficient inference on a range of text processing tasks.

cs.AI↗
arXiv2012

Asymptotic Learning on Bayesian Social Networks

Understanding information exchange and aggregation on networks is a central problem in theoretical economics, probability and statistics. We study a standard model of economic agents on the nodes of a social network graph who learn a binary "state of the world" S, from initial signals, by repeatedly observing each other's best guesses. Asymptotic learning is said to occur on a family of graphs G_n = (V_n, E_n), with |V_n| tending to infinity, if with probability tending to 1 as n tends to infinity all agents in G_n eventually estimate S correctly. We identify sufficient conditions for asymptotic learning and contruct examples where learning does not occur when the conditions do not hold.

math.PR↗
arXiv2012

Majority Dynamics and Aggregation of Information in Social Networks

Consider n individuals who, by popular vote, choose among q >= 2 alternatives, one of which is "better" than the others. Assume that each individual votes independently at random, and that the probability of voting for the better alternative is larger than the probability of voting for any other. It follows from the law of large numbers that a plurality vote among the n individuals would result in the correct outcome, with probability approaching one exponentially quickly as n tends to infinity. Our interest in this paper is in a variant of the process above where, after forming their initial opinions, the voters update their decisions based on some interaction with their neighbors in a social network. Our main example is "majority dynamics", in which each voter adopts the most popular opinion among its friends. The interaction repeats for some number of rounds and is then followed by a population-wide plurality vote. The question we tackle is that of "efficient aggregation of information": in which cases is the better alternative chosen with probability approaching one as n tends to infinity? Conversely, for which sequences of growing graphs does aggregation fail, so that the wrong alternative gets chosen with probability bounded away from zero? We construct a family of examples in which interaction prevents efficient aggregation of information, and give a condition on the social network which ensures that aggregation occurs. For the case of majority dynamics we also investigate the question of unanimity in the limit. In particular, if the voters' social network is an expander graph, we show that if the initial population is sufficiently biased towards a particular alternative then that alternative will eventually become the unanimous preference of the entire population.

math.ST↗
arXiv2012

An Abramov formula for stationary spaces of discrete groups

Let (G,mu) be a discrete group equipped with a generating probability measure, and let Gamma be a finite index subgroup of G. A mu-random walk on G, starting from the identity, returns to Gamma with probability one. Let theta be the hitting measure, or the distribution of the position in which the random walk first hits Gamma. We prove that the Furstenberg entropy of a (G,mu)-stationary space, with respect to the induced action of (Gamma,theta), is equal to the Furstenberg entropy with respect to the action of (G,mu), times the index of Gamma in G. The index is shown to be equal to the expected return time to Gamma. As a corollary, when applied to the Furstenberg-Poisson boundary of (G,mu), we prove that the random walk entropy of (Gamma,theta) is equal to the random walk entropy of (G,mu), times the index of Gamma in G.

math.DS↗
arXiv2012

Bundling Customers: How to Exploit Trust Among Customers to Maximize Seller Profit

We consider an auction of identical digital goods to customers whose valuations are drawn independently from known distributions. Myerson's classic result identifies the truthful mechanism that maximizes the seller's expected profit. Under the assumption that in small groups customers can learn each others' valuations, we show how Myerson's result can be improved to yield a higher payoff to the seller using a mechanism that offers groups of customers to buy bundles of items.

cs.GT↗
arXiv2011

Social Learning in a Changing World

We study a model of learning on social networks in dynamic environments, describing a group of agents who are each trying to estimate an underlying state that varies over time, given access to weak signals and the estimates of their social network neighbors. We study three models of agent behavior. In the "fixed response" model, agents use a fixed linear combination to incorporate information from their peers into their own estimate. This can be thought of as an extension of the DeGroot model to a dynamic setting. In the "best response" model, players calculate minimum variance linear estimators of the underlying state. We show that regardless of the initial configuration, fixed response dynamics converge to a steady state, and that the same holds for best response on the complete graph. We show that best response dynamics can, in the long term, lead to estimators with higher variance than is achievable using well chosen fixed responses. The "penultimate prediction" model is an elaboration of the best response model. While this model only slightly complicates the computations required of the agents, we show that in some cases it greatly increases the efficiency of learning, and on complete graphs is in fact optimal, in a strong sense.

cs.SI↗
arXiv2011

Adaptively Learning the Crowd Kernel

We introduce an algorithm that, given n objects, learns a similarity matrix over all n^2 pairs, from crowdsourced data alone. The algorithm samples responses to adaptively chosen triplet-based relative-similarity queries. Each query has the form "is object 'a' more similar to 'b' or to 'c'?" and is chosen to be maximally informative given the preceding responses. The output is an embedding of the objects into Euclidean space (like MDS); we refer to this as the "crowd kernel." SVMs reveal that the crowd kernel captures prominent and subtle features across a number of domains, such as "is striped" among neckties and "vowel vs. consonant" among letters.

cs.LG↗