Paul Valiant

On the complexity of two-player win-lose games

The efficient computation of Nash equilibria is one of the most formidable challenges in computational complexity today. The problem remains open for two-player games. We show that the complexity of two-player Nash equilibria is unchanged when all outcomes are restricted to be 0 or 1. That is, win-or-lose games are as complex as the general case for two-player games.

The Power of Linear Estimators

For a broad class of practically relevant distribution properties, which includes entropy and support size, nearly all of the proposed estimators have an especially simple form. Given a set of independent samples from a discrete distribution, these estimators tally the vector of summary statistics -- the number of domain elements seen once, twice, etc. in the sample -- and output the dot product between these summary statistics, and a fixed vector of coefficients. We term such estimators \emph{linear}. This historical proclivity towards linear estimators is slightly perplexing, since, despite many efforts over nearly 60 years, all proposed such estimators have significantly sub optimal convergence, compared to the bounds shown in [VV11]. Our main result, in some sense vindicating this insistence on linear estimators, is that for any property in this broad class, there exists a near-optimal linear estimator. Additionally, we give a practical and polynomial-time algorithm for constructing such estimators for any given parameters. While this result does not yield explicit bounds on the sample complexities of these estimation tasks, we leverage the insights provided by this result to give explicit constructions of near-optimal linear estimators for three properties: entropy,

Optimal algorithms for testing closeness of discrete distributions

L_1

Testing Symmetric Properties of Distributions

distance to uniformity, and for pairs of distributions,

How to Construct a Correct and Scalable iBGP Configuration

L_1

Estimating the Unseen: Improved Estimators for Entropy and Other Properties

distance. Our entropy estimator, when given

The tensor product of two codes is not necessarily robustly testable

O(\frac{n}{\eps \log n})

Size and Treewidth Bounds for Conjunctive Queries

independent samples from a distribution of support at most

Quantifying unobserved protein-coding variants in human populations provides a roadmap for large-scale sequencing projects

n,

Evolvability of Real Functions

will estimate the entropy of the distribution to within additive accuracy