As I announced in class today, I will not be lecturing this Thursday 9/22. This is because in a stroke of good luck, we have two excellent talks on differential privacy, given by Adam Smith and Sofya Raskhodnikova. Note that Sofya’s talk will be in a different time and place than I announced in class!

Adam’s talk will be at noon on 9/22 in Levine 307.

Sofya’s talk will be at 1:30 on 9/22 on Towne 315 (This is where and when we usually have class!)

**Speaker: Adam Smith**

**Title: “Differential Privacy in Statistical Estimation Problems”**

Consider an agency holding a large database of sensitive personal information — medical records, census survey answers, or web search records, for example. The agency would like to discover and publicly release global characteristics of the data (say, to inform policy and business decisions) while protecting the privacy of individuals’ records. This problem is known variously as “statistical disclosure control”, “privacy-preserving data mining” or simply “database privacy”.

This talk will describe differential privacy, a notion which emerged from a recent line of work in theoretical computer science that seeks to formulate and satisfy rigorous definitions of privacy for such statistical databases.

After a brief introduction to the topic, I will discuss recent results on differentially private versions of basic algorithms from statistics. I will discuss both a general result that works for any asymptotically normal statistical estimator (based on a STOC 2011 paper) and some results tailored to convex optimization problems (unpublished work, joint with A. G. Thakurta).

**Speaker: Sofya Raskhodnikova**

**Title: “Testing and Reconstruction of Lipschitz Functions with Applications to Differential Privacy”**

A function f : D -> R has Lipschitz constant c if d_R(f(x), f(y)) <= c d_D(x, y) for all x, y in D, where d_R and d_D denote the distance functions on the range and domain of f, respectively. We say a function is Lipschitz if it has Lipschitz constant 1. (Note that rescaling by a factor of 1/c converts a function with a Lipschitz constant c into a Lipschitz function.) Intuitively, a Lipschitz constant of f is a bound on how sensitive f is to small changes in its input. Lipschitz constants are important in mathematical analysis, the theory of differential equations and other areas of mathematics and computer science. However, in general, it is computationally infeasible to find a Lipschitz constant of a given function f or even to verify that f is c-Lipschitz for a given number c.

We initiate the study of testing (which is a relaxation of the decision problem described above) and local reconstruction of the Lipschitz property of functions. A property tester, given a proximity parameter epsilon, has to distinguish functions with the property (in this case, Lipschitz) from functions that are epsilon-far from having the property, that is, differ from every function with the property on at least an epsilon fraction of the domain. A local filter reconstructs a desired property (in this case, Lipschitz) in the following sense: given an arbitrary function f and a query x, it returns g(x), where the resulting function g satisfies the property, changing f only when necessary. If f has the property, g must be equal to f.

We design efficient testers and local reconstructors for functions over domains of the form {1,…,n}^d, equipped with L1 distance, and give corresponding lower bounds. The testers that we developed have applications to programs analysis. The reconstructors have applications to data privacy. The application to privacy is based on the fact that a function f of entries in a database of sensitive information can be released with noise of magnitude proportional to a Lipschitz constant of f, while preserving the privacy of individuals whose data is stored in the database (Dwork, McSherry, Nissim and Smith, TCC 2006). We give a differentially private mechanism, based on local filters, for releasing a function f when a purported Lipschitz constant of f is provided by a distrusted client.

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