On Approximating the Entropy of Polynomial Mappings

We investigate the complexity of Polynomial Entropy Approximation (PEA): Given a low-degree polynomial mapping p : F^n-> F^m, where F is a finite field, approximate the output entropy H(p(U_n)), where U_n is the uniform distribution on F^n and H may be any of several entropy measures. We show: Approximating the Shannon entropy of degree 3 polynomials p : F_2^n->F_2^m over F_2 to within an additive constant (or even n^{.9}) is complete for SZKPL, the class of problems having statistical zero-knowledge proofs where the honest verifier and its simulator are computable in logarithmic space. (SZKPL contains most of the natural problems known to be in the full class SZKP.) For prime fields F\neq F_2 and homogeneous quadratic polynomials p : F^n->F^m, there is a probabilistic polynomial-time algorithm that distinguishes the case that p(U_n) has entropy smaller than k from the case that p(U_n) has min-entropy (or even Renyi entropy) greater than (2+o(1))k. For degree d polynomials p : F_2^n->F_2^m, there is a polynomial-time algorithm that distinguishes the case that p(U_n) has max-entropy smaller than k (where the max-entropy of a random variable is the logarithm of its support size) from the case that p(U_n) has max-entropy at least (1+o(1))k^d (for fixed d and large k).Engineering and Applied Science

Privately Releasing Conjunctions and the Statistical Query Barrier

Suppose we would like to know all answers to a set of statistical queries C on a data set up to small error, but we can only access the data itself using statistical queries. A trivial solution is to exhaustively ask all queries in C. Can we do any better? + We show that the number of statistical queries necessary and sufficient for this task is---up to polynomial factors---equal to the agnostic learning complexity of C in Kearns' statistical query (SQ) model. This gives a complete answer to the question when running time is not a concern. + We then show that the problem can be solved efficiently (allowing arbitrary error on a small fraction of queries) whenever the answers to C can be described by a submodular function. This includes many natural concept classes, such as graph cuts and Boolean disjunctions and conjunctions. While interesting from a learning theoretic point of view, our main applications are in privacy-preserving data analysis: Here, our second result leads to the first algorithm that efficiently releases differentially private answers to of all Boolean conjunctions with 1% average error. This presents significant progress on a key open problem in privacy-preserving data analysis. Our first result on the other hand gives unconditional lower bounds on any differentially private algorithm that admits a (potentially non-privacy-preserving) implementation using only statistical queries. Not only our algorithms, but also most known private algorithms can be implemented using only statistical queries, and hence are constrained by these lower bounds. Our result therefore isolates the complexity of agnostic learning in the SQ-model as a new barrier in the design of differentially private algorithms

Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory

We continue our study of the dynamics of mappings with small topological degree on (projective) complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic ``equilibrium'' measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalize results that were known in the invertible case and is, to our knowledge, one among not very many instances in which a natural invariant measure for a non-invertible dynamical system is well-understood.Comment: v3. Exposition improved. Final version, to appear in Ann. Scient. de l'EN