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

Information-based complexity, feedback and dynamics in convex programming

We study the intrinsic limitations of sequential convex optimization through the lens of feedback information theory. In the oracle model of optimization, an algorithm queries an {\em oracle} for noisy information about the unknown objective function, and the goal is to (approximately) minimize every function in a given class using as few queries as possible. We show that, in order for a function to be optimized, the algorithm must be able to accumulate enough information about the objective. This, in turn, puts limits on the speed of optimization under specific assumptions on the oracle and the type of feedback. Our techniques are akin to the ones used in statistical literature to obtain minimax lower bounds on the risks of estimation procedures; the notable difference is that, unlike in the case of i.i.d. data, a sequential optimization algorithm can gather observations in a {\em controlled} manner, so that the amount of information at each step is allowed to change in time. In particular, we show that optimization algorithms often obey the law of diminishing returns: the signal-to-noise ratio drops as the optimization algorithm approaches the optimum. To underscore the generality of the tools, we use our approach to derive fundamental lower bounds for a certain active learning problem. Overall, the present work connects the intuitive notions of information in optimization, experimental design, estimation, and active learning to the quantitative notion of Shannon information.Comment: final version; to appear in IEEE Transactions on Information Theor