4 research outputs found
On Polynomial Approximations to AC^0
We make progress on some questions related to polynomial approximations of AC^0. It is known, from the works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that any AC^0 circuit of size s and depth d has an epsilon-error probabilistic polynomial over the reals of degree (log (s/epsilon))^{O(d)}. We improve this upper bound to (log s)^{O(d)}* log(1/epsilon), which is much better for small values of epsilon.
We give an application of this result by using it to resolve a question posed by Tal (ECCC 2014): we show that (log s)^{O(d)}* log(1/epsilon)-wise independence fools AC^0, improving on Tal\u27s strengthening of Braverman\u27s theorem (J. ACM 2010) that (log (s/epsilon))^{O(d)}-wise independence fools AC^0. Up to the constant implicit in the O(d), our result is tight. As far as we know, this is the first PRG construction for AC^0 that achieves optimal dependence on the error epsilon.
We also prove lower bounds on the best polynomial approximations to AC^0. We show that any polynomial approximating the OR function on n bits to a small constant error must have degree at least ~Omega(sqrt{log n}). This result improves exponentially on a recent lower bound demonstrated by Meka, Nguyen, and Vu (arXiv 2015)
A Moment-Matching Approach to Testable Learning and a New Characterization of Rademacher Complexity
A remarkable recent paper by Rubinfeld and Vasilyan (2022) initiated the
study of \emph{testable learning}, where the goal is to replace hard-to-verify
distributional assumptions (such as Gaussianity) with efficiently testable ones
and to require that the learner succeed whenever the unknown distribution
passes the corresponding test. In this model, they gave an efficient algorithm
for learning halfspaces under testable assumptions that are provably satisfied
by Gaussians.
In this paper we give a powerful new approach for developing algorithms for
testable learning using tools from moment matching and metric distances in
probability. We obtain efficient testable learners for any concept class that
admits low-degree \emph{sandwiching polynomials}, capturing most important
examples for which we have ordinary agnostic learners. We recover the results
of Rubinfeld and Vasilyan as a corollary of our techniques while achieving
improved, near-optimal sample complexity bounds for a broad range of concept
classes and distributions.
Surprisingly, we show that the information-theoretic sample complexity of
testable learning is tightly characterized by the Rademacher complexity of the
concept class, one of the most well-studied measures in statistical learning
theory. In particular, uniform convergence is necessary and sufficient for
testable learning. This leads to a fundamental separation from (ordinary)
distribution-specific agnostic learning, where uniform convergence is
sufficient but not necessary.Comment: 34 page
Interactive Proofs for Social Graphs
We consider interactive proofs for social graphs, where the verifier has only oracle access to the graph and can query for the neighbor of a vertex , given and . In this model, we construct a doubly-efficient public-coin two-message interactive protocol for estimating the size of the graph to within a multiplicative factor . The verifier performs queries to the graph, where is the mixing time of the graph and is the average degree of the graph. The prover runs in quasi-linear time in the number of nodes in the graph.
Furthermore, we develop a framework for computing the quantiles of essentially any (reasonable) function of vertices/edges of the graph. Using this framework, we can estimate many health measures of social graphs such as the clustering coefficients and the average degree, where the verifier performs only a small number of queries to the graph.
Using the Fiat-Shamir paradigm, we are able to transform the above protocols to a non-interactive argument in the random oracle model. The result is that social media companies (e.g., Facebook, Twitter, etc.) can publish, once and for all, a short proof for the size or health of their social network. This proof can be publicly verified by any single user using a small number of queries to the graph