TR12-152 Authors: Anindya De, Ilias Diakonikolas, Rocco Servedio

Publication: 9th November 2012 16:10

Downloads: 3974

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We initiate the study of \emph{inverse} problems in approximate uniform generation, focusing on uniform generation of satisfying assignments of various types of Boolean functions. In such an inverse problem, the algorithm is given uniform random satisfying assignments of an unknown function $f$ belonging to a class $\C$ of Boolean functions (such as linear threshold functions or polynomial-size DNF formulas), and the goal is to output a probability distribution $D$ which is $\epsilon$-close, in total variation distance, to the uniform distribution over $f^{-1}(1)$. Problems of this sort comprise a natural type of unsupervised learning problem in which the unknown distribution to be learned is the uniform distribution over satisfying assignments of an unknown function $f \in \C.$

{\bf Positive results:}

We prove a general positive result establishing sufficient conditions for efficient inverse approximate uniform generation for a class $\C$. We define a new type of algorithm called a \emph{densifier} for $\C$, and show (roughly speaking) how to combine (i) a densifier, (ii) an approximate counting / uniform generation algorithm, and (iii) a Statistical Query learning algorithm, to obtain an inverse approximate uniform generation algorithm. We apply this general result to obtain a poly$(n,1/\eps)$-time inverse approximate uniform generation algorithm for the class of $n$-variable linear threshold functions (halfspaces); and a quasipoly$(n,1/\eps)$-time inverse approximate uniform generation algorithm for the class of $\poly(n)$-size DNF formulas.

{\bf Negative results:} We prove a general negative result establishing that the existence of certain types of signature schemes in cryptography implies the hardness of certain inverse approximate uniform generation problems. We instantiate this negative result with known signature schemes from the cryptographic literature to prove (under a plausible cryptographic hardness assumption) that there are no {subexponential}-time inverse approximate uniform generation algorithms for 3-CNF formulas; for intersections of two halfspaces; for degree-2 polynomial threshold functions; and for monotone 2-CNF formulas.

Finally, we show that there is no general relationship between the complexity of the ``forward'' approximate uniform generation problem and the complexity of the inverse problem for a class $\C$ -- it is possible for either one to be easy while the other is hard. In one direction, we show that the existence of certain types of Message Authentication Codes (MACs) in cryptography implies the hardness of certain corresponding inverse approximate uniform generation problems, and we combine this general result with recent MAC constructions from the cryptographic literature to show (under a plausible cryptographic hardness assumption) that there is a class $\C$ for which the ``forward'' approximate uniform generation problem is easy but the inverse approximate uniform generation problem is computationally hard. In the other direction, we also show (assuming the GRAPH ISOMORPHISM problem is computationally hard) that there is a problem for which inverse approximate uniform generation is easy but ``forward'' approximate uniform generation is computationally hard.