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The approximate degree of a Boolean function f∶{0,1}ⁿ→{0,1} is the minimum degree of a real polynomial p that approximates f pointwise: |f(x)−p(x)|≤1/3 for all x∈{0,1}ⁿ. For any δ>0, we construct DNF and CNF formulas of polynomial size with approximate ...

Stochastic boolean function evaluation (SBFE) is the problem of determining the value of a given boolean function f on an unknown input x, when each bit $$x_i$$xi of x can only be determined by paying a given associated cost $$c_i$$ci. Further, x is ...

We present an iterative approach to constructing pseudorandom generators, based on the repeated application of mild pseudorandom restrictions. We use this template to construct pseudorandom generators for combinatorial rectangles and read-once CNFs and ...

We show that any $k$-wise independent probability distribution on $\{0,1\}^n$ $O(m^{2.2}$ $2^{-\sqrt{k}/10})$-fools any boolean function computable by an $m$-clause disjunctive normal form (DNF) (or conjunctive normal form (CNF)) formula on $n$ ...

We consider the complexity of properly learning concept classes, i.e. when the learner must output a hypothesis of the same form as the unknown concept. We present the following new upper and lower bounds on well-known concept classes:*We show that ...

We study the average number of well-chosen labeled examples that are required for a helpful teacher to uniquely specify a target function within a concept class. This "average teaching dimension" has been studied in learning theory and combinatorics ...

A challenging problem within machine learning is how to make good inferences from data sets in which pieces of information are missing. While it is valuable to have algorithms that perform well for specific domains, to gain a fundamental understanding ...

We present two related results about the learnability of disjunctive normal form (DNF) formulas. First we show that a common approach for learning arbitrary DNF formulas requires exponential time. We then contrast this with a polynomial time algorithm ...