# Hardness Amplification and the Approximate Degree of Constant-Depth Circuits

### Citation:

Mark Bun and Justin Thaler. 2015. “Hardness Amplification and the Approximate Degree of Constant-Depth Circuits.” International Colloquium on Automata, Languages, and Programming (ICALP 2015) BG. ArXiv Version
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### Abstract:

We establish a generic form of hardness amplification for the approximability of constant-depth Boolean circuits by polynomials. Specifically, we show that if a Boolean circuit cannot be pointwise approximated by low-degree polynomials to within constant error in a certain one-sided sense, then an OR of disjoint copies of that circuit cannot be pointwise approximated even with very high error. As our main application, we show that for every sequence of degrees d(n), there is an explicit depththree circuit F : {−1, 1} n → {−1, 1} of polynomial-size such that any degree-d polynomial cannot pointwise approximate F to error better than 1 − exp −Ω( ˜ nd−3/2 ) . As a consequence of our main result, we obtain an exp −Ω( ˜ n 2/5 ) upper bound on the the discrepancy of a function in AC0, and an exp Ω( ˜ n 2/5 ) lower bound on the threshold weight of AC0, improving over the previous best results of exp −Ω(n 1/3 ) and exp Ω(n 1/3 ) respectively. Our techniques also yield a new lower bound of Ω n 1/2/ log(d−2)/2 (n) on the approximate degree of the AND-OR tree of depth d, which is tight up to polylogarithmic factors for any constant d, as well as new bounds for read-once DNF formulas. In turn, these results imply new lower bounds on the communication and circuit complexity of these classes, and demonstrate strong limitations on existing PAC learning algorithms.

Last updated on 10/13/2016