Abstract
We prove two new results about the inability of lowdegree polynomials to uniformly approximate constantdepth circuits, even to slightlybetterthantrivial error. First, we prove a tight Omega~(n^{1/2}) lower bound on the threshold degree of the SURJECTIVITY function on n variables. This matches the best known threshold degree bound for any AC^0 function, previously exhibited by a much more complicated circuit of larger depth (Sherstov, FOCS 2015). Our result also extends to a 2^{Omega~(n^{1/2})} lower bound on the signrank of an AC^0 function, improving on the previous best bound of 2^{Omega(n^{2/5})} (Bun and Thaler, ICALP 2016).
Second, for any delta>0, we exhibit a function f : {1,1}^n > {1,1} that is computed by a circuit of depth O(1/delta) and is hard to approximate by polynomials in the following sense: f cannot be uniformly approximated to error epsilon=12^{Omega(n^{1delta})}, even by polynomials of degree n^{1delta}. Our recent prior work (Bun and Thaler, FOCS 2017) proved a similar lower bound, but which held only for error epsilon=1/3.
Our result implies 2^{Omega(n^{1delta})} lower bounds on the complexity of AC^0 under a variety of basic measures such as discrepancy, margin complexity, and threshold weight. This nearly matches the trivial upper bound of 2^{O(n)} that holds for every function. The previous best lower bound on AC^0 for these measures was 2^{Omega(n^{1/2})} (Sherstov, FOCS 2015). Additional applications in learning theory, communication complexity, and cryptography are described.
BibTeX  Entry
@InProceedings{bun_et_al:LIPIcs:2019:11270,
author = {Mark Bun and Justin Thaler},
title = {{The LargeError Approximate Degree of AC^0}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
pages = {55:155:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771252},
ISSN = {18688969},
year = {2019},
volume = {145},
editor = {Dimitris Achlioptas and L{\'a}szl{\'o} A. V{\'e}gh},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/11270},
URN = {urn:nbn:de:0030drops112709},
doi = {10.4230/LIPIcs.APPROXRANDOM.2019.55},
annote = {Keywords: approximate degree, discrepancy, margin complexity, polynomial approximations, secret sharing, threshold circuits}
}
Keywords: 

approximate degree, discrepancy, margin complexity, polynomial approximations, secret sharing, threshold circuits 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019) 
Issue Date: 

2019 
Date of publication: 

17.09.2019 