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We prove two new results about the inability of low-degree polynomials to uniformly approximate constant-depth circuits, even to slightly-better-than-trivial 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 sign-rank 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=1-2^{-Omega(n^{1-delta})}, even by polynomials of degree n^{1-delta}. 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^{1-delta})} 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.
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Identification of mixtures of discrete product distributions in near-optimal sample and time complexity
We consider the problem of \emph{identifying,} from statistics, a distribution of discrete random variables $$X_1 \ldots,X_n$$ that is a mixture of $$k$$ product distributions. The best previous sample complexity for $$n \in O(k)$$ was $$(1/\zeta)^{O(k^2 \log k)}$$ (under a mild separation assumption parameterized by $$\zeta$$). The best known lower bound was $$\exp(\Omega(k))$$. It is known that $$n\geq 2k-1$$ is necessary and sufficient for identification. We show, for any $$n\geq 2k-1$$, how to achieve sample complexity and run-time complexity $$(1/\zeta)^{O(k)}$$. We also extend the known lower bound of $$e^{\Omega(k)}$$ to match our upper bound across a broad range of $$\zeta$$. Our results are obtained by combining (a) a classic method for robust tensor decomposition, (b) a novel way of bounding the condition number of key matrices called Hadamard extensions, by studying their action only on flattened rank-1 tensors.
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- Award ID(s):
- 2321079
- PAR ID:
- 10608820
- Editor(s):
- Agrawal, Shipra; Roth, Aaron
- Publisher / Repository:
- Proceedings of Machine Learning Research
- Date Published:
- Volume:
- 247
- ISSN:
- 2640-3498
- Page Range / eLocation ID:
- 2071-2091
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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