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Revision #3 to TR13-151 | 28th April 2014 20:57

Hardness Amplification and the Approximate Degree of Constant-Depth Circuits

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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 depth-three circuit $F: \{-1,1\}^n \to \{-1,1\}$ of polynomial-size such that any degree-$d$ polynomial cannot pointwise approximate $F$ to error better than $1-\exp\left(-\tilde{\Omega}(nd^{-3/2})\right)$. As a consequence of our main result, we obtain an $\exp\left(-\tilde{\Omega}(n^{2/5})\right)$ upper bound on the the discrepancy of a function in AC$^0$, and an $\exp\left(\tilde{\Omega}(n^{2/5})\right)$ lower bound on the threshold weight of AC$^0$, improving over the previous best results of $\exp\left(-\Omega(n^{1/3})\right)$ and $\exp\left(\Omega(n^{1/3})\right)$ respectively.
Our techniques also yield a new lower bound of $\Omega\left(n^{1/2}/\log^{(d-2)/2}(n)\right)$ 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.



Changes to previous version:

Reorganized summary of results and added discussion of subsequent work. Included a more general observation relating one-sided approximate degree to approximate degree.


Revision #2 to TR13-151 | 25th December 2013 07:21

Hardness Amplification and the Approximate Degree of Constant-Depth Circuits





Revision #2
Authors: Mark Bun, Justin Thaler
Accepted on: 25th December 2013 07:21
Downloads: 653
Keywords: 


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 depth-three circuit $F: \{-1,1\}^n \to \{-1,1\}$ of polynomial-size such that any degree-$d$ polynomial cannot pointwise approximate $F$ to error better than $1-\exp\left(-\tilde{\Omega}(nd^{-3/2})\right)$. As a consequence of our main result, we obtain an $\exp\left(-\tilde{\Omega}(n^{2/5})\right)$ upper bound on the the discrepancy of a function in \acz, and an $\exp\left(-\tilde{\Omega}(n^{2/5})\right)$ lower bound on the threshold weight of \acz, improving over the previous best results of $\exp\left(-\Omega(n^{1/3})\right)$ and $\exp\left(\Omega(n^{1/3})\right)$ respectively.
Our techniques also yield a new lower bound of $\Omega\left(n^{1/2}/\log^{(d-2)/2}(n)\right)$ 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.



Changes to previous version:

Updated to discuss prior work on upper bounds for AND-OR trees via quantum query algorithms.


Revision #1 to TR13-151 | 8th November 2013 17:00

Hardness Amplification and the Approximate Degree of Constant-Depth Circuits


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 depth-three circuit $F: \{-1,1\}^n \to \{-1,1\}$ of polynomial-size such that any degree-$d$ polynomial cannot pointwise approximate $F$ to error better than $1-\exp\left(-\tilde{\Omega}(nd^{-3/2})\right)$. As a consequence of our main result, we obtain an $\exp\left(-\tilde{\Omega}(n^{2/5})\right)$ upper bound on the the discrepancy of a function in AC$^0$, and an $\exp\left(\tilde{\Omega}(n^{2/5})\right)$ lower bound on the threshold weight of AC$^0$, improving over the previous best results of $\exp\left(-\Omega(n^{1/3})\right)$ and $\exp\left(\Omega(n^{1/3})\right)$ respectively.
Our techniques also yield a new lower bound of $\Omega\left(n^{1/2}/\log^{(d-2)/2}(n)\right)$ 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.



Changes to previous version:

Fixed a typo and removed macros from the abstract.


Paper:

TR13-151 | 7th November 2013 06:24

Hardness Amplification and the Approximate Degree of Constant-Depth Circuits


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 depth-three circuit $F: \{-1,1\}^n \to \{-1,1\}$ of polynomial-size such that any degree-$d$ polynomial cannot pointwise approximate $F$ to error better than $1-\exp\left(-\tilde{\Omega}(nd^{-3/2})\right)$. As a consequence of our main result, we obtain an $\exp\left(-\tilde{\Omega}(n^{2/5})\right)$ upper bound on the the discrepancy of a function in \acz, and an $\exp\left(-\tilde{\Omega}(n^{2/5})\right)$ lower bound on the threshold weight of \acz, improving over the previous best results of $\exp\left(-\Omega(n^{1/3})\right)$ and $\exp\left(\Omega(n^{1/3})\right)$ respectively.
Our techniques also yield a new lower bound of $\Omega\left(n^{1/2}/\log^{(d-2)/2}(n)\right)$ 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.



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