On polynomial approximations to AC

Abstract

Classical AC 0 approximation results show that any AC 0 circuit of size s and depth d has an ε-error probabilistic polynomial over the reals of degree (log(s/ε)) O(d) . We improve this upper bound to (log s) O(d) · log(1/ε), which is much better for small values of ε. We then use this result to show that (log s) O(d) · log(1/ε)-wise independence fools AC 0 circuits of size s and depth d up to error at most ε, improving on Tal's strengthening of Braverman's result that (log(s/ε)) O(d)wise independence suffices. To our knowledge, this is the first PRG construction for AC 0 that achieves optimal dependence on the error ε. We also prove lower bounds on the best polynomial approximations to AC 0 . We show that any polynomial approximating the OR function on n bits to a small constant error must have degree at least Ω( log n). This result improves exponentially on a result of Meka, Nguyen, and Vu (Theory Comput. 2016). K E Y W O R D Santi-concentration for polynomials, bounded independence, constant-depth circuits, probabilistic polynomials MOTIVATION AND RESULTSIn this paper, we study AC 0 circuits, the family of circuits of constant depth and polynomial size (in the input length) with unbounded-fanin AND and OR gates. We use AC 0 (s, d) to denote the family of AC 0 circuits of size s and depth d.

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“…Oliveira and Santhanam [22] use a variant over the field F p to prove lower bounds for AC 0 [p] compression protocols 2 for the Majority function. Recently, Harsha and the second author [17] used a variant over the reals to prove that (log s) O(d) • log(1/ε)-wise independence ε-fools AC 0 circuits of size s and depth d, improving upon results of Braverman [8] and Tal [30].…”

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“…To prove this, we construct approximating polynomials as in Section 3 for size-s depth-d AC 0 [MOD p ] formulas of degree O((1/d)p log s) d−1 . The construction is almost exactly the same, except that we use F p -analogs of Lemma 3.6 (which is true with the weaker degree bound of (p − 1) • log(1/ε) [16]) and Lemma 3.7 (which holds as stated over all fields [5]). The Smolensky-Szegedy degree lower bound for approximating the Majority function (Lemma 3.2) is true over all fields.…”

“…That is, for inputs a ∈ Y and b ∈ N , the 0-weights of C (a) and C (b) have 0-weights that are either at most exp(−pt) • exp(−γ pt) or at least exp(−pt) • exp(γ pt). In particular, they are multiplicatively separated by 1 + γ where γ ≈ (pt) • γ, 5 which is much greater than γ if pt is much greater than 1. In an analogous way, we can construct a depth-1 circuit (made up of ANDs this time) that amplifies a multiplicative gap of (1 + γ) in the 0-weights to a multiplicative gap of (1 + γ ) in the weights.…”

On the probabilistic degree of OR over the reals

“…Classical results in polynomial approximation of Boolean functions [5,15,16] show that the OR function over n variables, denoted by OR n , has -error probabilistic degree at most O (log n ⋅ log(1∕ )). This basic construction for the OR function is then recursively used to show that any function computed by an AC 0 circuit of size s and depth d has -error probabilistic degree at most (log s) O( ) ⋅ log(1∕ ) (see work by Harsha and Srinivasan [8] for recent improvements). These results can then be used to prove [12,13] a (slightly weaker) version of Håstad's celebrated theorem [7] that parity does not have subexponential-sized AC 0 circuits.…”

“…However, until recently, it was not clear whether any dependence on n is necessary in P-eg (OR n ) over the reals. 2 In recent papers of Meka, Nguyen and Vu [11] and Harsha and Srinivasan [8], it was shown using anti-concentration of low-degree polynomials that the P-eg 1∕4 (OR n ) =Ω( √ log n). The main objective of this paper is to obtain a better understanding of the -error probabilistic degree of OR n , P-eg (OR n ).…”