Improved Bounds for Reduction to Depth 4 and Depth 3

Abstract

Koiran [8] showed that if an n-variate polynomial fn of degree d (with d = n O(1) ) is computed by a circuit of size s, then it is also computed by a homogeneous circuit of depth four and of size 2 O( √ d log(n) log(s)) . Using this result, Gupta, Kamath, Kayal and Saptharishi [7] found an upper bound for the size of a depth three circuit computing fn. We improve here Koiran's bound. Indeed, we show that it is possible to transform an arithmetic circuit into a depth four circuit of size 2 O √ d log(ds) log(n) . Then, mimicking the proof in [7], it also implies an 2 O √ d log(ds) log(n)upper bound for depth three circuits. This new bound is not far from optimal in the sense that Gupta, Kamath, Kayal and Saptharishi [6] also showed a 2 Ω( √ d) lower bound for the size of homogeneous depth four circuits such that gates at the bottom have fan-in at most √ d. Finally, we show that this last lower bound also holds if the fan-in is at least √ d.

On the Power of Homogeneous Depth 4 Arithmetic Circuits

“…The lower bounds of [GKKS13a] were later improved to 2 Ω( √ n log n) in a follow up work of Kayal, Saha, Saptharishi [KSS13]. These results were all the more remarkable in the light of the results of Koiran [Koi12] and Tavenas [Tav13] who had in fact showed that 2 ω( √ n log n) lower bounds even for homogeneous ΣΠΣΠ…”

“…In particular they imply that the depth reduction results of Koiran [Koi12] and Tavenas [Tav13] are tight even for reductions to general homogeneous depth 4 circuits (without the restriction of bounded bottom fanin).…”

“…Thus, in order to separate VNP from VP, it would suffice to show a super-polynomial lower bound for just circuits of depth O(log 2 n). In an intriguing line of recent works in this direction, Agrawal-Vinay [AV08], Koiran [Koi12] and Tavenas [Tav13] built upon the results of Valiant et al [VSBR83] and showed that much stronger depth reductions are possible. In order to separate VNP form VP, it would suffice to prove strong enough (n ω( √ n) ) lower bounds for just homogeneous depth 4 circuits.…”

“…This is contrary to what is known for Boolean circuits, where we know exponential lower bounds for bounded depth circuits. This seemed surprising until the depth reduction results of Agrawal-Vinay [AV08] and later Koiran [Koi12] and Tavenas [Tav13], which demontrated that in some sense, homogeneous depth 4 circuits capture the inherent complexity of general arithmetic circuits. In a breakthrough result in 2012, Gupta, Kamath, Kayal and Saptharishi [GKKS13a], made the first major progress on the problem of obtaining lower bounds for bounded depth circuits, by proving 2 Ω( √ n) lower bounds for an explicit polynomial of degree n in n O(1) variables computed by a homogeneous depth 4 circuit, where the fan-in of the product gates at the bottom level of the depth 4 circuits is bounded by √ n. For ease of exposition, let us denote the class of depth 4 circuits with bottom fanin √ n by ΣΠΣΠ [ √ n] circuits.…”

“…On one hand, these results give us extremely strong lower bounds for an interesting class of depth 4 homogeneous circuits. On the other hand, since these lower bounds also hold for polynomials in VP and for homogeneous formulas [FLMS13,KS], it follows that the depth reduction results of Koiran [Koi12] and Tavenas [Tav13] to the class of homogeneous ΣΠΣΠ [ √ n] circuits are tight and cannot be improved even for homoegeneous formulas.…”