The Limits of Depth Reduction for Arithmetic Formulas: It's All About the Top Fan-In

Abstract

In recent years, a very exciting and promising method for proving lower bounds for arithmetic circuits has been proposed. This method combines the method of depth reduction developed in the works of Agrawal-Vinay [AV08], Koiran [Koi12] and Tavenas [Tav13], and the use of the shifted partial derivative complexity measure developed in the works of Kayal [Kay12] and Gupta et al [GKKS13a]. These results inspired a flurry of other beautiful results and strong lower bounds for various classes of arithmetic circuits, in particular a recent work of Kayal et al [KSS13] showing superpolynomial lower bounds for regular arithmetic formulas via an improved depth reduction for these formulas. It was left as an intriguing question if these methods could prove superpolynomial lower bounds for general (homogeneous) arithmetic formulas, and if so this would indeed be a breakthrough in arithmetic circuit complexity.In this paper we study the power and limitations of depth reduction and shifted partial derivatives for arithmetic formulas. We do it via studying the class of depth 4 homogeneous arithmetic circuits. We show: (1) the first superpolynomial lower bounds for the class of homogeneous depth 4 circuits with top fan-in o(log n). The core of our result is to show improved depth reduction for these circuits. This class of circuits has received much attention for the problem of polynomial identity testing. We give the first nontrivial lower bounds for these circuits for any top fan-in ≥ 2. (2) We show that improved depth reduction is not possible when the top fan-in is Ω(log n). In particular this shows that the depth reduction procedure of Koiran and Tavenas [Koi12,Tav13] cannot be improved even for homogeneous formulas, thus strengthening the results of Fournier et al [FLMS13] who showed that depth reduction is tight for circuits, and answering some of the main open questions of [KSS13,FLMS13]. Our results in particular suggest that the method of improved depth reduction and shifted partial derivatives may not be powerful enough to prove superpolynomial lower bounds for (even homogeneous) arithmetic formulas.

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“…For, say, d ∈ o(lg N/ lg lg N), and for sufficiently large N, any α > 0 works.) All previous lower bound results using the shifted partial derivative method also obtain similar statements [9,6,17,16].…”

“…This was further improved by Kayal, Saha, and Saptharishi [13] who gave a family of explicit polynomials in VNP for which the shifted partial derivative complexity is (nearly) as large as possible 1 and hence showed a lower bound of N Ω(d/t) for the top fan-in of ΣΠ [O(d/t)] ΣΠ [t] circuits computing these polynomials. Later, a similar result for a polynomial in VP was proved in [6] and this was subsequently strengthened by Kumar and Saraf [17], who gave a polynomial computable by homogeneous ΠΣΠ circuits such that any ΣΠ [O(d/t)] ΣΠ [t] circuits computing it must have top fan-in N Ω(d/t) . Finally, using a variant of the shifted partial derivative measure, Kayal et al [12] and Kumar and Saraf [16] were able to prove similar lower bounds for general depth-4 homogeneous circuits as well.…”

On the Power of Homogeneous Depth 4 Arithmetic Circuits

“…The notion of shifted partial derivatives was introduced in [Kay12] and was subsequently used as a complexity measure in proving several recent lower bound results [FLMS13,GKKS13a,KSS13,KS13,KS]. In this paper, we use a variant of the method which first introduced in [KLSS14].…”

“…The best lower bound known for homogeneous depth 4 circuits computing a poly in VP is the lower bound of n Ω(log n) by [LSS,KLSS14]. Recall that when one introduces the restriction on bounded bottom fanin, then stronger exponential lower bounds are indeed known [FLMS13,KS]. This fact is also related to the next bullet point below.…”

“…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.…”