Technical Perspective: Low-depth arithmetic circuits

by Agrawal and Vinay,1 which they called a “chasm at depth 4” appeared in 2008. Its clear message: proving lower bounds for (even homogeneous) depth-4 circuits is as hard (or, for optimists, as useful) as proving lower bounds on general circuits. Extending the celebrated depth reduction technique of Valiant, Skyum, Berkowitz and Rackoff, this paper (with subsequent improvements of Tavenas and Koiran) shows that any arithmetic circuit of size s computing a degree d polynomial can also be computed by a homogeneous depth-4 circuit of size sΟ(√d ). For example, proving a subexponential nω(√ n ) lower bound for computing the permanent on n × n matrices, on such a weak constant-depth circuit would separate VP from VNP! But recall, even for depth-3 we were stuck. Next, a series of papers, mainly by subsets of the present authors and a few others got us extremely close to this goal! Using deep ideas and results from algebraic geometry originating from the work of Hilbert in commutative algebra, they have extended the method of partial derivatives to the much stronger “shifted partial derivatives” and combined with other ideas including very fine combinatorial analysis were able to reach the chasm, but not cross it. More precisely they proved lower bounds of n√ n ) or both determinant and permanent. Note that for the determinant this lower bound is tight, and changing the Ω to ω in this expression for the permanent would thus separate the two and hence separate VP from VNP. So far for depth 4. The following paper proves that the very same chasm actually exists in depth 3, at least over fields of characteristic 0 like the rational numbers. More precisely, as above, every size s arithmetic circuit computing a polynomial of degree d can be computed by a depth-3 circuit of size sΟ(√d ) (which unA series of important works in the 1980s on constant-depth Boolean circuits gives a very good picture of their limitations, including tight exponential lower bounds on extremely simple functions like the symmetric functions. The basic message is that constant-depth (and polynomial size) is an extremely weak class of algorithms. Strangely (and specifically over large enough fields) this intuition fails completely for arithmetic circuits. In 1980, Ben-Or already made the important simple observation that the symmetric polynomials can be computed in depth-3 by a quadratic size formula! So, the challenge to prove exponential lower bounds for this simple model was on. Such bounds were proved under further restrictions (like homogeneity) by Nisan-Wigderson, who introduced important techniques of partial derivatives and random restrictions to the study arithmetic circuits. While the best general lower bound is quadratic (matching Ben-Or’s result for symmetric polynomials), there was still a belief that constant depth is a weak model and we should easily prove much better bounds, for harder functions like permanent and even determinant. Still, no such progress followed for over a decade. A surprising and influential paper