Lower bounds for bounded depth arithmetic circuits

Abstract

Proving lower bounds for arithmetic circuits is a problem of fundamental importance intheoretical computer science. In recent years, an approach to this problem has emergedvia the depth reduction results of Agrawal and Vinay [AV08], which show that strongenough lower bounds for extremely structured bounded depth circuits (even homogeneousdepth-4 circuits) suffice for general arithmetic circuits lower bounds. In this dissertation,we study homogeneous depth-4 and homogeneous depth-5 arithmetic circuitswith a view towards proving strong lower bounds, and understanding the optimality ofthe depth reduction results. Some of our main results are as follows.• We show a hierarchy theorem for bottom fan-in for homogeneous depth-4 circuitswith bounded bottom fan-in. More formally, we show that there for a wide rangeof choice of parameter t, there is a homogeneous polynomial in n variables ofdegree d = nΘ(1) which can be computed by a homogeneous depth-4 circuit ofbottom fan-in t, but any homogeneous depth-4 circuit of bottom fan-in at mostt/20 must have top fan-in nΩ(d/t)• We show that there is an explicit polynomial family such that any homogeneousdepth-4 arithmetic circuit computing it must have super-polynomial size. Thesewere the first superpolynomial lower bounds for homogeneous depth-4 circuitswith no restriction on top or bottom fan-in. Simultaneously and independently,a similar lower bound was also proved by Kayal et al [KLSS14b].• We show that any homogeneous depth-4 circuit computing the iterated matrixmultiplication polynomial in n variables and degree d = nΘ(1) must have size atleast nΩ(√d). This shows that the upper bounds of depth reduction from generalarithmetic circuits to homogeneous depth-4 circuits are almost optimal, up to aconstant in the exponent. Moreover, these were the first nΩ(√d) lower boundsfor homogeneous depth-4 circuits over all fields. Prior to our work, Kayal etal. [KLSS14a] had shown such a lower bound over the fields of characteristic zero.• We show that there is a family of polynomials in n variables and degree d =O(log2 n) which can be computed by linear size homogeneous depth-5 circuitsand polynomial size non-homogeneous depth-3 circuits but require homogeneousdepth-4 circuits of size nΩ(√d). In addition to indicating the power of increaseddepth, and non-homogeneity, these results also show that for the range of parametersconsidered here, the upper bounds for the depth reduction results [AV08,Koi12, Tav15] are close to optimal in a very strong sense : a general depth reductionto homogeneous depth-4 circuits of size nΩ(√d) is not possible even forhomogeneous depth-5 circuits of linear size.• We show an exponential lower bound for homogeneous depth-5 circuits computingan explicit polynomial over all finite fields of constant size. For any non-binaryfield, these were the first such super-polynomial lower bounds, and prior to ourwork, even cubic lower bounds were not known for homogeneous depth-5 circuits.On the way to our proofs, we study the complexity of some natural polynomial families(for instance, homogeneous depth-4, depth-5 circuits, iterated matrix multiplication)with respect to many existing partial derivative based complexity measures, and alsodefine and analyze some new variants of these measures [KS14, KS15b].