The Computational Power of Depth Five Arithmetic Circuits

Abstract

Surprising and beautiful depth reduction results show that sufficiently strong lower bounds for bounded-depth arithmetic circuits imply superpolynomial lower bounds for general arithmetic circuits. Motivated by this, in the last few years, there has been renewed interest in the question of proving superpolynomial lower bounds for bounded-depth circuits, starting with homogeneous depth-4 circuits. Following a sequence of recent results, we now know an $n^{\Omega(\sqrt{d})}$ lower bound for homogeneous depth-4 circuits, for an explicit polynomial of degree $d$ in $n$ variables. It is also known that any asymptotic improvement in the exponent of this lower bound implies a superpolynomial lower bound for general arithmetic circuits. An intriguing fact is that in spite of all this recent progress which seems to bring us to the edge of the chasm at depth-4, it appears to shine very little light on the question of lower bounds for even slight generalizations of homogeneous depth-4 circuits. Indeed, superquadratic lower bounds are not known for even nonhomogeneous depth-4 circuits, or homogeneous depth-5 circuits, and supercubic lower bounds are not known for nonhomogeneous depth-3 circuits. In this paper, we study homogeneous depth-5 circuits, with the aim of proving superpolynomial lower bounds for them and understanding their computational power and limitations when compared to homogeneous depth-4 circuits. We prove the following results. Depth-$5$ versus depth-$4$ circuits. We show that there is a family of polynomials $\{P_n\}$, where $P_n$ is a polynomial in $n$ variables of degree at most $d=O(\log^2n)$, such that $P_n$ can be computed by linear sized homogeneous depth-5 circuits; $P_n$ can be computed by $\operatorname{poly}(n)$ sized nonhomogeneous depth-3 circuits; and any homogeneous depth-4 circuit computing $P_n$ must have size at least $n^{\Omega(\sqrt{d})}$. This shows that the parameters for the depth reduction results of M. Agrawal and V. Vinay, P. Koiran, and S. Tavenas are tight for extremely restricted classes of arithmetic circuits, for instance homogeneous depth-5 circuits and nonhomogeneous depth-3 circuits, and over an appropriate range of parameters. This also qualitatively improves a result of Kumar and Saraf, which showed that the parameters of depth reductions are optimal for algebraic branching programs. Depth-$5$ circuits over small fields. We show that there is an explicit family $\{P_d\}$ of polynomials, where $P_d$ is of degree $d$ in $n=d^{O(1)}$ variables, such that over all finite fields $\mathbb{F}_q$, any homogeneous depth-5 circuit which computes $P_d$ must have size at least $\exp(\Omega_q(\sqrt{d}))$. To the best of our knowledge, this is the first superpolynomial lower bound for this class for any field $\mathbb{F}_q\neq\mathbb{F}_2$. Our proof builds on the ideas developed on the way to proving lower bounds for homogeneous depth-4 circuits and for nonhomogeneous depth-3 circuits over finite fields. Our key insight is to look at the space of shifted partial derivatives of a polynomial as a space of functions from $\mathbb{F}_q^n\rightarrow\mathbb{F}_q$ as opposed to looking at them as a space of formal polynomials and builds over a tighter analysis of the lower bound of Kumar and Saraf.