Lecture 10 - Depth Reduction III

In the last lecture, we saw how to reduce the depth of any circuit of size $s$ computing a polynomial of degree $d$ to a circuit of size $\text{poly}(s,d)$ and depth $O(\log d(\log s+\log d))$ computing the same polynomial.

In particular, we have the following conclusion:

Corollary 1: If $f$ is a polynomial in $n$ variables of degree $d=\mathsf{poly}(n)$ such that $f$ cannot be computed by any circuit of size $\text{poly}(n)$ and depth $O({\log }^{2}n)$, then $f$ is not in $\mathsf{VP}$.

In this lecture, we will see how to reduce the depth even further, all the way to depths 4 and 3, albeit with a siginificant increase in the size of the circuit. This is useful for (at least) two reasons:

1. Circuits of constant depth should (in principle) be easier to analyze.
2. One could hope to get exponential lower bounds on the size of circuits of constant depth computing a hard polynomial.

Whenever we talk about a constant depth circuit (or formula), we will always assume that the circuit has alternating layers of sum and product gates, having the top gate being a sum gate. This convention makes sense as we want to talk about circuits that can compute several polynomials, as opposed to circuits which only compute reducible polynomials.

Note that a depth 2 circuit, denoted by $\mathrm{\Sigma }\mathrm{\Pi }$ circuit, is simply a sum of monomials, and therefore it is easy to give an exponential lower bound on the size of a $\mathrm{\Sigma }\mathrm{\Pi }$ circuit computing a hard polynomial. Since every polynomial is uniquely defined by its monomials (and their coefficients), the size of a $\mathrm{\Sigma }\mathrm{\Pi }$ circuit computing a polynomial is at least the number of monomials in the polynomial. The hard polynomials in this case are all polynomials with exponentially many monomials. A depth 2 circuit is also known as the sparse representation of a polynomial.

Since the complexity of a polynomial in depth 2 circuits is completely understood, and even very easy polynomial families require exponential size depth 2 circuits, we cannot hope to have efficient depth reduction to depth 2 circuits. So, the next question would be: can we reduce the depth of a circuit to depth 3 or 4?

We begin with the surprising result that the answer is yes, for both depth 4 and depth 3 circuits! The first depth reduction to depth 4 was given by Agrawal and Vinay in 2008, and the first depth reduction to depth 3 was given by Gupta, Kamath, Kayal, and Saptharishi in 2014.

Depth Reduction to Depth 4

We begin by showing how to reduce the degree of a circuit all the way to depth 4. We denote a homogeneous depth 4 circuit by $\mathsf{\text{hom-}}\mathrm{\Sigma }\mathrm{\Pi }\mathrm{\Sigma }\mathrm{\Pi }$ circuit. Moreover, we denote by $\mathsf{\text{hom-}}\mathrm{\Sigma }{\mathrm{\Pi }}^{[d/t]}\mathrm{\Sigma }{\mathrm{\Pi }}^{[t]}$ a homogeneous depth 4 circuit where the top product gate has fanin $d/t$ and the bottom product gate has fanin $t$. The top fanin of the circuit is the fanin of the top sum gate.

Theorem 1 ([AV'08, Koi'12, Tav'15]): Let $f$ be a form in $n$ variables of degree $d$ computed by a homogeneous circuit of size $s$. Then, for any $0<t\le d$, there is a $\mathsf{\text{hom-}}\mathrm{\Sigma }{\mathrm{\Pi }}^{[d/t]}\mathrm{\Sigma }{\mathrm{\Pi }}^{[t]}$ circuit of size $s^{O(t+d/t)}$ and top fanin $s^{O(d/t)}$ computing $f$.

Proof: Once again, we can assume w.l.o.g. that $s\ge n$. By VSBR, we know that there is a homogeneous circuit $\mathrm{\Psi }$ of size $\mathsf{poly}(s,d)$ and depth $O(\log (d))$ computing $f$, with the following properties:

1. The circuit $\mathrm{\Psi }$ has alternating layers of sum and product gates.
2. The top gate of $\mathrm{\Psi }$ is a sum gate.
3. Each product gate $v\in \mathrm{\Psi }$ has fanin at most $5$, and each child $u$ of $v$ satisfies $\mathrm{deg}(u)\le \mathrm{deg}(v)/2$.

We will now show how to convert $\mathrm{\Psi }$ into a $\mathsf{\text{hom-}}\mathrm{\Sigma }{\mathrm{\Pi }}^{[d/t]}\mathrm{\Sigma }{\mathrm{\Pi }}^{[t]}$ circuit.

Depth Reduction to Depth 3