Lecture 15

In this lecture we finished the proof of the Switching Lemma and showed how it implied Håstad's Theorem: Computing Parity by depth-$k$ circuits (unbounded fan-in) requires size $2^{\Omega \left(n^{1/(k-1)}\right)}$.

We then stated and sketched the proof of the Linial-Mansour-Nisan (LMN) Theorem: If $f$ is computed by depth-$k$, size-$S$ circuits, then there is a multilinear real polynomial $p$ of degree at most $t=O(\log (S/\epsilon ){)}^k$ which approximates $f$ in the following sense:

E${}_x\left[\right(f(x_1,...,x_n)-p(x_1,...,x_n){)}^2]\le \epsilon$,

where $x=(x_1,...x_n)$ is a uniformly random string.

By the way, Håstad subsequently slightly sharpened this; he improved $t$ to

$O(\log (S/\epsilon ){)}^{k-1}\cdot min\{\log S,\log (1/\epsilon )\}$.