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 $2^{\Omega(n^{1/(k-1)})}$.

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${}_{x}[(f(x_1, ..., x_n) - p(x_1, ..., x_n))^2] \leq \epsilon$,

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

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

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