We give a “regularity lemma” for degree-$d$ polynomial threshold functions (PTFs) over the Boolean cube $\{-1,1\}^n$. Roughly speaking, this result shows that every degree-$d$ PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a “regular” PTF is a PTF $\sign(p(x))$ where the influence of each variable on the polynomial $p(x)$ is a small fraction of the total influence of $p$.

As an application of this regularity lemma, we prove that for any constants $d \geq 1, \epsilon > 0$, every degree-$d$ PTF over $n$ variables can be approximated to accuracy $\epsilon$ by a constant-degree PTF that has integer weights of total magnitude $O_{\epsilon,d}(n^d)$. This weight bound is shown to be optimal up to logarithmic factors