Nearly optimal solutions for the Chow Parameters Problem and low-weight approximation of halfspaces

The \emph{Chow parameters} of a Boolean function $f: {-1,1}ⁿ \to {-1,1}$ are its $n+1$ degree-0 and degree-1 Fourier coefficients. It has been known since 1961 (Chow, Tannenbaum) that the (exact values of the) Chow parameters of any linear threshold function $f$ uniquely specify $f$ within the space of all Boolean functions, but until recently (O'Donnell and Servedio) nothing was known about efficient algorithms for \emph{reconstructing} $f$ (exactly or approximately) from exact or approximate values of its Chow parameters. We refer to this reconstruction problem as the \emph{Chow Parameters Problem.} Our main result is a new algorithm for the Chow Parameters Problem which, given (sufficiently accurate approximations to) the Chow parameters of any linear threshold function $f$, runs in time $\tilde{O}(n^2)\cdot (1/\eps)^{{O(\log2(1/\eps))}$} and with high probability outputs a representation of an LTF $f'$ that is $\eps$-close to $f$. The only previous algorithm (O'Donnell and Servedio) had running time $\poly(n) \cdot 2^{{2{\tilde{O}(1/\eps2)}}.$} As a byproduct of our approach, we show that for any linear threshold function $f$ over ${-1,1}^n$, there is a linear threshold function $f'$ which is $\eps$-close to $f$ and has all weights that are integers at most $\sqrt{n} \cdot (1/\eps)^{{O(\log2(1/\eps))}$.} This significantly improves the best previous result of Diakonikolas and Servedio which gave a $\poly(n) \cdot 2^{{\tilde{O}(1/\eps{2/3})}$} weight bound, and is close to the known lower bound of $\max{\sqrt{n},$ $(1/\eps)^{\Omega(\log \log (1/\eps))}}$ (Goldberg, Servedio). Our techniques also yield improved algorithms for related problems in learning theory.