SQ Lower Bounds for Learning Mixtures of Linear Classifiers

Our main result is a Statistical Query (SQ) lower bound suggesting that known algorithms for this problem are essentially best possible,even for the special case of uniform mixtures.In particular, we show that the complexity of any SQ algorithm for the problem is $n^{\mathrm{poly}(1/\Delta) \log(r)}$,where $\Delta$ is a lower bound on the pairwise $\ell_2$-separation between the $\mathbf{v}_{\ell}$'s.The key technical ingredient underlying our result is a new construction of spherical designs on the unit sphere that may be of independent interest.