  Sketching is a dimensionality reduction technique where one compresses a
matrix by linear combinations that are chosen at random. A line of work has
shown how to sketch the Hessian to speed up each iteration in a second order
method, but such sketches usually depend only on the matrix at hand, and in a
number of cases are even oblivious to the input matrix. One could instead hope
to learn a distribution on sketching matrices that is optimized for the
specific distribution of input matrices. We show how to design learned sketches
for the Hessian in the context of second order methods. We prove that a smaller
sketching dimension of the column space of a tall matrix is possible, given an
oracle that can predict the indices of the rows of large leverage score. We
design such an oracle for various datasets, and this leads to a faster
convergence of the well-studied iterative Hessian sketch procedure, which
applies to a wide range of problems in convex optimization. We show empirically
that learned sketches, compared with their "non-learned" counterparts, do
improve the approximation accuracy for important problems, including LASSO and
matrix estimation with nuclear norm constraints.
