Learning-Augmented Sketches for Hessians

Sketching is a dimensionality reduction technique where one compresses a matrix by linear combinations that are typically 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, where we learn potentially different sketches for the different iterations of an optimization procedure. We show empirically that learned sketches, compared with their "non-learned" counterparts, improve the approximation accuracy for important problems, including LASSO, SVM, and matrix estimation with nuclear norm constraints. Several of our schemes can be proven to perform no worse than their unlearned counterparts. Additionally, we show that a smaller sketching dimension of the column space of a tall matrix is possible, assuming an oracle for predicting rows which have a large leverage score.