Learning-Augmented Sketches for Hessians
This work addresses the efficiency of convex optimization algorithms for researchers and practitioners, but it is incremental as it builds on existing sketching techniques by incorporating learning.
The paper tackles the problem of improving Hessian sketching for second-order optimization methods by learning sketching matrices tailored to specific data distributions, resulting in a smaller required sketching dimension and faster convergence for iterative Hessian sketch procedures, with empirical improvements in approximation accuracy for problems like LASSO and matrix estimation.
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.