Learning-Augmented Sketches for Hessians

TL;DR

This paper introduces learned sketching techniques for Hessians in second order optimization methods, improving convergence and approximation accuracy by leveraging data-specific information.

Contribution

It proposes a novel approach to design learned sketches for Hessians, with theoretical guarantees and empirical validation on optimization problems.

Findings

Smaller sketching dimensions are achievable with an oracle predicting leverage scores.

Learned sketches lead to faster convergence in Hessian-based methods.

Empirical results show improved approximation accuracy on LASSO and matrix estimation.

Abstract

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.