Optimal Iterative Sketching with the Subsampled Randomized Hadamard Transform

TL;DR

This paper analyzes the performance of iterative Hessian sketching using subsampled randomized Hadamard transforms, providing precise convergence rates and showing improvements over Gaussian projections in least-squares problems.

Contribution

The paper introduces a novel second-moment formula for inverse projected matrices and derives closed-form expressions for optimal step-sizes and convergence rates, extending random matrix theory.

Findings

Haar and randomized Hadamard matrices have identical asymptotic convergence rates.

The convergence rates with these transforms outperform Gaussian projections asymptotically.

New formulas enable precise analysis of iterative sketching algorithms.

Abstract

Random projections or sketching are widely used in many algorithmic and learning contexts. Here we study the performance of iterative Hessian sketch for least-squares problems. By leveraging and extending recent results from random matrix theory on the limiting spectrum of matrices randomly projected with the subsampled randomized Hadamard transform, and truncated Haar matrices, we can study and compare the resulting algorithms to a level of precision that has not been possible before. Our technical contributions include a novel formula for the second moment of the inverse of projected matrices. We also find simple closed-form expressions for asymptotically optimal step-sizes and convergence rates. These show that the convergence rate for Haar and randomized Hadamard matrices are identical, and asymptotically improve upon Gaussian random projections. These techniques may be applied to other algorithms that employ randomized dimension reduction.