  In second-order optimization, a potential bottleneck can be computing the
Hessian matrix of the optimized function at every iteration. Randomized
sketching has emerged as a powerful technique for constructing estimates of the
Hessian which can be used to perform approximate Newton steps. This involves
multiplication by a random sketching matrix, which introduces a trade-off
between the computational cost of sketching and the convergence rate of the
optimization algorithm. A theoretically desirable but practically much too
expensive choice is to use a dense Gaussian sketching matrix, which produces
unbiased estimates of the exact Newton step and which offers strong
problem-independent convergence guarantees. We show that the Gaussian sketching
matrix can be drastically sparsified, significantly reducing the computational
cost of sketching, without substantially affecting its convergence properties.
This approach, called Newton-LESS, is based on a recently introduced sketching
technique: LEverage Score Sparsified (LESS) embeddings. We prove that
Newton-LESS enjoys nearly the same problem-independent local convergence rate
as Gaussian embeddings, not just up to constant factors but even down to lower
order terms, for a large class of optimization tasks. In particular, this leads
to a new state-of-the-art convergence result for an iterative least squares
solver. Finally, we extend LESS embeddings to include uniformly sparsified
random sign matrices which can be implemented efficiently and which perform
well in numerical experiments.
