Sparse sketches with small inversion bias

Abstract: For a tall $n\times d$ matrix $A$ and a random $m\times n$ sketching matrix $S$, the sketched estimate of the inverse covariance matrix $(A^\top A)^{-1}$ is typically biased: $E[(\tilde A^\top\tilde A)^{-1}]\ne(A^\top A)^{-1}$, where $\tilde A=SA$. This phenomenon, which we call inversion bias, arises, e.g., in statistics and distributed optimization, when averaging multiple independently constructed estimates of quantities that depend on the inverse covariance. We develop a framework for analyzing inversion bias, based on our proposed concept of an $(\epsilon,\delta)$-unbiased estimator for random matrices. We show that when the sketching matrix $S$ is dense and has i.i.d. sub-gaussian entries, then after simple rescaling, the estimator $(\frac m{m-d}\tilde A^\top\tilde A)^{-1}$ is $(\epsilon,\delta)$-unbiased for $(A^\top A)^{-1}$ with a sketch of size $m=O(d+\sqrt d/\epsilon)$. This implies that for $m=O(d)$, the inversion bias of this estimator is $O(1/\sqrt d)$, which is much smaller than the $\Theta(1)$ approximation error obtained as a consequence of the subspace embedding guarantee for sub-gaussian sketches. We then propose a new sketching technique, called LEverage Score Sparsified (LESS) embeddings, which uses ideas from both data-oblivious sparse embeddings as well as data-aware leverage-based row sampling methods, to get $\epsilon$ inversion bias for sketch size $m=O(d\log d+\sqrt d/\epsilon)$ in time $O(\text{nnz}(A)\log n+md^2)$, where nnz is the number of non-zeros. The key techniques enabling our analysis include an extension of a classical inequality of Bai and Silverstein for random quadratic forms, which we call the Restricted Bai-Silverstein inequality; and anti-concentration of the Binomial distribution via the Paley-Zygmund inequality, which we use to prove a lower bound showing that leverage score sampling sketches generally do not achieve small inversion bias.