The Complexity of Dynamic Least-Squares Regression

TL;DR

This paper establishes the computational complexity boundaries for dynamic least-squares regression, revealing significant differences in update times depending on the dynamic setting and accuracy, and connects these results to the Online Matrix Vector Conjecture.

Contribution

It provides the first sharp complexity separations for dynamic LSR and introduces a novel reduction from exact to approximate OMv, advancing understanding in fine-grained complexity.

Findings

Sharp separation between fully and partially dynamic LSR update times.

Demonstrates hardness of high-accuracy dynamic LSR under certain conditions.

Links approximate OMv hardness to dynamic regression problems.

Abstract

We settle the complexity of dynamic least-squares regression (LSR), where rows and labels $(\mathbf{A}^{(t)}, \mathbf{b}^{(t)})$ can be adaptively inserted and/or deleted, and the goal is to efficiently maintain an $\epsilon$-approximate solution to $\min_{\mathbf{x}^{(t)}} \| \mathbf{A}^{(t)} \mathbf{x}^{(t)} - \mathbf{b}^{(t)} \|_2$ for all $t\in [T]$. We prove sharp separations ($d^{2-o(1)}$ vs. $\sim d$) between the amortized update time of: (i) Fully vs. Partially dynamic $0.01$-LSR; (ii) High vs. low-accuracy LSR in the partially-dynamic (insertion-only) setting. Our lower bounds follow from a gap-amplification reduction -- reminiscent of iterative refinement -- rom the exact version of the Online Matrix Vector Conjecture (OMv) [HKNS15], to constant approximate OMv over the reals, where the $i$-th online product $\mathbf{H}\mathbf{v}^{(i)}$ only needs to be computed to $0.1$-relative error. All previous fine-grained reductions from OMv to its approximate versions only show hardness for inverse polynomial approximation $\epsilon = n^{-\omega(1)}$ (additive or multiplicative) . This result is of independent interest in fine-grained complexity and for the investigation of the OMv Conjecture, which is still widely open.