The Complexity of Dynamic Least-Squares Regression

We settle the complexity of dynamic least-squares regression (LSR), where rows and labels $\left(\mathbf{A}^{(t)}, \mathbf{b}^{(t)}\right)$ can be adaptively inserted and/or deleted, and the goal is to efficiently maintain an $\epsilon$-approximate solution to $\min _{\mathbf{x}^{(t)}}\left\|\mathbf{A}^{(t)} \mathbf{x}^{(t)}-\mathbf{b}^{(t)}\right\|_{2}$ for all $t \in[T]$. We prove sharp separations $\left(d^{2-o(1)}\right.$ vs. $\left.\sim d\right)$ 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-from the exact version of the Online Matrix Vector Conjecture (OMv) [HKNS15], to constant approximate OMv over the reals, where the i-th online product $\mathrm{Hv}^{(i)}$ 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.