The Complexity of Dynamic Least-Squares Regression

We settle the complexity of dynamic least-squares regression (LSR), where rows and labels (A(t), b(t)) can be adaptively inserted and/or deleted, and the goal is to efficiently maintain an ϵ-approximate solution to minx(t)||A(t) x(t)-b(t)||_2 for all t ∈[T]. We prove sharp separations (d2-o(1). vs. .∼ 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-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 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 ϵ= n-Ω(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.