  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.
