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.

翻译：我们确定了动态最小二乘回归（LSR）的复杂度，其中行和标签$(\mathbf{A}^{(t)}, \mathbf{b}^{(t)})$可自适应地插入和/或删除，目标是为所有$t\in [T]$高效维护$\min_{\mathbf{x}^{(t)}} \| \mathbf{A}^{(t)} \mathbf{x}^{(t)} - \mathbf{b}^{(t)} \|_2$的$\epsilon$-近似解。我们证明了以下情形间平摊更新时间的显著分离（$d^{2-o(1)}$ 与 $\sim d$）：（i）完全动态与部分动态$0.01$-LSR；（ii）部分动态（仅插入）设置下高精度与低精度LSR。我们的下界源于一种间隙放大归约——类似于迭代精化——将精确版本的在线矩阵向量猜想（OMv）[HKNS15]归约到实数上的常数近似OMv问题，其中第$i$个在线乘积$\mathbf{H}\mathbf{v}^{(i)}$只需计算至$0.1$相对误差。此前所有从OMv到其近似版本的细粒度归约仅能证明逆多项式近似$\epsilon = n^{-\omega(1)}$（加法或乘法）下的困难性。该结果在细粒度复杂性理论及仍广泛开放的OMv猜想研究中具有独立意义。