The Complexity of Dynamic Least-Squares Regression

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• github.com/pengbinghui/dynamicl2regression
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Abstract

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 -approximate solution to _x^(t) \| A^(t) x^(t) - b^(t) \|_2 for all t [T]. We prove sharp separations (d^2-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 -- 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 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 = n^-(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.

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