The Complexity of Dynamic Least-Squares Regression

Shunhua Jiang, Binghui Peng, Omri Weinstein

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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.

Original languageAmerican English
Title of host publicationProceedings - 2023 IEEE 64th Annual Symposium on Foundations of Computer Science, FOCS 2023
PublisherIEEE Computer Society
Number of pages23
ISBN (Electronic)9798350318944
StatePublished - 2023
Event64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023 - Santa Cruz, United States
Duration: 6 Nov 20239 Nov 2023

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428


Conference64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023
Country/TerritoryUnited States
CitySanta Cruz

Bibliographical note

Publisher Copyright:
© 2023 IEEE.


  • Numerical linear algebra
  • dynamic algorithms
  • fine-grained complexity


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