Dynamic Least-Squares Regression

Binghui Peng, Shunhua Jiang, OMRI WEINSTEIN

Keywords: Least squares regression, dynamic algorithm

Abstract: In large-scale supervised learning, after a model is trained with an initial dataset, a common challenge is how to exploit new incremental data without re-training the model from scratch. Motivated by this problem, we revisit the canonical problem of dynamic least-squares regression (LSR), where the goal is to learn a linear model over incremental training data. In this setup, data and labels $(\mathbf{A}^{(t)}, \mathbf{b}^{(t)}) \in \mathbb{R}^{t \times d}\times \mathbb{R}^t$ evolve in an online fashion ($t\gg d$), and the goal is to efficiently maintain an (approximate) solution of $\min_{\mathbf{x}^{(t)}} \| \mathbf{A}^{(t)} \mathbf{x}^{(t)} - \mathbf{b}^{(t)} \|_2$ for all $t\in [T]$. Our main result is a dynamic data structure which maintains an arbitrarily small constant approximate solution to dynamic LSR with amortized update time $O(d^{1+o(1)})$, almost matching the running time of the static (sketching-based) solution. By contrast, for exact (or $1/\mathrm{poly}(n)$-accuracy) solutions, we show a separation between the models, namely, that dynamic LSR requires $\Omega(d^{2-o(1)})$ amortized update time under the OMv Conjecture (Henzinger et al., STOC'15). Our data structure is fast, conceptually simple, easy to implement, and our experiments demonstrate their practicality on both synthetic and real-world datasets.

One-sentence Summary: Near optimal algorithm and hardness results for dynamic least squares regressions.

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2201.00228/code)