Worth Their Weight: Randomized and Regularized Block Kaczmarz Algorithms without Preprocessing
Gil Goldshlager, Jiang Hu, Lin Lin

TL;DR
This paper introduces ReBlocK, a regularized version of the randomized block Kaczmarz algorithm, which converges reliably without preprocessing and outperforms existing methods on certain large-scale least-squares problems.
Contribution
It provides the first analysis of RBK with uniform sampling, introduces regularization to control bias and variance, and demonstrates improved performance over traditional methods.
Findings
ReBlocK converges to a weighted least-squares solution without preprocessing.
Regularization effectively controls bias and variance in RBK.
ReBlocK outperforms RBK and minibatch SGD on problems with rapidly decaying singular values.
Abstract
Due to the ever growing amounts of data leveraged for machine learning and scientific computing, it is increasingly important to develop algorithms that sample only a small portion of the data at a time. In the case of linear least-squares, the randomized block Kaczmarz method (RBK) is an appealing example of such an algorithm, but its convergence is only understood under sampling distributions that require potentially prohibitively expensive preprocessing steps. To address this limitation, we analyze RBK when the data is sampled uniformly, showing that its iterates converge in a Monte Carlo sense to a least-squares solution. Unfortunately, for general problems the bias of the weighted least-squares solution and the variance of the iterates can become arbitrarily large. We show that these quantities can be rigorously controlled by incorporating regularization into…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Code & Models
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsStochastic Gradient Optimization Techniques · Facility Location and Emergency Management · Statistical Methods and Inference
