Correcting the bias in least squares regression with volume-rescaled sampling

Abstract

Consider linear regression where the examples are generated by an unknown distribution on $R^d\times R$. Without any assumptions on the noise, the linear least squares solution for any i.i.d. sample will typically be biased w.r.t. the least squares optimum over the entire distribution. However, we show that if an i.i.d. sample of any size k is augmented by a certain small additional sample, then the solution of the combined sample becomes unbiased. We show this when the additional sample consists of d points drawn jointly according to the input distribution that is rescaled by the squared volume spanned by the points. Furthermore, we propose algorithms to sample from this volume-rescaled distribution when the data distribution is only known through an i.i.d sample.

Cited by in corpus (7)

• Exact expressions for double descent and implicit regularization via surrogate random design
• Exact sampling of determinantal point processes with sublinear time preprocessing
• Unbiased estimators for random design regression
• Query Complexity of Least Absolute Deviation Regression via Robust Uniform Convergence
• Bayesian experimental design using regularized determinantal point processes
• Debiasing Distributed Second Order Optimization with Surrogate Sketching and Scaled Regularization
• Sampling from a $k$-DPP without looking at all items