Correcting the bias in least squares regression with volume-rescaled sampling
This addresses bias correction in regression for statistical and machine learning applications, offering a novel sampling-based method that is incremental in improving estimation accuracy.
The paper tackles the bias in linear least squares regression by showing that augmenting an i.i.d. sample with a small additional sample drawn from a volume-rescaled distribution yields an unbiased solution, and proposes algorithms to sample from this distribution when only an i.i.d. sample is available.
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.