Papers › Data subsampling for Poisson regression with pth-root-link

Data subsampling for Poisson regression with pth-root-link

30 Oct 2024arXiv:2410.22872archive 2025-07-28

Han Cheng Lie, Alexander Munteanu

We develop and analyze data subsampling techniques for Poisson regression, the standard model for count data y∈ℕ. In particular, we consider the Poisson generalized linear model with ID- and square root-link functions. We consider the method of coresets, which are small weighted subsets that approximate the loss function of Poisson regression up to a factor of 1±ε. We show Ω(n) lower bounds against coresets for Poisson regression that continue to hold against arbitrary data reduction techniques up to logarithmic factors. By introducing a novel complexity parameter and a domain shifting approach, we show that sublinear coresets with 1±ε approximation guarantee exist when the complexity parameter is small. In particular, the dependence on the number of input points can be reduced to polylogarithmic. We show that the dependence on other input parameters can also be bounded sublinearly, though not always logarithmically. In particular, we show that the square root-link admits an O(log(yₘₐₓ)) dependence, where yₘₐₓ denotes the largest count presented in the data, while the ID-link requires a Θ(√(yₘₐₓ/log(yₘₐₓ))) dependence. As an auxiliary result for proving the tightness of the bound with respect to yₘₐₓ in the case of the ID-link, we show an improved bound on the principal branch of the Lambert W₀ function, which may be of independent interest. We further show the limitations of our analysis when $p$th degree root-link functions for p≥3 are considered, which indicate that other analytical or computational methods would be required if such a generalization is even possible.

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