We develop and analyze data subsampling techniques for Poisson regression, the standard model for count data $y\in\mathbb{N}$. 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 $1\pm\varepsilon$. We show $\Omega(n)$ 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 $1\pm\varepsilon$ 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 $O(\log(y_{\max}))$ dependence, where $y_{\max}$ denotes the largest count presented in the data, while the ID-link requires a $\Theta(\sqrt{y_{\max}/\log(y_{\max})})$ dependence. As an auxiliary result for proving the tightness of the bound with respect to $y_{\max}$ in the case of the ID-link, we show an improved bound on the principal branch of the Lambert $W_0$ function, which may be of independent interest. We further show the limitations of our analysis when $p$th degree root-link functions for $p\geq 3$ are considered, which indicate that other analytical or computational methods would be required if such a generalization is even possible.

翻译：我们针对计数数据$y\in\mathbb{N}$的标准模型——泊松回归，开发并分析了数据子采样技术。具体而言，我们考虑具有恒等链接函数和平方根链接函数的泊松广义线性模型。我们研究了核心集方法，即通过小型加权子集以$1\pm\varepsilon$的近似比逼近泊松回归的损失函数。我们证明了泊松回归的核心集存在$\Omega(n)$的下界，且该下界在对数因子范围内对任意数据约简技术均成立。通过引入新颖的复杂度参数和域偏移方法，我们证明当复杂度参数较小时，存在具有$1\pm\varepsilon$近似保证的亚线性核心集。特别地，对输入点数量的依赖可降至多对数级别。我们证明了对其他输入参数的依赖也可实现亚线性有界，尽管并非总是对数级别。具体而言，平方根链接具有$O(\log(y_{\max}))$的依赖度（其中$y_{\max}$表示数据中的最大计数值），而恒等链接需要$\Theta(\sqrt{y_{\max}/\log(y_{\max})})$的依赖度。作为证明恒等链接情形下$y_{\max}$相关界限紧性的辅助结果，我们改进了Lambert $W_0$函数主分支的界，该结果可能具有独立研究价值。进一步地，我们展示了当考虑$p\geq 3$的$p$次方根链接函数时现有分析的局限性，这表明若需实现此类泛化，可能需要其他解析或计算方法。