Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation

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For any real numbers B \ge 1 and \delta \in (0, 1) and function f: [0, B] \rightarrow \mathbb{R}, let d_{B; \delta} (f) \in \mathbb{Z}_{> 0} denote the minimum degree of a polynomial p(x) satisfying \sup_{x \in [0, B]} \big| p(x) - f(x) \big| < \delta. In this paper, we provide precise asymptotics for d_{B; \delta} (e^{-x}) and d_{B; \delta} (e^{x}) in terms of both B and \delta, improving both the previously known upper bounds and lower bounds. In particular, we show d_{B; \delta} (e^{-x}) = \Theta\left( \max \left\{ \sqrt{B \log(\delta^{-1})}, \frac{\log(\delta^{-1}) }{ \log(B^{-1} \log(\delta^{-1}))} \right\}\right), \text{ and} d_{B; \delta} (e^{x}) = \Theta\left( \max \left\{ B, \frac{\log(\delta^{-1}) }{ \log(B^{-1} \log(\delta^{-1}))} \right\}\right).
Polynomial approximations for e^{-x} and e^x have applications to the design of algorithms for many problems, and our degree bounds show both the power and limitations of these algorithms.
We focus in particular on the Batch Gaussian Kernel Density Estimation problem for n sample points in \Theta(\log n) dimensions with error \delta = n^{-\Theta(1)}. We show that the running time one can achieve depends on the square of the diameter of the point set, B, with a transition at B = \Theta(\log n) mirroring the corresponding transition in d_{B; \delta} (e^{-x}):
- When B=o(\log n), we give the first algorithm running in time n^{1 + o(1)}.
- When B = \kappa \log n for a small constant \kappa>0, we give an algorithm running in time n^{1 + O(\log \log \kappa^{-1} /\log \kappa^{-1})}. The \log \log \kappa^{-1} /\log \kappa^{-1} term in the exponent comes from analyzing the behavior of the leading constant in our computation of d_{B; \delta} (e^{-x}).
- When B = \omega(\log n), we show that time n^{2 - o(1)} is necessary assuming SETH.