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Register a new userNear-Optimal Coresets of Kernel Density Estimates
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Added by
Wai Ming Tai
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We construct near-optimal coresets for kernel density estimates for points in $mathbb{R}^d$ when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size $O(sqrt{d}/varepsiloncdot sqrt{log 1/varepsilon} )$, and we show a near-matching lower bound of size $Omega(min{sqrt{d}/varepsilon, 1/varepsilon^2})$. When $dgeq 1/varepsilon^2$, it is known that the size of coreset can be $O(1/varepsilon^2)$. The upper bound is a polynomial-in-$(1/varepsilon)$ improvement when $d in [3,1/varepsilon^2)$ and the lower bound is the first known lower bound to depend on $d$ for this problem. Moreover, the upper bound restriction that the kernel is positive definite is significant in that it applies to a wide-variety of kernels, specifically those most important for machine learning. This includes kernels for information distances and the sinc kernel which can be negative.
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We study the construction of coresets for kernel density estimates. That is we show how to approximate the kernel density estimate described by a large point set with another kernel density estimate with a much smaller point set. For characteristic kernels (including Gaussian and Laplace kernels), our approximation preserves the $L_infty$ error between kernel density estimates within error $epsilon$, with coreset size $2/epsilon^2$, but no other aspects of the data, including the dimension, the diameter of the point set, or the bandwidth of the kernel common to other approximations. When the dimension is unrestricted, we show this bound is tight for these kernels as well as a much broader set. This work provides a careful analysis of the iterative Frank-Wolfe algorithm adapted to this context, an algorithm called emph{kernel herding}. This analysis unites a broad line of work that spans statistics, machine learning, and geometry. When the dimension $d$ is constant, we demonstrate much tighter bounds on the size of the coreset specifically for Gaussian kernels, showing that it is bounded by the size of the coreset for axis-aligned rectangles. Currently the best known constructive bound is $O(frac{1}{epsilon} log^d frac{1}{epsilon})$, and non-constructively, this can be improved by $sqrt{log frac{1}{epsilon}}$. This improves the best constant dimension bounds polynomially for $d geq 3$.
Given a point set $Psubset mathbb{R}^d$, a kernel density estimation for Gaussian kernel is defined as $overline{mathcal{G}}_P(x) = frac{1}{left|Pright|}sum_{pin P}e^{-leftlVert x-p rightrVert^2}$ for any $xinmathbb{R}^d$. We study how to construct a small subset $Q$ of $P$ such that the kernel density estimation of $P$ can be approximated by the kernel density estimation of $Q$. This subset $Q$ is called coreset. The primary technique in this work is to construct $pm 1$ coloring on the point set $P$ by the discrepancy theory and apply this coloring algorithm recursively. Our result leverages Banaszczyks Theorem. When $d>1$ is constant, our construction gives a coreset of size $Oleft(frac{1}{varepsilon}right)$ as opposed to the best-known result of $Oleft(frac{1}{varepsilon}sqrt{logfrac{1}{varepsilon}}right)$. It is the first to give a breakthrough on the barrier of $sqrt{log}$ factor even when $d=2$.
Many clustering algorithms are guided by certain cost functions such as the widely-used $k$-means cost. These algorithms divide data points into clusters with often complicated boundaries, creating difficulties in explaining the clustering decision. In a recent work, Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML20) introduced explainable clustering, where the cluster boundaries are axis-parallel hyperplanes and the clustering is obtained by applying a decision tree to the data. The central question here is: how much does the explainability constraint increase the value of the cost function? Given $d$-dimensional data points, we show an efficient algorithm that finds an explainable clustering whose $k$-means cost is at most $k^{1 - 2/d}mathrm{poly}(dlog k)$ times the minimum cost achievable by a clustering without the explainability constraint, assuming $k,dge 2$. Combining this with an independent work by Makarychev and Shan (ICML21), we get an improved bound of $k^{1 - 2/d}mathrm{polylog}(k)$, which we show is optimal for every choice of $k,dge 2$ up to a poly-logarithmic factor in $k$. For $d = 2$ in particular, we show an $O(log kloglog k)$ bound, improving exponentially over the previous best bound of $widetilde O(k)$.
Coreset is usually a small weighted subset of $n$ input points in $mathbb{R}^d$, that provably approximates their loss function for a given set of queries (models, classifiers, etc.). Coresets become increasingly common in machine learning since existing heuristics or inefficient algorithms may be improved by running them possibly many times on the small coreset that can be maintained for streaming distributed data. Coresets can be obtained by sensitivity (importance) sampling, where its size is proportional to the total sum of sensitivities. Unfortunately, computing the sensitivity of each point is problem dependent and may be harder to compute than the original optimization problem at hand. We suggest a generic framework for computing sensitivities (and thus coresets) for wide family of loss functions which we call near-convex functions. This is by suggesting the $f$-SVD factorization that generalizes the SVD factorization of matrices to functions. Example applications include coresets that are either new or significantly improves previous results, such as SVM, Logistic regression, M-estimators, and $ell_z$-regression. Experimental results and open source are also provided.
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