2 Search Results for "Aggarwal, Amol"

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DOI: 10.4230/LIPIcs.CCC.2024.35

Finer-Grained Hardness of Kernel Density Estimation

Authors: Josh Alman and Yunfeng Guan

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract

In batch Kernel Density Estimation (KDE) for a kernel function f : ℝ^m × ℝ^m → ℝ, we are given as input 2n points x^{(1)}, …, x^{(n)}, y^{(1)}, …, y^{(n)} ∈ ℝ^m in dimension m, as well as a vector v ∈ ℝⁿ. These inputs implicitly define the n × n kernel matrix K given by K[i,j] = f(x^{(i)}, y^{(j)}). The goal is to compute a vector v ∈ ℝⁿ which approximates K w, i.e., with || Kw - v||_∞ < ε ||w||₁. For illustrative purposes, consider the Gaussian kernel f(x,y) : = e^{-||x-y||₂²}. The classic approach to this problem is the famous Fast Multipole Method (FMM), which runs in time n ⋅ O(log^m(ε^{-1})) and is particularly effective in low dimensions because of its exponential dependence on m. Recently, as the higher-dimensional case m ≥ Ω(log n) has seen more applications in machine learning and statistics, new algorithms have focused on this setting: an algorithm using discrepancy theory, which runs in time O(n / ε), and an algorithm based on the polynomial method, which achieves inverse polynomial accuracy in almost linear time when the input points have bounded square diameter B < o(log n). A recent line of work has proved fine-grained lower bounds, with the goal of showing that the "curse of dimensionality" arising in FMM is necessary assuming the Strong Exponential Time Hypothesis (SETH). Backurs et al. [NeurIPS 2017] first showed the hardness of a variety of Empirical Risk Minimization problems including KDE for Gaussian-like kernels in the case with high dimension m = Ω(log n) and large scale B = Ω(log n). Alman et al. [FOCS 2020] later developed new reductions in roughly this same parameter regime, leading to lower bounds for more general kernels, but only for very small error ε < 2^{- log^{Ω(1)} (n)}. In this paper, we refine the approach of Alman et al. to show new lower bounds in all parameter regimes, closing gaps between the known algorithms and lower bounds. For example: - In the setting where m = Clog n and B = o(log n), we prove Gaussian KDE requires n^{2-o(1)} time to achieve additive error ε < Ω(m/B)^{-m}, matching the performance of the polynomial method up to low-order terms. - In the low dimensional setting m = o(log n), we show that Gaussian KDE requires n^{2-o(1)} time to achieve ε such that log log (ε^{-1}) > ̃ Ω ((log n)/m), matching the error bound achievable by FMM up to low-order terms. To our knowledge, no nontrivial lower bound was previously known in this regime. Our approach also generalizes to any parameter regime and any kernel. For example, we achieve similar fine-grained hardness results for any kernel with slowly-decaying Taylor coefficients such as the Cauchy kernel. Our new lower bounds make use of an intricate analysis of the "counting matrix", a special case of the kernel matrix focused on carefully-chosen evaluation points. As a key technical lemma, we give a novel approach to bounding the entries of its inverse by using Schur polynomials from algebraic combinatorics.

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Josh Alman and Yunfeng Guan. Finer-Grained Hardness of Kernel Density Estimation. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 35:1-35:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{alman_et_al:LIPIcs.CCC.2024.35,
  author =	{Alman, Josh and Guan, Yunfeng},
  title =	{{Finer-Grained Hardness of Kernel Density Estimation}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{35:1--35:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.35},
  URN =		{urn:nbn:de:0030-drops-204311},
  doi =		{10.4230/LIPIcs.CCC.2024.35},
  annote =	{Keywords: Kernel Density Estimation, Fine-Grained Complexity, Schur Polynomials}
}

Document

DOI: 10.4230/LIPIcs.CCC.2022.22

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

Authors: Amol Aggarwal and Josh Alman

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract

For any real numbers B ≥ 1 and δ ∈ (0,1) and function f: [0,B] → ℝ, let d_{B; δ}(f) ∈ ℤ_{> 0} denote the minimum degree of a polynomial p(x) satisfying sup_{x ∈ [0,B]} |p(x) - f(x)| < δ. In this paper, we provide precise asymptotics for d_{B; δ}(e^{-x}) and d_{B; δ}(e^x) in terms of both B and δ, improving both the previously known upper bounds and lower bounds. In particular, we show d_{B; δ}(e^{-x}) = Θ(max{√{B log(δ^{-1})}, log(δ^{-1})/{log(B^{-1} log(δ^{-1}))}}), and d_{B; δ}(e^{x}) = Θ(max{B, log(δ^{-1})/{log(B^{-1} log(δ^{-1}))}}), and we explicitly determine the leading coefficients in most parameter regimes. Polynomial approximations for e^{-x} and e^x have applications to the design of algorithms for many problems, including in scientific computing, graph algorithms, machine learning, and statistics. 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 Θ(log n) dimensions with error δ = n^{-Θ(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 = Θ(log n) mirroring the corresponding transition in d_{B; δ}(e^{-x}): - When B = o(log n), we give the first algorithm running in time n^{1 + o(1)}. - When B = κ log n for a small constant κ > 0, we give an algorithm running in time n^{1 + O(log log κ^{-1} /log κ^{-1})}. The log log κ^{-1} /log κ^{-1} term in the exponent comes from analyzing the behavior of the leading constant in our computation of d_{B; δ}(e^{-x}). - When B = ω(log n), we show that time n^{2 - o(1)} is necessary assuming SETH.

Cite as

Amol Aggarwal and Josh Alman. Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{aggarwal_et_al:LIPIcs.CCC.2022.22,
  author =	{Aggarwal, Amol and Alman, Josh},
  title =	{{Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{22:1--22:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.22},
  URN =		{urn:nbn:de:0030-drops-165846},
  doi =		{10.4230/LIPIcs.CCC.2022.22},
  annote =	{Keywords: polynomial approximation, kernel density estimation, Chebyshev polynomials}
}

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