License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2021.53
URN: urn:nbn:de:0030-drops-135921
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13592/
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Garg, Ankit ; Kothari, Robin ; Netrapalli, Praneeth ; Sherif, Suhail

No Quantum Speedup over Gradient Descent for Non-Smooth Convex Optimization

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LIPIcs-ITCS-2021-53.pdf (0.5 MB)

Abstract

We study the first-order convex optimization problem, where we have black-box access to a (not necessarily smooth) function f:ℝⁿ → ℝ and its (sub)gradient. Our goal is to find an ε-approximate minimum of f starting from a point that is distance at most R from the true minimum. If f is G-Lipschitz, then the classic gradient descent algorithm solves this problem with O((GR/ε)²) queries. Importantly, the number of queries is independent of the dimension n and gradient descent is optimal in this regard: No deterministic or randomized algorithm can achieve better complexity that is still independent of the dimension n.
In this paper we reprove the randomized lower bound of Ω((GR/ε)²) using a simpler argument than previous lower bounds. We then show that although the function family used in the lower bound is hard for randomized algorithms, it can be solved using O(GR/ε) quantum queries. We then show an improved lower bound against quantum algorithms using a different set of instances and establish our main result that in general even quantum algorithms need Ω((GR/ε)²) queries to solve the problem. Hence there is no quantum speedup over gradient descent for black-box first-order convex optimization without further assumptions on the function family.

BibTeX - Entry

@InProceedings{garg_et_al:LIPIcs.ITCS.2021.53,
  author =	{Ankit Garg and Robin Kothari and Praneeth Netrapalli and Suhail Sherif},
  title =	{{No Quantum Speedup over Gradient Descent for Non-Smooth Convex Optimization}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{53:1--53:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{James R. Lee},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13592},
  URN =		{urn:nbn:de:0030-drops-135921},
  doi =		{10.4230/LIPIcs.ITCS.2021.53},
  annote =	{Keywords: Quantum algorithms, Gradient descent, Convex optimization}
}

Keywords: Quantum algorithms, Gradient descent, Convex optimization
Collection: 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)
Issue Date: 2021
Date of publication: 04.02.2021

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