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

Authors Ankit Garg, Robin Kothari, Praneeth Netrapalli, Suhail Sherif

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ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Convex optimization
Keywords
  • Quantum algorithms
  • Gradient descent
  • Convex optimization

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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.

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Ankit Garg, Robin Kothari, Praneeth Netrapalli, and Suhail Sherif. No Quantum Speedup over Gradient Descent for Non-Smooth Convex Optimization. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 53:1-53:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ITCS.2021.53

Author Details

Ankit Garg
  • Microsoft Research India, Bangalore, India
Robin Kothari
  • Microsoft Quantum and Microsoft Research, Redmond, WA, USA
Praneeth Netrapalli
  • Microsoft Research India, Bangalore, India
Suhail Sherif
  • Microsoft Research India, Bangalore, India
  • Tata Institute of Fundamental Research, Mumbai, India

Acknowledgements

We thank Sébastien Bubeck, Ronald de Wolf, and András Gilyén for helpful conversations about this work. RK thanks Vamsi Pritham Pingali for many helpful conversations about multivariable calculus. We would also like to thank Matthias Kleinmann for asking us some questions about Lemma 17 that helped us improve its presentation.

References

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