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- W3134003265 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 e-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/e)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/e)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/e) 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/e)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." @default.
- W3134003265 created "2021-03-15" @default.
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- W3134003265 date "2021-01-01" @default.
- W3134003265 modified "2023-09-27" @default.
- W3134003265 title "No Quantum Speedup over Gradient Descent for Non-Smooth Convex Optimization" @default.
- W3134003265 doi "https://doi.org/10.4230/lipics.itcs.2021.53" @default.
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