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- W4313304943 abstract "Polyak-{L}ojasiewicz (PL) [Polyak, 1963] condition is a weaker condition than the strong convexity but suffices to ensure a global convergence for the Gradient Descent algorithm. In this paper, we study the lower bound of algorithms using first-order oracles to find an approximate optimal solution. We show that any first-order algorithm requires at least ${Omega}left(frac{L}{mu}logfrac{1}{varepsilon}right)$ gradient costs to find an $varepsilon$-approximate optimal solution for a general $L$-smooth function that has an $mu$-PL constant. This result demonstrates the optimality of the Gradient Descent algorithm to minimize smooth PL functions in the sense that there exists a ``hard'' PL function such that no first-order algorithm can be faster than Gradient Descent when ignoring a numerical constant. In contrast, it is well-known that the momentum technique, e.g. [Nesterov, 2003, chap. 2] can provably accelerate Gradient Descent to ${O}left(sqrt{frac{L}{hat{mu}}}logfrac{1}{varepsilon}right)$ gradient costs for functions that are $L$-smooth and $hat{mu}$-strongly convex. Therefore, our result distinguishes the hardness of minimizing a smooth PL function and a smooth strongly convex function as the complexity of the former cannot be improved by any polynomial order in general." @default.
- W4313304943 created "2023-01-06" @default.
- W4313304943 creator A5016399094 @default.
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- W4313304943 date "2022-12-27" @default.
- W4313304943 modified "2023-09-26" @default.
- W4313304943 title "On the Lower Bound of Minimizing Polyak-{L}ojasiewicz Functions" @default.
- W4313304943 doi "https://doi.org/10.48550/arxiv.2212.13551" @default.
- W4313304943 hasPublicationYear "2022" @default.
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