On the Complexity of Finite-Sum Smooth Optimization under the Polyak-{\L}ojasiewicz Condition

• URL: http://arxiv.org/abs/2402.02569v1
• Date: Sun, 4 Feb 2024 17:14:53 GMT
• Title: On the Complexity of Finite-Sum Smooth Optimization under the Polyak-{\L}ojasiewicz Condition
• Authors: Yunyan Bai, Yuxing Liu, Luo Luo
• Abstract summary: This paper considers the optimization problem of the form $min_bf xinmathbb Rd f(bf x)triangleq frac1nsum_i=1n f_i(bf x)$, where $f(cdot)$ satisfies the Polyak--Lojasiewicz (PL) condition with parameter $mu$ and $f_i(cdot)_i=1n$ is $L$-mean-squared smooth.
• Score: 14.781921087738967
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper considers the optimization problem of the form $\min_{{\bf x}\in{\mathbb R}^d} f({\bf x})\triangleq \frac{1}{n}\sum_{i=1}^n f_i({\bf x})$, where $f(\cdot)$ satisfies the Polyak--{\L}ojasiewicz (PL) condition with parameter $\mu$ and $\{f_i(\cdot)\}_{i=1}^n$ is $L$-mean-squared smooth. We show that any gradient method requires at least $\Omega(n+\kappa\sqrt{n}\log(1/\epsilon))$ incremental first-order oracle (IFO) calls to find an $\epsilon$-suboptimal solution, where $\kappa\triangleq L/\mu$ is the condition number of the problem. This result nearly matches upper bounds of IFO complexity for best-known first-order methods. We also study the problem of minimizing the PL function in the distributed setting such that the individuals $f_1(\cdot),\dots,f_n(\cdot)$ are located on a connected network of $n$ agents. We provide lower bounds of $\Omega(\kappa/\sqrt{\gamma}\,\log(1/\epsilon))$, $\Omega((\kappa+\tau\kappa/\sqrt{\gamma}\,)\log(1/\epsilon))$ and $\Omega\big(n+\kappa\sqrt{n}\log(1/\epsilon)\big)$ for communication rounds, time cost and local first-order oracle calls respectively, where $\gamma\in(0,1]$ is the spectral gap of the mixing matrix associated with the network and~$\tau>0$ is the time cost of per communication round. Furthermore, we propose a decentralized first-order method that nearly matches above lower bounds in expectation.

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