We study the random reshuffling (RR) method for smooth nonconvex optimization problems with a finite-sum structure. Though this method is widely utilized in practice such as the training of neural networks, its convergence behavior is only understood in several limited settings. In this paper, under the well-known Kurdyka-Lojasiewicz (KL) inequality, we establish strong limit-point convergence results for RR with appropriate diminishing step sizes, namely, the whole sequence of iterates generated by RR is convergent and converges to a single stationary point in an almost sure sense. In addition, we derive the corresponding rate of convergence, depending on the KL exponent and the suitably selected diminishing step sizes. When the KL exponent lies in $[0,\frac12]$, the convergence is at a rate of $\mathcal{O}(t^{-1})$ with $t$ counting the iteration number. When the KL exponent belongs to $(\frac12,1)$, our derived convergence rate is of the form $\mathcal{O}(t^{-q})$ with $q\in (0,1)$ depending on the KL exponent. The standard KL inequality-based convergence analysis framework only applies to algorithms with a certain descent property. We conduct a novel convergence analysis for the non-descent RR method with diminishing step sizes based on the KL inequality, which generalizes the standard KL framework. We summarize our main steps and core ideas in an informal analysis framework, which is of independent interest. As a direct application of this framework, we also establish similar strong limit-point convergence results for the reshuffled proximal point method.

翻译：我们用一个有限和总结构来研究平滑非康文优化问题的随机重整方法。 虽然这种方法在神经网络培训等实践中被广泛使用, 但只在几个有限的环境中才理解其趋同行为。 在本文中, 在众所周知的 Kurdyka- Lojasiewicz (KL) 不平等之下, 我们为 RR 设定了强大的限制点趋同结果, 并相应缩小步骤大小, 即 RR 生成的迭代的整个序列会趋同, 并接近一个几乎肯定的固定点。 此外, 我们得出相应的趋同率, 取决于 KL 指数和适当选择的缩放大小 。 当 KL 指数位于 $ $0,\ frafcs a mathcalcall {O] (t\\\ 1} 美元, 和 $xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx