This paper proposes {\sf AAPG-SPIDER}, an Adaptive Accelerated Proximal Gradient (AAPG) method with variance reduction for minimizing composite nonconvex finite-sum functions. It integrates three acceleration techniques: adaptive stepsizes, Nesterov's extrapolation, and the recursive stochastic path-integrated estimator SPIDER. While targeting stochastic finite-sum problems, {\sf AAPG-SPIDER} simplifies to {\sf AAPG} in the full-batch, non-stochastic setting, which is also of independent interest. To our knowledge, {\sf AAPG-SPIDER} and {\sf AAPG} are the first learning-rate-free methods to achieve optimal iteration complexity for this class of \textit{composite} minimization problems. Specifically, {\sf AAPG} achieves the optimal iteration complexity of $\mathcal{O}(N \epsilon^{-2})$, while {\sf AAPG-SPIDER} achieves $\mathcal{O}(N + \sqrt{N} \epsilon^{-2})$ for finding $\epsilon$-approximate stationary points, where $N$ is the number of component functions. Under the Kurdyka-Lojasiewicz (KL) assumption, we establish non-ergodic convergence rates for both methods. Preliminary experiments on sparse phase retrieval and linear eigenvalue problems demonstrate the superior performance of {\sf AAPG-SPIDER} and {\sf AAPG} compared to existing methods.

翻译：本文提出了一种用于最小化复合非凸有限和函数的自适应加速近端梯度方差缩减方法——{\sf AAPG-SPIDER}。该方法集成了三种加速技术：自适应步长、Nesterov外推法以及递归随机路径积分估计器SPIDER。虽然针对随机有限和问题，但在全批量非随机设定下，{\sf AAPG-SPIDER}可简化为同样具有独立研究价值的{\sf AAPG}方法。据我们所知，{\sf AAPG-SPIDER}与{\sf AAPG}是首类针对该\textit{复合}最小化问题实现最优迭代复杂度的免学习率方法。具体而言，在寻找$\epsilon$近似驻点时，{\sf AAPG}达到$\mathcal{O}(N \epsilon^{-2})$的最优迭代复杂度，而{\sf AAPG-SPIDER}达到$\mathcal{O}(N + \sqrt{N} \epsilon^{-2})$，其中$N$为分量函数个数。在Kurdyka-Lojasiewicz (KL)假设下，我们建立了两种方法的非遍历收敛速率。在稀疏相位恢复和线性特征值问题上的初步实验表明，{\sf AAPG-SPIDER}与{\sf AAPG}相比现有方法具有优越性能。