关于非光滑且相对弱凸的最小化问题
On Nonsmooth and Relatively Weakly Convex Minimization
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中文总结 AI 辅助
本文研究非光滑非凸复合优化问题，提出满足次梯度上界条件的确定性Bregman邻近次梯度方法，放宽正则项凸性要求，还拓展出随机场景下收敛速率为$\text{O}(\text{ε}^{-4})$的基于模型的最小化方法及随机预言机访问场景的框架。
中文摘要 AI 辅助
复合优化在现代机器学习与信号处理中占据核心地位，它能在数据保真度与结构特性之间实现自然平衡。本文研究两个组成部分均为非光滑非凸的复合优化问题。我们首先提出一种确定性Bregman邻近次梯度方法，该方法在次梯度上界条件下收敛。此方法放宽了对正则项凸性的标准要求，因此可适配更广泛的应用场景。为将其拓展到随机场景，我们在相对Lipschitz条件下提出了一种基于模型的最小化方法，并证明其收敛速率为$\boldsymbol{\text{O}}(\boldsymbol{\text{ε}}^{-4})$。我们还将带收敛保证的该框架拓展到了距离生成函数及其梯度仅能通过随机预言机访问的场景。
英文摘要
Composite optimization plays a central role in modern machine learning and signal processing, as it offers a natural balance between data fidelity and structural properties. In this paper, we study composite optimization in the setting where both components are nonsmooth and nonconvex. We start with a deterministic Bregman proximal subgradient method that converges under subgradient upper-bound conditions. This approach relaxes the standard requirement on the convexity of the regularization term, thus accommodating a broader range of applications. To extend this to the stochastic regime, we develop a model-based minimization method under a relative Lipschitz condition and establish a convergence rate of $\mathcal{O}(\varepsilon^{-4})$. We also extend the framework with convergence guarantees to the setting where the distance generating function and its gradient are accessible only through a stochastic oracle.
