期望形式广义Stiefel流形上的随机优化
Stochastic optimization over expectation-formulated generalized Stiefel manifold
AI总结：
本文针对期望形式广义Stiefel流形上的随机优化问题，提出保持可微阶数的约束消解惩罚函数，建立其与原问题的一阶和二阶稳定点全局等价性，并设计带自适应步长的随机梯度算法及其加速变体，证明了$\mathcal{O}(\varepsilon^{-4})$样本复杂度。
摘要：
本文考虑一类定义在期望形式广义Stiefel流形（SOEGS）上的随机优化问题，其中目标函数$f$连续可微。我们提出了一种带有定制惩罚项的新型约束消解惩罚函数（CDFDP），该函数保持与$f$相同的可微阶数。理论分析建立了CDFCP与SOEGS之间的全局等价性，即在温和条件下二者具有相同的一阶和二阶稳定点。这些等价性结果使得各种随机优化方法可以直接用于求解SOEGS。特别地，我们通过引入自适应步长策略，开发了一种随机梯度算法及其加速变体。此外，我们证明了它们找到CDFCP的$\varepsilon$-稳定点所需的样本复杂度为$\mathcal{O}(\varepsilon^{-4})$。全面的数值实验表明了我们所提算法的高效性和鲁棒性。
英文摘要：
In this paper, we consider a class of stochastic optimization problems over the expectation-formulated generalized Stiefel manifold (SOEGS), where the objective function $f$ is continuously differentiable. We propose a novel constraint dissolving penalty function with a customized penalty term (CDFDP), which maintains the same order of differentiability as $f$. Our theoretical analysis establishes the global equivalence between CDFCP and SOEGS in the sense that they share the same first-order and second-order stationary points under mild conditions. These results on equivalence enable the direct implementation of various stochastic optimization approaches to solve SOEGS. In particular, we develop a stochastic gradient algorithm and its accelerated variant by incorporating an adaptive step size strategy. Furthermore, we prove their $\mathcal{O}(\varepsilon^{-4})$ sample complexity for finding an $\varepsilon$-stationary point of CDFCP. Comprehensive numerical experiments show the efficiency and robustness of our proposed algorithms.
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