提出一种新型随机近端算法，可在非正曲率空间中高效求解单调向量场零点。
Mean-square and sublinear convergence of a stochastic proximal point algorithm in metric spaces of nonpositive curvature
- 在非正曲率空间中设计随机近端算法，利用强单调性保证收敛
- 给出均方和几乎必然收敛的显式速率，且与数据分布无关
- 适用于优化、机器学习中的非欧空间问题，适合理论研究者
我们在一般非线性Hadamard空间中定义了一种随机近端点算法，用于逼近随机扰动单调向量场平均值的零点。推广了P. Bianchi的工作，在可分希尔伯特-Hadamard空间中，假设所有切空间均可等距嵌入希尔伯特空间（涵盖但不局限于Hadamard流形），在合适强单调性假设下证明了该方法的收敛性。我们的收敛证明完全有效，可构造出迭代过程趋于唯一解的均方和几乎必然收敛速率，且这些速率高度均匀，独立于大部分迭代相关的数据、空间或分布。在此一般性下，这些速率在希尔伯特空间背景下亦属首次。还讨论了在Yosida逼近满足额外二阶矩条件下的次线性非渐近保证，以及随机凸优化的特殊情况。
原文摘要 · Abstract (English)
We define a stochastic variant of the proximal point algorithm in the general setting of nonlinear Hadamard spaces for approximating zeros of the mean of a stochastically perturbed monotone vector field. Generalizing previous work by P. Bianchi, we prove the convergence of this method under a suitable strong monotonicity assumption in (separable) Hilbert-Hadamard spaces, that is assuming that all tangent spaces isometrically embed into Hilbert spaces (covering, but not being limited to, the setting of Hadamard manifolds). Moreover, our convergence proof is fully effective and allows for the construction of explicit rates of convergence for the iteration towards the (unique) solution both in mean and almost surely. These rates are moreover highly uniform, being independent of most data surrounding the iteration, space or distribution. In that generality, these rates are novel already in the context of Hilbert spaces. Sublinear nonasymptotic guarantees under additional second-moment conditions on the Yosida approximates and special cases of stochastic convex minimization are discussed.
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