arXiv：cs.LG· Yuecai Han, Jianming Xu·· 3 小时前
基于分数布朗运动驱动的神经网络逐样本反向传播方法
A samplewise backpropagation method for neural networks driven by fractional Brownian motion
AI 导读
研究者提出一种由分数布朗运动驱动的分数阶随机神经网络，通过引入离散随机最大值原理构建伴随递归，并证明了投影逐样本随机梯度下降的均方收敛性。数值实验涵盖闭式收敛测试、带不确定性量化的噪声回归、长记忆时间序列生成及结构化扰动下的图像分类，结果显示分数阶驱动在长记忆恢复或鲁棒性上优于布朗运动与确定性基线。
正文
Abstract:In this paper, we develop a fractional stochastic neural network with residual dynamics driven by fractional Brownian motion. By introducing a discrete stochastic maximum principle for the network, we construct the corresponding adjoint recursion. For deterministic network parameters, we prove mean square convergence of projected samplewise stochastic gradient descent. Numerical experiments include a closed form convergence test, noisy regression with uncertainty quantification, long memory time series generation and image classification under structured perturbations. The results identify settings in which fractional drivers improve long memory recovery or robustness relative to Brownian and deterministic baselines.
| Comments: | 29 pages, 3 figures, 6 tables |
| Subjects: | Optimization and Control (math.OC); Machine Learning (cs.LG) |
| MSC classes: | 60G22, 65K05, 68T07, 93E20 |
| Cite as: | arXiv:2606.29438 [math.OC] |
| (or arXiv:2606.29438v2 [math.OC] for this version) | |
| https://doi.org/10.48550/arXiv.2606.29438 arXiv-issued DOI via DataCite |
Submission history
From: Jianming Xu [view email]
[v1]
Sun, 28 Jun 2026 14:55:00 UTC (1,445 KB)
[v2]
Thu, 8 Oct 2026 07:26:11 UTC (136 KB)
来源：arXiv：cs.LG · arxiv.org