动态调整分数阶梯度下降的指数，提升非凸优化收敛速度与稳定性。
More Optimal Fractional-Order Stochastic Gradient Descent for Non-Convex Optimization Problems
- 基于模型敏感度和有效维度自适应调节分数阶指数
- 在高斯与α稳定噪声下收敛更快，参数估计更稳健
- 适合需要稳定优化的复杂建模任务
分数阶随机梯度下降（FOSGD）利用分数阶指数捕捉优化过程中的长时记忆效应。然而，其应用常受限于指数调优困难与不稳定性。本文提出2SED分数阶随机梯度下降（2SEDFOSGD），将双尺度有效维度（2SED）算法与FOSGD结合，实现数据驱动的分数阶指数自适应。通过追踪模型敏感度与有效维度，2SEDFOSGD动态调节指数，抑制振荡并加速收敛。理论上，该方法在非凸优化问题中保留了分数阶记忆优势，避免了原始方法中缓慢或不稳定的缺陷。在自回归（AR）模型上，针对高斯噪声和α-稳定噪声场景的实证评估表明，相比基线方法，其收敛速度更快，参数估计更鲁棒，凸显了维度感知型分数阶技术在高级建模与估计任务中的潜力。
原文摘要 · Abstract (English)
Fractional-order stochastic gradient descent (FOSGD) leverages fractional exponents to capture long-memory effects in optimization. However, its utility is often limited by the difficulty of tuning and stabilizing these exponents. We propose 2SED Fractional-Order Stochastic Gradient Descent (2SEDFOSGD), which integrates the Two-Scale Effective Dimension (2SED) algorithm with FOSGD to adapt the fractional exponent in a data-driven manner. By tracking model sensitivity and effective dimensionality, 2SEDFOSGD dynamically modulates the exponent to mitigate oscillations and hasten convergence. Theoretically, for onoconvex optimization problems, this approach preserves the advantages of fractional memory without the sluggish or unstable behavior observed in naïve fractional SGD. Empirical evaluations in Gaussian and $α$-stable noise scenarios using an autoregressive (AR) model highlight faster convergence and more robust parameter estimates compared to baseline methods, underscoring the potential of dimension-aware fractional techniques for advanced modeling and estimation tasks.
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