用神经网络在线学习随机数据，理论证明了学习效果的上限。
Continuous-time Online Learning via Mean-Field Neural Networks: Regret Analysis in Diffusion Environments
- 用均值场神经网络持续更新参数，适应随机数据生成过程。
- 在非凸情况下，后悔值随时间线性增长，受数据波动和正则化影响。
- 适合研究在线学习、随机优化或深度学习理论的读者。
我们研究连续时间在线学习问题，其中数据由未知系数的扩散过程生成。学习者采用两层神经网络，以非前瞻方式持续更新参数。学习动态的均值场极限对应于适应数据滤波的随机Wasserstein梯度流。我们建立了均值场极限与有限粒子系统的后悔界。分析中利用了对数Sobolev不等式、Polyak-Lojasiewicz条件、Malliavin微积分及时间一致的传播混沌。在位移凸条件下，获得恒定静态后悔界；在一般非凸情形下，推导出显式的线性后悔界，刻画了数据变化、熵探索和二次正则化的效应。最后，仿真展示了在线方法的优势，以及网络宽度和正则化参数的影响。
原文摘要 · Abstract (English)
We study continuous-time online learning where data are generated by a diffusion process with unknown coefficients. The learner employs a two-layer neural network, continuously updating its parameters in a non-anticipative manner. The mean-field limit of the learning dynamics corresponds to a stochastic Wasserstein gradient flow adapted to the data filtration. We establish regret bounds for both the mean-field limit and finite-particle system. Our analysis leverages the logarithmic Sobolev inequality, Polyak-Lojasiewicz condition, Malliavin calculus, and uniform-in-time propagation of chaos. Under displacement convexity, we obtain a constant static regret bound. In the general non-convex setting, we derive explicit linear regret bounds characterizing the effects of data variation, entropic exploration, and quadratic regularization. Finally, our simulations demonstrate the outperformance of the online approach and the impact of network width and regularization parameters.
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