We consider stochastic gradient descents on the space of large symmetric matrices of suitable functions that are invariant under permuting the rows and columns using the same permutation. We establish deterministic limits of these random curves as the dimensions of the matrices go to infinity while the entries remain bounded. Under a ``small noise'' assumption the limit is shown to be the gradient flow of functions on graphons whose existence was established in~\cite{oh2021gradient}. We also consider limits of stochastic gradient descents with added properly scaled reflected Brownian noise. The limiting curve of graphons is characterized by a family of stochastic differential equations with reflections and can be thought of as an extension of the classical McKean-Vlasov limit for interacting diffusions to the graphon setting. The proofs introduce a family of infinite-dimensional exchangeable arrays of reflected diffusions and a novel notion of propagation of chaos for large matrices of diffusions converging to such arrays in a suitable sense.

翻译：我们考虑在大型对称矩阵空间上，对满足“行与列经相同置换后函数值不变”条件的适当函数进行随机梯度下降。当矩阵维度趋于无穷而矩阵元素保持有界时，我们建立了这些随机曲线的确定性极限。在“小噪声”假设下，该极限被证明是图朗函数空间上的梯度流——其存在性已在文献~\cite{oh2021gradient} 中得到证明。我们还研究了添加适当尺度反射布朗噪声的随机梯度下降过程的极限。图朗极限曲线由一族带反射的随机微分方程刻画，可视为经典相互作用扩散系统的麦基恩-弗拉索夫极限在图朗框架下的拓展。证明过程中引入了一类无限维可交换反射扩散阵列，并提出了一种新颖的混沌传播概念，用于描述收敛于此类阵列的大型扩散矩阵系统。