论文标题
矩阵和Graphon McKean-Vlasov限制的随机优化
Stochastic optimization on matrices and a graphon McKean-Vlasov limit
论文作者
论文摘要
我们考虑在使用相同的置换量的排列行和列下不变的大型对称矩阵的空间上的随机梯度下降。我们建立了这些随机曲线的确定性限制，因为矩阵的尺寸移至无穷大，而条目保持界限。在``小噪声''假设下，极限被证明是在〜\ cite {oh2021gradient}中建立的图形上的函数梯度流。我们还考虑了随机梯度下降的限制，并增加了适当缩放的布朗噪声。图形的限制曲线的特征在于具有反射的随机微分方程家族，可以被认为是经典的McKean-Vlasov经典限制，以相互作用扩散到Graphon设置。这些证明引入了一个无限二维的可兑换扩散阵列的家族，并在适当的意义上介绍了混乱的新颖概念，以使大型扩散矩阵融合到了此类阵列。
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.
