We study training algorithms with data following a Gaussian mixture model. For a specific family of such algorithms, we present a non-asymptotic result, connecting the evolution of the model to a surrogate dynamical system, which can be easier to analyze. The proof of our result is based on the celebrated Gordon comparison theorem. Using our theorem, we rigorously prove the validity of the dynamic mean-field (DMF) expressions in the asymptotic scenarios. Moreover, we suggest an iterative refinement scheme to obtain more accurate expressions in non-asymptotic scenarios. We specialize our theory to the analysis of training a perceptron model with a generic first-order (full-batch) algorithm and demonstrate that fluctuation parameters in a non-asymptotic domain emerge in addition to the DMF kernels.

翻译：本研究针对数据服从高斯混合模型的训练算法展开分析。针对此类算法的特定族系，我们提出了一个非渐近性结果，将模型演化过程与一个更易于分析的替代动力系统相关联。该结果的证明基于著名的Gordon比较定理。运用该定理，我们严格证明了动态平均场（DMF）表达式在渐近场景下的有效性。此外，我们提出了一种迭代优化方案，以获得非渐近场景下更精确的表达式。我们将该理论专门应用于分析采用通用一阶（全批次）算法训练感知机模型的过程，并证明在非渐近域中除了DMF核函数外还会出现波动参数。