发表机构
University of Chicago（芝加哥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过匹配平稳能量与局部OU极限建立常步长随机逼近的矩精确高斯混合，给出Wasserstein误差界，并验证了可观测协方差与SGD预测。
AI 中文摘要
常步长学习的局部高斯模型可预测输出变异性与期望损失，但仅凭弱收敛不足以证明这些矩预测的合理性。我们通过将平稳能量与局部奥恩斯坦-乌伦贝克极限相匹配，建立了矩精确的高斯混合，排除了弱收敛不可见的二次尾部质量。对于步长 $a$，二阶Wasserstein误差为 $o(\sqrt a)$，且在不变测度上一致成立，利用了各测度实际根权重。假设条件结合了约束、下降、有限多个双曲平衡点以及根连续性，并具有有限方差创新。该结果可得出可观测协方差、期望目标间隙和一阶均值偏移，同时允许奇异协方差、兼容鞍点及无极限的权重。对于由独立标准化Student $t_3$ 坐标的固定可逆变换给出的加性噪声，对称性给出了阶最优的 $\sqrt a$ 平滑检验界。数值输运计算展示了根特定协方差的价值；受控SGD研究评估了不同步长、批大小和模型几何下的可观测预测。
英文摘要
Local Gaussian models of constant-step learning predict output variability and expected losses, but weak convergence alone does not justify these moment predictions. We establish moment-accurate Gaussian mixtures by matching stationary energy with local Ornstein--Uhlenbeck limits, ruling out quadratic tail mass invisible to weak convergence. For step size $a$, the second-order Wasserstein error is $o(\sqrt a)$, uniformly over invariant laws, using each law's actual root weights. The assumptions combine confinement, descent, finitely many hyperbolic equilibria and root continuity with finite-variance innovations. The result yields observable covariances, expected objective gaps and first-order mean shifts, while allowing singular covariances, compatible saddles and weights without a limit. For additive noise given by a fixed invertible transform of independent standardized Student $t_3$ coordinates, symmetry gives an order-sharp $\sqrt a$ smooth-test bound. Numerical transport calculations demonstrate the value of root-specific covariances; controlled SGD studies assess observable predictions across step sizes, batch sizes and model geometries.
