谱重尾起始的完整Adam定理
A Full Adam Theorem for Spectral Heavy-Tail Onset
- Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该论文在封闭高斯Stein-Hermite教师-学生模型中证明了谱重尾起始的完整Adam定理，给出了精确的命中时间定律，并证明了更强定理的不可能性及线性网络动力学的局限性。
AI中文摘要：
我们证明了在封闭高斯Stein-Hermite教师-学生状态演化模型中，谱重尾起始的完整Adam定理。该定理从实际的全批量Adam递推开始，通过Stein-Hermite微积分推导出总体梯度，证明了有限宽度协方差集中性，将多步Adam动量转换为精确的非中心高斯符号核，通过基齐次化定理控制对角Adam分母，从Hermite边传递定理推导出正则变化投影更新响应，将响应通过精确Gram更新推动，并证明了具有匹配上下命中界的近似目标KL收缩。最终定律为(\tau_\tau=\Theta(\Delta_1^{-\gamma}d^\rho\log(\Psi_0/\varepsilon)))，其中(\\(\Delta_1\\))是第一尖峰-体谱间隙。该结果是完整的，其精确意义在于：从Adam的动量和分母到谱命中定律的每一步都在封闭状态演化模型内形式化。我们还证明了更强的任意梯度Adam定理是不可能的，并且精确的两步线性网络损失动力学无法识别因子谱或重尾命中时间。
英文摘要：
We prove a full Adam theorem for spectral heavy-tail onset in a closed Gaussian Stein-Hermite teacher-student state-evolution model. The theorem begins with the actual full-batch Adam recurrences, derives the population gradient by Stein-Hermite calculus, proves finite-width covariance concentration, converts multi-step Adam momentum into an exact non-centered Gaussian sign kernel, controls the diagonal Adam denominator by a basis-homogenization theorem, derives a regularly varying projected update response from a Hermite edge-transfer theorem, pushes the response through the exact Gram update, and proves approximate-target KL contraction with matching upper and lower hitting bounds. The final law is (τ_\varepsilon=Θ(Δ_1^{-γ}d^ρ\log(Ψ_0/\varepsilon))), where (Δ_1) is the first spike-bulk spectral gap. The result is full in the following precise sense: every step from Adam's momentum and denominator to the spectral hitting law is formalized inside the closed state-evolution model. We also prove that a stronger arbitrary-gradient Adam theorem is impossible, and that exact two-step linear-network loss dynamics do not identify factor spectra or heavy-tail hitting times.
