Coordinatewise Epsilon for Adam under Heavy-Tailed Noise

Adam estimates second moments coordinatewise, but typically adds the same scalar $ϵ$ to every denominator. We study how this shared $ϵ$ affects convergence when noise and curvature vary across coordinates. Under coordinatewise finite-$p$ noise moments, $p\in (1,2]$, choosing ${ϵ}_i$ from the noise and curvature bounds for each coordinate attains the heavy-tailed ${\ell }_1$-stationarity benchmark $O(T^{-\frac{p-1}{3p-2}})$, matching the known optimal time exponent for smooth nonconvex stochastic optimization. For $1<p<2$, Adam with a shared ${ϵ}_t$ admits a lower bound of order $\Omega (T^{-\frac{p-1}{2p}})$. This lower bound holds for every predictable scalar choice of ${\eta }_t$, ${\beta }_{1,t}$, ${\beta }_{2,t}$, and ${ϵ}_t$, precluding a dimension-uniform benchmark rate for this class.