Long-Run Dynamics of AdaGrad-Norm: Trajectory Stability and Asymptotic Stationarity

AdaGrad-Norm is widely studied, yet its long-run behavior in smooth nonconvex optimization remains delicate. For the standard current-denominator recursion, the same stochastic gradient determines both the update and its normalizer, rendering the effective stepsize nonpredictable. At the square-root normalization, the quadratic smoothness budget admits only logarithmic control rather than a uniform summability bound. We establish trajectory stability and full-sequence asymptotic stationarity directly for this unmodified recursion. Under global smoothness, non-flatness at infinity, conditional unbiasedness, and an affine conditional second-moment bound, we prove \(\E[\sup_{n\geq1} g(θ_n)]