Adam as Magnitude-Weighted Sign Descent: Dimension-Free Convergence under Weak Assumptions

We develop a convergence analysis of fully bias-corrected Adam through its *magnitude-weighted stochastic sign* representation. The key step controls the accumulated wrong-sign loss while retaining the joint randomness of Adam's update magnitude, momentum, and gradient noise. Under $L$-smoothness and conditional $p$-affine second moments, we prove that the expected average ${\ell }_2$ gradient norm is at most $C{T}^{-1/4}$, where $C$ has no dependence on explicit dimension factor and the rate has no extra $\log T$; under global point-pair $(L_0,L_1,q)$-smoothness, a stopped analysis yields the same exponent with high probability. We further instantiate the theory for fan-in-scaled LayerNorm/RMSNorm networks with fixed depth and increasing hidden width. For a concrete fixed-affine family, explicit width–horizon conditions guarantee that the Adam iterates remain in a width-uniform parameter region throughout training, yielding a convergence constant independent of the total parameter count. To our knowledge, this is the first Adam convergence-rate result establishing dimension-free optimization for an explicit growing family of normalized networks.