Activation-Adjacent Layer Normalization: Effects on Approximation Power

Layer normalization (LN) and root mean square normalization (RMSNorm) remain input-dependent and nonlinear at inference, so inserting either next to an activation need not preserve represented functions. We study when activation-normalized (AN) and preactivation-normalized (PN) shallow networks can approximate the function represented by a prescribed width-$N$ one-hidden-layer MLP. Viewing RMSNorm as input-dependent scaling, we use width augmentation to locally linearize normalization on compact feature sets. For AN networks, an unbounded activation range allows width $N+1$, while every continuous nonconstant activation admits width $N+r$ with error $O_f(r^{-1})$. For PN networks, affine parameters or, without them, degree-one positive homogeneity recover width $N+1$; general continuous activations admit width $N+r$ with error tending to zero. The corresponding LN constructions use centered coordinate pairs, yielding widths $N+2$ and $N+2r$, respectively. We prove that $N+1$ for RMSNorm and $N+2$ for LN are also the worst-case bound for approximations, and show that applicable universal-approximation theorems for ordinary MLPs transfer to these normalized architectures. Numerical experiments recover the predicted approximation rates for both normalizations across all five settings and illustrate the shifted-ReLU finite-width obstruction.