Spectral Sign-Balanced Update Blocks

Integrate this at a fixed-size update-combination point rather than inside every individual matrix multiply. The cheapest MVP is a LoRA or adapter block with $n$ candidate matrices $B_i$ of identical shape; alternatively use $n$ expert residual branches whose outputs are added before the next layer. First normalize each candidate by estimating $s_i=\lVert B_iB_i^\top\rVert_{\mathrm{op}}$ with 3--5 power iterations, then implement the symmetric operator implicitly as $A_i v=B_i(B_i^\top v)/s_i$. Initialize all signs to random values. Run coordinate local search: for each $i$, temporarily flip $x_i$, estimate the largest eigenvalue of $S=\sum_i x_iA_i$ using 5 power iterations, keep the flip if the estimated operator norm decreases, and repeat 2--5 passes with several random restarts. Do not materialize $A_i$; compute matrix-vector products through $B_i$ and $B_i^\top$. Freeze the selected signs during training, or recompute them every few hundred optimizer steps using detached weights. The exact $O(\sqrt n)$ guarantee and universal constant $C$ come from the Matrix Spencer theorem; the local-search coloring and spectral estimates are empirical approximations, not the paper's partial-coloring algorithm. Start with CIFAR-10 ResNet-18 and eight rank-8 adapters, comparing random signs, all-positive signs, coordinatewise magnitude minimization, and the spectral method at equal trainable parameter count. Measure validation accuracy, gradient and activation spectral norms, loss spikes, and wall-clock overhead. A useful signal is lower worst-direction curvature or interference and improved accuracy or stability at the same adapter size; reject the method if local search fails to beat random signs after accounting for recomputation cost.