SOAR: Smooth Online Activation Routing for Stable Neural Learning from Evolving Streams

Online neural learning requires models that update after each incoming example, remain calibrated under distributional change, and avoid brittle gradient transmission. The original version of this work used a small static benchmark, a shallow model, few random seeds, and no significance tests. This revision reformulates the problem as genuine prequential online learning and introduces Smooth Online Activation Routing (SOAR), a Hedge-style routing algorithm over smooth activation experts trained by single-instance adaptive updates. SOAR links activation smoothness with online regret through a prediction-with-expert-advice bound: the mixture incurs only logarithmic-exponential-weight overhead relative to the best activation expert in hindsight, while each expert uses diagonal adaptive online gradients. We evaluate two-hidden-layer online neural networks on eight streams, fourteen methods, thirteen comparators, and eight random seeds, yielding 896 fully reproducible runs. The benchmark includes four real scikit-learn data sets and four synthetic streaming tasks with stationary, nonlinear, gradual-drift, and sudden-drift regimes. Across all dataset-seed blocks, SOAR obtains the best mean final accuracy (92.56%), best mean negative log-likelihood (0.199), and best prequential accuracy (86.83%). Paired Wilcoxon and Holm-corrected tests show significant improvements over ReLU, LeakyReLU, ELU, Softplus, Adam, SGD, and the compact ablation; improvements over the strongest fixed TeLU activation are positive but not statistically significant. The results support a nuanced conclusion: adaptive smooth activation routing is a robust online-learning mechanism and the best overall method in this controlled benchmark suite, but fixed smooth activations remain highly competitive on small static data.