Rowan Martnishn

Rowan Martnishn

Production deep learning systems across enterprise domains operate under constraints that academic benchmarks routinely obscure: labeled data is expensive, inference budgets are tight, and models that cannot explain their behavior are difficult to trust and maintain. We present ChainzRule (CR), a neural architecture replacing typical activations with learnable polynomial layers governed by Differential Regularization (DREG), a layer-wise Jacobian penalty computed analytically during the forward pass at standard inference cost. The core claim is that bounding intermediate derivatives forces the network toward low-frequency, structurally stable representations, simultaneously reducing dependence on labeled data volume, improving robustness to distribution shift, and providing a measurable, gradient-based handle on model behavior. Evaluated across five domains, CR achieves $85.71\% \pm 2.01\%$ on Pima Diabetes (statistically superior to SVM and XGBoost), $46.20\% \pm 0.37\%$ on SST-5 sentiment classification with a frozen encoder (superior to RNTN using approximately 5\% of its training data), $55.79\%$ on SST-5 with a fine-tuned BERT backbone (versus BERT-base linear head at $54.9\%$), $70.17\%$ on Yelp Full ordinal regression with 3.2M parameters versus a 10-model average of $66.35\%$, and $+2.32\%$ mean corruption accuracy on CIFAR-10-C. All results with reported $p$-values fall below the $α= 0.05$ threshold after Bonferroni correction. CR maintains a gradient tail ratio $τ$ (p99/mean) of $1.01$--$1.02$ against $1.07$--$1.09$ for all typical activation function baselines across every data fraction, a structural invariant we propose as the mechanistic driver of sample efficiency and a deployment-time proxy for model reliability.

Rowan Martnishn, Sean Anderson

As machine learning models grow in complexity, they increasingly struggle with three conflicting demands: the need for high accuracy, the requirement for hardware efficiency, and the necessity of functional stability. Traditional architectures often achieve performance at the expense of spiky or unpredictable behavior, where small changes in input lead to massive swings in output -- a critical flaw for real-world deployment in sensitive environments. This paper introduces ChainzRule (CR), a novel neural architecture designed to harmonize these competing goals. ChainzRule replaces standard piecewise-linear activations with a Polynomial Engine governed by Differential Regularization (DREG). Unlike traditional methods that impose global, coarse-grained constraints on a model's Lipschitz constant, DREG acts as a targeted regularization on intermediate derivatives. This approach suppresses extreme sensitivity without attenuating the representational power inherent in the Polynomial Engine. In head-to-head "Fair Fight" benchmarks, ChainzRule outperformed standard models while using 15.5x fewer parameters. On the MNIST dataset, it reduced peak gradient volatility by an average of 23.1%, ensuring a smoother and more predictable manifold. On Yelp Full ordinal regression under explicit DREG regularization, ChainzRule achieves 70.17% accuracy, validating that derivative-aware regularization is compatible with competitive performance on realistic tasks. By embedding gradient awareness into the architecture via DREG, ChainzRule demonstrates that stability and accuracy need not be competing objectives.