ChainzRule: Sample-Efficient, Robust Deep Learning Across Tabular, NLP, and Vision Tasks

Rowan Martnishn

cs.LGMay 23, 2026

ChainzRule: Sample-Efficient, Robust Deep Learning Across Tabular, NLP, and Vision Tasks

Authors: Rowan Martnishn

Organizations: Sentivity AI

Abstract

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.

Explore similar work

Jun 6, 2026cs.LG

Layer-wise Derivative Controlled Networks Achieve Competitive Accuracy and Gradient Stability Across Data Regimes

Derivative-controlled networks based on ChainzRule (CR) combine cubic polynomial layers with a lightweight forward-mode per-layer Jacobian penalty (DREG). In this second paper of a multi-part series, we evaluate the generalization properties of CR across data regimes. We ablate the shape of the DREG coefficient schedule, demonstrating that the optimal annealing range depends on representation noise. On the Pima Diabetes dataset, CR achieves strong low-data performance and maintains a consistent accuracy advantage over baselines from 5% to 100% training data, supported by exceptionally stable gradient tail ratios (

\sim

1.01--1.02 vs. 1.07--1.09 for ReLU networks). Extensions to SST-5 show competitive or superior results in both frozen-embedding and BERT fine-tuned regimes, including outperforming prior BERT baselines despite substantially less training data. These results are statistically significant: CR achieves superior accuracy over the strongest published baselines we could identify on both datasets (

p < 0.05

). These results establish that layer-wise derivative control induces a structural inductive bias toward low-frequency, stable representations that generalizes robustly across tabular and NLP domains, data volumes, and representation qualities. The gradient tail ratio serves as a reliable, label-free diagnostic of generalization capability.

Rowan Martnishn

Rectified Linear Unit Networks Regularization Methods Gradient

Ask PDF

May 14, 2026cs.LG

Layer-wise Derivative Controlled Networks

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.

Rowan Martnishn, Sean Anderson

Regularization Methods Polynomials Volatility

Ask PDF

Jun 22, 2026cs.LG

DREG: A Layer-Wise Jacobian Regularization as a General-Purpose Penalty

We present a large-scale empirical study isolating the contributions of the Derivative Regularization penalty (DREG). Across a fully-crossed factorial sweep of 960 experiments spanning 4 activations, 6 regularizers, 8 datasets, and 5 random seeds, we ask: when, where, and why does DREG work? Our results establish three principal findings. First, DREG achieves the highest overall and clean-regime accuracy among all regularizers evaluated (significantly so against the unregularized baseline, Weight Decay, and IGPen; Wilcoxon

p \leq 0.031

). It ranks second in noise robustness behind Spectral Normalization (SN) - the only two layer-wise regularizers in the study. Second, DREG is globally the best-performing regularizer under GELU, the default activation in modern transformer architectures, particularly on both messy vision and messy NLP benchmarks, suggesting direct applicability to frontier deep learning settings. Third, DREG's advantage over competing regularizers is most pronounced under data scarcity, consistent with its role as a geometric inductive bias that substitutes for the regularizing effect of data volume. Throughout, DREG is applied with a single fixed hyperparameter

λ= 10^{-2.5}

and no per-dataset tuning, supporting its characterization as a plug-and-play regularizer for neural networks with nontrivial Jacobian structure. These findings are consistent with DREG's design: concentrating regularization pressure on layers where the activation derivative is largest, rather than constraining the network uniformly.

Rowan Martnishn

Regularization Methods Penalties

Ask PDF