HAND: A Biologically-Inspired Activation Function that Improves Generalisation and Sample Efficiency in Image Classification

Activation functions play a central role in neural networks by shaping internal representations. Recently, learning binary activation representations has attracted significant attention due to their advantages in computational and memory efficiency, as well as interpretability. However, training neural networks with Heaviside activations remains challenging, as their non-differentiability obstructs standard gradient-based optimization. In this paper, we propose Heavy Tailed Activation Function (HTAF), a smooth approximation to the Heaviside function that enables stable training with gradient-based optimization. We construct HTAF as a sigmoid hyperbolic tangent composite function and theoretically show that it maintains a large gradient mass around zero inputs while exhibiting slower gradient decay in the tail regions. We show that Spiking Neural Networks, Binary Neural Networks and Deep Heaviside neural Networks can be trained stably using HTAF with gradient-based optimization. Finally, we introduce Implicit Concept Bottleneck Models (ICBMs), an interpretable image model that leverages HTAF to induce discrete feature representations. Extensive experiments across various architectures and image datasets demonstrate that ICBM enables stable discretization while achieving prediction performance comparable to or better than standard models.

We study robust generalization under spurious correlations: tasks where a shortcut feature is correlated with the true label in training but anti-correlated in an adversarial held-out split. Varying the spurious ratio $r$ (the fraction of training examples where shortcut = true label) and model capacity, we find a counterintuitive result: data imbalance promotes generalization in sufficiently capable models. On a synthetic task where the true label is sum parity of an integer sequence and the shortcut is the parity of the maximum-valued element, a 2-layer, 2-head transformer generalized (reached $100\%$ adversarial accuracy) in 0% of seeds at $r{=}0.50$ but 77% of seeds at $r{=}0.90$. The effect is absent in 1-layer models, where imbalance instead traps the model on the shortcut. Through mechanistic analysis -- gradient conflict dynamics, circuit evolution, and QK/OV circuit ablations -- we characterize a mechanistic pathway consistent with imbalance promoting generalization.

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.