Mingi Kang

Activation functions are fundamental to deep neural networks, governing gradient flow, optimization stability, and representational capacity. Within historic deep architectures, while ReLU has been the dominant choice for the activation function, modern transformer-based models increasingly are adopting smoother alternatives such as GELU and other self-gated alternatives. Despite their empirical success, the mathematical relationships among these functions and the principles underlying their effectiveness remains only partially understood. We introduce IGLU, a parametric activation function derived as a scale mixture of GELU gates under a half-normal mixing distribution. This derivation yields a closed-form expression whose gating component is exactly the Cauchy CDF, providing a principled one-parameter family that continuously interpolates between identity-like and ReLU-like behavior via a single sharpness parameter σ. Unlike GELU’s Gaussian gate, IGLU’s heavy-tailed Cauchy gate decays polynomially in the negative tail, guaranteeing non-zero gradients for all finite inputs and offering greater robustness to vanishing gradients. We further introduce IGLU-Approx, a computationally efficient rational approximation of IGLU expressed entirely in terms of ReLU operations that eliminates transcendental function evaluation. Through evaluations on CIFAR-10, CIFAR-100, and WikiText-103 across ResNet-20, ViT-Tiny, and GPT-2 Small, IGLU achieves competitive or superior performance on both vision and language datasets against ReLU and GELU baselines, with IGLU-Approx recovering this performance at substantially reduced computational cost. In particular, we show that employing a heavy-tailed gate leads to considerable performance gains in heavily imbalanced classification datasets.

The shift from Convolutional Neural Networks to Transformers has reshaped computer vision, yet these two architectural families are typically viewed as fundamentally distinct. We argue that convolution and self-attention, despite their apparent differences, can be unified within a single k-nearest neighbor aggregation framework. The critical insight is that both operations are special cases of neighbor selection and aggregation; convolution selects neighbors by spatial proximity, while attention selects by feature similarity, revealing they exist on a continuous spectrum. We introduce Convolutional Nearest Neighbors (ConvNN), a unified framework that formalizes this connection. Crucially, ConvNN serves as a drop-in replacement for convolutional and attention layers, enabling systematic exploration of the intermediate spectrum between these two extremes. We validate the framework’s coherence on CIFAR-10 and CIFAR-100 classification tasks across two complementary architectures: (1) Hybrid branching in VGG improves accuracy on both CIFAR datasets by combining spatial-proximity and feature-similarity selection; and (2) ConvNN in ViT outperforms standard attention and other attention variants on both datasets. Extensive ablations on k values and architectural variants reveal that interpolating along this spectrum provides regularization benefits by balancing local and global receptive fields. Our work provides a unifying framework that dissolves the apparent distinction between convolution and attention, with implications for designing more principled and interpretable vision architectures.