# Universal Smoothness via Bernstein Polynomials: A Constructive Approximation Approach for Activation Functions

arXiv: https://arxiv.org/abs/2605.02591 · HTML: https://mlsanity.com/paper/2605.02591/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 33 references that are in the map.
- First posted on arXiv 4 May 2026.

## Builds on (most cited first, 19 of 19)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
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- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [DeepSeek-V2: A Strong, Economical, and Efficient Mixture-of-Experts Language Model](https://mlsanity.com/paper/2405.04434/) (2024)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Transformer in Transformer](https://mlsanity.com/paper/2103.00112/) (2021)
- [Activation Functions in Deep Learning: A Comprehensive Survey and Benchmark](https://mlsanity.com/paper/2109.14545/) (2021)
- [On the Impact of the Activation Function on Deep Neural Networks Training](https://mlsanity.com/paper/1902.06853/) (2019)
- [Lipschitz Recurrent Neural Networks](https://mlsanity.com/paper/2006.12070/) (2020)
- [Lipschitz constant estimation of Neural Networks via sparse polynomial optimization](https://mlsanity.com/paper/2004.08688/) (2020)
- [Continuously Differentiable Exponential Linear Units](https://mlsanity.com/paper/1704.07483/) (2017)
- [From Knowing to Doing Precisely: A General Self-Correction and Termination Framework for VLA models](https://mlsanity.com/paper/2602.01811/) (2026)

## Abstract (arXiv)

The efficacy of deep neural networks is heavily reliant on the design of non-linear activation functions, yet existing approaches often struggle to balance optimization stability with computational efficiency. While piecewise linear functions offer inference speed, they suffer from optimization instability due to non-differentiability at the origin, whereas smooth counterparts typically incur significant computational overhead through their reliance on transcendental operations. To address these limitations, this paper proposes a general smoothing framework based on constructive approximation theory and introduces the Bernstein Linear Unit (BerLU). This novel activation function utilizes Bernstein polynomials to construct a differentiable quadratic transition region that effectively eliminates singularities while maintaining a piecewise linear structure. Theoretical analysis demonstrates that the proposed method guarantees strictly continuous differentiability and a non-expansive Lipschitz constant of one, which ensures stable gradient propagation and prevents the gradient explosion problems common in deep architectures. Comprehensive empirical evaluations across representative Vision Transformer and Convolutional Neural Network architectures confirm that this approach consistently outperforms state-of-the-art baselines on standard image classification benchmarks while delivering superior computational and memory efficiency.