# When narrower is better: the narrow width limit of Bayesian parallel branching neural networks

arXiv: https://arxiv.org/abs/2407.18807 · HTML: https://mlsanity.com/paper/2407.18807/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 47 references that are in the map.
- First posted on arXiv 26 Jul 2024.

## Builds on (most cited first, 30 of 37)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [Tensor Programs V: Tuning Large Neural Networks via Zero-Shot Hyperparameter Transfer](https://mlsanity.com/paper/2203.03466/) (2022)
- [Generalization and Representational Limits of Graph Neural Networks](https://mlsanity.com/paper/2002.06157/) (2020)
- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [What graph neural networks cannot learn: depth vs width](https://mlsanity.com/paper/1907.03199/) (2019)
- [Graph Neural Tangent Kernel: Fusing Graph Neural Networks with Graph Kernels](https://mlsanity.com/paper/1905.13192/) (2019)
- [Contextual Stochastic Block Models](https://mlsanity.com/paper/1807.09596/) (2018)
- [Revisiting Graph based Collaborative Filtering: A Linear Residual Graph Convolutional Network Approach](https://mlsanity.com/paper/2001.10167/) (2020)
- [Neural Networks as Kernel Learners: The Silent Alignment Effect](https://mlsanity.com/paper/2111.00034/) (2021)
- [Self-Consistent Dynamical Field Theory of Kernel Evolution in Wide Neural Networks](https://mlsanity.com/paper/2205.09653/) (2022)
- [A PAC-Bayesian Approach to Generalization Bounds for Graph Neural Networks](https://mlsanity.com/paper/2012.07690/) (2020)
- [Statistical Mechanics of Deep Linear Neural Networks: The Back-Propagating Kernel Renormalization](https://mlsanity.com/paper/2012.04030/) (2020)
- [Expressiveness and Approximation Properties of Graph Neural Networks](https://mlsanity.com/paper/2204.04661/) (2022)
- [On Provable Benefits of Depth in Training Graph Convolutional Networks](https://mlsanity.com/paper/2110.15174/) (2021)
- [Towards Understanding the Generalization of Graph Neural Networks](https://mlsanity.com/paper/2305.08048/) (2023)
- [Feature-Learning Networks Are Consistent Across Widths At Realistic Scales](https://mlsanity.com/paper/2305.18411/) (2023)
- [Generalization in Graph Neural Networks: Improved PAC-Bayesian Bounds on Graph Diffusion](https://mlsanity.com/paper/2302.04451/) (2023)
- [Predicting the outputs of finite deep neural networks trained with noisy gradients](https://mlsanity.com/paper/2004.01190/) (2020)
- [Implicit bias of SGD in L2-regularized linear DNNs: One-way jumps from high to low rank](https://mlsanity.com/paper/2305.16038/) (2023)
- [The effective noise of Stochastic Gradient Descent](https://mlsanity.com/paper/2112.10852/) (2021)
- [The Onset of Variance-Limited Behavior for Networks in the Lazy and Rich Regimes](https://mlsanity.com/paper/2212.12147/) (2022)
- [Graph Convolutional Gaussian Processes](https://mlsanity.com/paper/1905.05739/) (2019)
- [The Effect of Network Width on Stochastic Gradient Descent and Generalization: an Empirical Study](https://mlsanity.com/paper/1905.03776/) (2019)
- [Towards Deepening Graph Neural Networks: A GNTK-based Optimization Perspective](https://mlsanity.com/paper/2103.03113/) (2021)

## Built on by (most cited first, 1 of 1)

- [Minimum Width of Deep Narrow Networks for Universal Approximation](https://mlsanity.com/paper/2511.06837/) (2025)

## Abstract (arXiv)

The infinite width limit of random neural networks is known to result in Neural Networks as Gaussian Process (NNGP) (Lee et al. (2018)), characterized by task-independent kernels. It is widely accepted that larger network widths contribute to improved generalization (Park et al. (2019)). However, this work challenges this notion by investigating the narrow width limit of the Bayesian Parallel Branching Neural Network (BPB-NN), an architecture that resembles neural networks with residual blocks. We demonstrate that when the width of a BPB-NN is significantly smaller compared to the number of training examples, each branch exhibits more robust learning due to a symmetry breaking of branches in kernel renormalization. Surprisingly, the performance of a BPB-NN in the narrow width limit is generally superior to or comparable to that achieved in the wide width limit in bias-limited scenarios. Furthermore, the readout norms of each branch in the narrow width limit are mostly independent of the architectural hyperparameters but generally reflective of the nature of the data. We demonstrate such phenomenon primarily in the branching graph neural networks, where each branch represents a different order of convolutions of the graph; we also extend the results to other more general architectures such as the residual-MLP and demonstrate that the narrow width effect is a general feature of the branching networks. Our results characterize a newly defined narrow-width regime for parallel branching networks in general.