# Generalization and Risk Bounds for Recurrent Neural Networks

arXiv: https://arxiv.org/abs/2411.02784 · HTML: https://mlsanity.com/paper/2411.02784/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 48 references that are in the map.
- First posted on arXiv 5 Nov 2024.

## Builds on (most cited first, 18 of 18)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [A large annotated corpus for learning natural language inference](https://mlsanity.com/paper/1508.05326/) (2015)
- [Google's Neural Machine Translation System: Bridging the Gap between Human and Machine Translation](https://mlsanity.com/paper/1609.08144/) (2016)
- [Show, Attend and Tell: Neural Image Caption Generation with Visual Attention](https://mlsanity.com/paper/1502.03044/) (2015)
- [Deep Visual-Semantic Alignments for Generating Image Descriptions](https://mlsanity.com/paper/1412.2306/) (2014)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Speech Recognition with Deep Recurrent Neural Networks](https://mlsanity.com/paper/1303.5778/) (2013)
- [xLSTM: Extended Long Short-Term Memory](https://mlsanity.com/paper/2405.04517/) (2024)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [On the Convergence Rate of Training Recurrent Neural Networks](https://mlsanity.com/paper/1810.12065/) (2018)
- [Model Complexity of Deep Learning: A Survey](https://mlsanity.com/paper/2103.05127/) (2021)
- [Generalization Error Bounds of Gradient Descent for Learning Over-parameterized Deep ReLU Networks](https://mlsanity.com/paper/1902.01384/) (2019)
- [All You Need is Beyond a Good Init: Exploring Better Solution for Training Extremely Deep Convolutional Neural Networks with Orthonormality and Modulation](https://mlsanity.com/paper/1703.01827/) (2017)
- [On Generalization Bounds of a Family of Recurrent Neural Networks](https://mlsanity.com/paper/1910.12947/) (2019)
- [Stabilizing Gradients for Deep Neural Networks via Efficient SVD Parameterization](https://mlsanity.com/paper/1803.09327/) (2018)
- [DeepSleepNet-Lite: A Simplified Automatic Sleep Stage Scoring Model with Uncertainty Estimates](https://mlsanity.com/paper/2108.10600/) (2021)

## Built on by (most cited first, 2 of 2)

- [Generalization Error Analysis for Selective State-Space Models Through the Lens of Attention](https://mlsanity.com/paper/2502.01473/) (2025)
- [Upper Bounds on the Generalization Error of Deep Learning Models via Local Robustness and Stability](https://mlsanity.com/paper/2606.16883/) (2026)

## Abstract (arXiv)

Recurrent Neural Networks (RNNs) have achieved great success in the prediction of sequential data. However, their theoretical studies are still lagging behind because of their complex interconnected structures. In this paper, we establish a new generalization error bound for vanilla RNNs, and provide a unified framework to calculate the Rademacher complexity that can be applied to a variety of loss functions. When the ramp loss is used, we show that our bound is tighter than the existing bounds based on the same assumptions on the Frobenius and spectral norms of the weight matrices and a few mild conditions. Our numerical results show that our new generalization bound is the tightest among all existing bounds in three public datasets. Our bound improves the second tightest one by an average percentage of 13.80% and 3.01% when the tanh and ReLU activation functions are used, respectively. Moreover, we derive a sharp estimation error bound for RNN-based estimators obtained through empirical risk minimization (ERM) in multi-class classification problems when the loss function satisfies a Bernstein condition.