# Generalization Bounds for Quantum Learning via R'enyi Divergences

arXiv: https://arxiv.org/abs/2505.11025 · HTML: https://mlsanity.com/paper/2505.11025/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 52 references that are in the map.
- First posted on arXiv 16 May 2025.

## Builds on (most cited first, 17 of 17)

- [Information-theoretic analysis of generalization capability of learning algorithms](https://mlsanity.com/paper/1705.07809/) (2017)
- [Information-Theoretic Generalization Bounds for SGLD via Data-Dependent Estimates](https://mlsanity.com/paper/1911.02151/) (2019)
- [Tightening Mutual Information Based Bounds on Generalization Error](https://mlsanity.com/paper/1901.04609/) (2019)
- [Sharpened Generalization Bounds based on Conditional Mutual Information and an Application to Noisy, Iterative Algorithms](https://mlsanity.com/paper/2004.12983/) (2020)
- [Generalization Bounds: Perspectives from Information Theory and PAC-Bayes](https://mlsanity.com/paper/2309.04381/) (2023)
- [Generalization Error Bounds Via R'enyi-, f-Divergences and Maximal Leakage](https://mlsanity.com/paper/1912.01439/) (2019)
- [A PAC-Bayesian Tutorial with A Dropout Bound](https://mlsanity.com/paper/1307.2118/) (2013)
- [Generalization Bounds via Information Density and Conditional Information Density](https://mlsanity.com/paper/2005.08044/) (2020)
- [Quantum Differential Privacy: An Information Theory Perspective](https://mlsanity.com/paper/2202.10717/) (2022)
- [A New Family of Generalization Bounds Using Samplewise Evaluated CMI](https://mlsanity.com/paper/2210.06422/) (2022)
- [Rate-Distortion Theoretic Generalization Bounds for Stochastic Learning Algorithms](https://mlsanity.com/paper/2203.02474/) (2022)
- [Variational Representations and Neural Network Estimation of R'enyi Divergences](https://mlsanity.com/paper/2007.03814/) (2020)
- [Quantum Differentially Private Sparse Regression Learning](https://mlsanity.com/paper/2007.11921/) (2020)
- [Rate-Distortion Theoretic Bounds on Generalization Error for Distributed Learning](https://mlsanity.com/paper/2206.02604/) (2022)
- [Quantum Local Differential Privacy and Quantum Statistical Query Model](https://mlsanity.com/paper/2203.03591/) (2022)
- [Information-theoretic generalization bounds for learning from quantum data](https://mlsanity.com/paper/2311.05529/) (2023)
- [Lower Bounds on Learning Pauli Channels with Individual Measurements](https://mlsanity.com/paper/2301.09192/) (2023)

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

- [Privacy Implies Stability: Information-Theoretic Generalization Bounds for Quantum Learning](https://mlsanity.com/paper/2602.01177/) (2026)

## Abstract (arXiv)

This work advances the theoretical understanding of quantum learning by establishing a new family of upper bounds on the expected generalization error of quantum learning algorithms, leveraging the framework introduced by Caro et al. (2024) and a new definition for the expected true loss. Our primary contribution is the derivation of these bounds in terms of quantum and classical R'enyi divergences, utilizing a variational approach for evaluating quantum R'enyi divergences, specifically the Petz and a newly introduced modified sandwich quantum R'enyi divergence. Analytically and numerically, we demonstrate the superior performance of the bounds derived using the modified sandwich quantum R'enyi divergence compared to those based on the Petz divergence. Furthermore, we provide probabilistic generalization error bounds using two distinct techniques: one based on the modified sandwich quantum R'enyi divergence and classical R'enyi divergence, and another employing smooth max R'enyi divergence.