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Stability and Generalization of Quantum Neural Networks

Jiaqi Yang, Wei Xie, Xiaohua Xu

TL;DR

This paper tackles the generalization behavior of quantum neural networks (QNNs) by importing algorithmic stability from classical learning theory. It develops high-probability generalization bounds based on uniform stability for SGD-trained QNNs and extends the analysis to depolarizing noise on NISQ devices, revealing that quantum noise can act as a regularizer. A refined, optimization-dependent bound via on-average stability complements the worst-case results and links generalization to initialization and gradient variance, with empirical results on real datasets supporting the theory. The work provides practical guidance on how to balance model complexity (e.g., number of trainable gates) against training dynamics (step sizes, iterations) to achieve robust generalization in quantum settings, and it highlights the potential of noise-aware training as a design knob.

Abstract

Quantum neural networks (QNNs) play an important role as an emerging technology in the rapidly growing field of quantum machine learning. While their empirical success is evident, the theoretical explorations of QNNs, particularly their generalization properties, are less developed and primarily focus on the uniform convergence approach. In this paper, we exploit an advanced tool in classical learning theory, i.e., algorithmic stability, to study the generalization of QNNs. We first establish high-probability generalization bounds for QNNs via uniform stability. Our bounds shed light on the key factors influencing the generalization performance of QNNs and provide practical insights into both the design and training processes. We next explore the generalization of QNNs on near-term noisy intermediate-scale quantum (NISQ) devices, highlighting the potential benefits of quantum noise. Moreover, we argue that our previous analysis characterizes worst-case generalization guarantees, and we establish a refined optimization-dependent generalization bound for QNNs via on-average stability. Numerical experiments on various real-world datasets support our theoretical findings.

Stability and Generalization of Quantum Neural Networks

TL;DR

This paper tackles the generalization behavior of quantum neural networks (QNNs) by importing algorithmic stability from classical learning theory. It develops high-probability generalization bounds based on uniform stability for SGD-trained QNNs and extends the analysis to depolarizing noise on NISQ devices, revealing that quantum noise can act as a regularizer. A refined, optimization-dependent bound via on-average stability complements the worst-case results and links generalization to initialization and gradient variance, with empirical results on real datasets supporting the theory. The work provides practical guidance on how to balance model complexity (e.g., number of trainable gates) against training dynamics (step sizes, iterations) to achieve robust generalization in quantum settings, and it highlights the potential of noise-aware training as a design knob.

Abstract

Quantum neural networks (QNNs) play an important role as an emerging technology in the rapidly growing field of quantum machine learning. While their empirical success is evident, the theoretical explorations of QNNs, particularly their generalization properties, are less developed and primarily focus on the uniform convergence approach. In this paper, we exploit an advanced tool in classical learning theory, i.e., algorithmic stability, to study the generalization of QNNs. We first establish high-probability generalization bounds for QNNs via uniform stability. Our bounds shed light on the key factors influencing the generalization performance of QNNs and provide practical insights into both the design and training processes. We next explore the generalization of QNNs on near-term noisy intermediate-scale quantum (NISQ) devices, highlighting the potential benefits of quantum noise. Moreover, we argue that our previous analysis characterizes worst-case generalization guarantees, and we establish a refined optimization-dependent generalization bound for QNNs via on-average stability. Numerical experiments on various real-world datasets support our theoretical findings.
Paper Structure (20 sections, 25 theorems, 101 equations, 4 figures, 1 table)

This paper contains 20 sections, 25 theorems, 101 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Setup
  4. Quantum Computation Basics and Notations
  5. Stability and Generalization Analysis of Randomized Algorithm
  6. Main Assumptions
  7. Main Results
  8. Uniform Stability-Based Generalization Bounds
  9. Towards Optimization-Dependent Generalization Bounds
  10. Experimental Evaluation
  11. Conclusion
  12. Acknowledgements
  13. Notions
  14. Auxiliary Lemmas
  15. Proofs of Main Results
  16. ...and 5 more sections

Key Result

Lemma 3.3

Let $A$ be a randomized algorithm, $\epsilon>0$ and $\delta\in(0,1)$.

Figures (4)

  • Figure 1: Generalization gap for varying numbers of trainable quantum gates: (a) MNIST, (b) Fashion MNIST.
  • Figure 2: Generalization gap for varying step sizes: (a) MNIST, (b) Fashion MNIST.
  • Figure 3: Generalization gap with varying noise levels: (a) MNIST, (b) Fashion MNIST.
  • Figure 4: On-average variance and generalization gap for varying random label probabilities: (a) MNIST, (b) Fashion MNIST.

Theorems & Definitions (54)

  • Definition 3.1: Uniform Stability
  • Definition 3.2: On-Average Stability
  • Lemma 3.3: Stability and Generalization
  • Remark 3.4
  • Definition 3.5: Stochastic Gradient Descent
  • Remark 3.8
  • Theorem 4.1: Uniform Stability Bound
  • Remark 4.2
  • Corollary 4.3
  • Remark 4.4
  • ...and 44 more