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Towards Understanding Generalization in DP-GD: A Case Study in Training Two-Layer CNNs

Zhongjie Shi, Puyu Wang, Chenyang Zhang, Yuan Cao

TL;DR

The paper investigates generalization under differential privacy in gradient-based learning, identifying a concrete binary classification setup where DP-GD can surpass standard GD. Using a two-layer Hub ReLU CNN with a noise-containing data model, the authors derive stage-based analyses showing noise memorization harms GD’s test performance, while DP-GD can cross activation thresholds and achieve strong training accuracy plus privacy with early stopping. They provide theoretical bounds on training loss, generalization error, and DP guarantees, and corroborate them with numerical experiments across varying SNRs. The results challenge the usual privacy-utility trade-off by demonstrating task-specific regimes where privacy-preserving training improves generalization.

Abstract

Modern deep learning techniques focus on extracting intricate information from data to achieve accurate predictions. However, the training datasets may be crowdsourced and include sensitive information, such as personal contact details, financial data, and medical records. As a result, there is a growing emphasis on developing privacy-preserving training algorithms for neural networks that maintain good performance while preserving privacy. In this paper, we investigate the generalization and privacy performances of the differentially private gradient descent (DP-GD) algorithm, which is a private variant of the gradient descent (GD) by incorporating additional noise into the gradients during each iteration. Moreover, we identify a concrete learning task where DP-GD can achieve superior generalization performance compared to GD in training two-layer Huberized ReLU convolutional neural networks (CNNs). Specifically, we demonstrate that, under mild conditions, a small signal-to-noise ratio can result in GD producing training models with poor test accuracy, whereas DP-GD can yield training models with good test accuracy and privacy guarantees if the signal-to-noise ratio is not too small. This indicates that DP-GD has the potential to enhance model performance while ensuring privacy protection in certain learning tasks. Numerical simulations are further conducted to support our theoretical results.

Towards Understanding Generalization in DP-GD: A Case Study in Training Two-Layer CNNs

TL;DR

The paper investigates generalization under differential privacy in gradient-based learning, identifying a concrete binary classification setup where DP-GD can surpass standard GD. Using a two-layer Hub ReLU CNN with a noise-containing data model, the authors derive stage-based analyses showing noise memorization harms GD’s test performance, while DP-GD can cross activation thresholds and achieve strong training accuracy plus privacy with early stopping. They provide theoretical bounds on training loss, generalization error, and DP guarantees, and corroborate them with numerical experiments across varying SNRs. The results challenge the usual privacy-utility trade-off by demonstrating task-specific regimes where privacy-preserving training improves generalization.

Abstract

Modern deep learning techniques focus on extracting intricate information from data to achieve accurate predictions. However, the training datasets may be crowdsourced and include sensitive information, such as personal contact details, financial data, and medical records. As a result, there is a growing emphasis on developing privacy-preserving training algorithms for neural networks that maintain good performance while preserving privacy. In this paper, we investigate the generalization and privacy performances of the differentially private gradient descent (DP-GD) algorithm, which is a private variant of the gradient descent (GD) by incorporating additional noise into the gradients during each iteration. Moreover, we identify a concrete learning task where DP-GD can achieve superior generalization performance compared to GD in training two-layer Huberized ReLU convolutional neural networks (CNNs). Specifically, we demonstrate that, under mild conditions, a small signal-to-noise ratio can result in GD producing training models with poor test accuracy, whereas DP-GD can yield training models with good test accuracy and privacy guarantees if the signal-to-noise ratio is not too small. This indicates that DP-GD has the potential to enhance model performance while ensuring privacy protection in certain learning tasks. Numerical simulations are further conducted to support our theoretical results.

Paper Structure

This paper contains 20 sections, 45 theorems, 199 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Setting
  4. Main Results
  5. Noise Memorization of GD
  6. Signal Learning of DP-GD
  7. Numerical Experiments
  8. Conclusion and Future Work
  9. Preliminary Lemmas
  10. Noise Memorization of GD
  11. Signal-noise Decomposition and the properties
  12. First Stage
  13. Second Stage
  14. Generalization Error Analysis
  15. Signal Learning of DP-GD
  16. ...and 5 more sections

Key Result

Theorem 1

Under Condition condition, for any $\epsilon >0$, denote $T_1= \widetilde{\Theta}\left(\frac{\kappa^{q-1} mn}{\eta \sigma_0^{q-2} (\sigma_p \sqrt{d})^q}\right)$, and $T_2= T_1+ \frac{36 nm^2}{\eta \sigma_p^2d}$. Then within $T^*=T_1+ \widetilde{O}\left( \frac{m^{3}n}{\eta \epsilon ||\bm{\mu}||_2^2}

Figures (1)

  • Figure 1: Training loss, test loss, and test accuracy of two-layer CNNs trained with GD and DP-GD. Results are shown for three noise levels ($\sigma_p =0.1, 0.3, 0.5$) with fixed signal strength ($\|\bm{\mu}\|_2=1$).

Theorems & Definitions (76)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • Theorem 3
  • Definition 3
  • Lemma 4
  • ...and 66 more