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Generalization Analysis and Method for Domain Generalization for a Family of Recurrent Neural Networks

Atefeh Termehchi, Ekram Hossain, Isaac Woungang

TL;DR

This paper proposes a method to analyze interpretability and out-of-domain (OOD) generalization for a family of recurrent neural networks (RNNs) and proposes a domain generalization method that reduces the OOD generalization error and improves the robustness to distribution shifts.

Abstract

Deep learning (DL) has driven broad advances across scientific and engineering domains. Despite its success, DL models often exhibit limited interpretability and generalization, which can undermine trust, especially in safety-critical deployments. As a result, there is growing interest in (i) analyzing interpretability and generalization and (ii) developing models that perform robustly under data distributions different from those seen during training (i.e. domain generalization). However, the theoretical analysis of DL remains incomplete. For example, many generalization analyses assume independent samples, which is violated in sequential data with temporal correlations. Motivated by these limitations, this paper proposes a method to analyze interpretability and out-of-domain (OOD) generalization for a family of recurrent neural networks (RNNs). Specifically, the evolution of a trained RNN's states is modeled as an unknown, discrete-time, nonlinear closed-loop feedback system. Using Koopman operator theory, these nonlinear dynamics are approximated with a linear operator, enabling interpretability. Spectral analysis is then used to quantify the worst-case impact of domain shifts on the generalization error. Building on this analysis, a domain generalization method is proposed that reduces the OOD generalization error and improves the robustness to distribution shifts. Finally, the proposed analysis and domain generalization approach are validated on practical temporal pattern-learning tasks.

Generalization Analysis and Method for Domain Generalization for a Family of Recurrent Neural Networks

TL;DR

This paper proposes a method to analyze interpretability and out-of-domain (OOD) generalization for a family of recurrent neural networks (RNNs) and proposes a domain generalization method that reduces the OOD generalization error and improves the robustness to distribution shifts.

Abstract

Deep learning (DL) has driven broad advances across scientific and engineering domains. Despite its success, DL models often exhibit limited interpretability and generalization, which can undermine trust, especially in safety-critical deployments. As a result, there is growing interest in (i) analyzing interpretability and generalization and (ii) developing models that perform robustly under data distributions different from those seen during training (i.e. domain generalization). However, the theoretical analysis of DL remains incomplete. For example, many generalization analyses assume independent samples, which is violated in sequential data with temporal correlations. Motivated by these limitations, this paper proposes a method to analyze interpretability and out-of-domain (OOD) generalization for a family of recurrent neural networks (RNNs). Specifically, the evolution of a trained RNN's states is modeled as an unknown, discrete-time, nonlinear closed-loop feedback system. Using Koopman operator theory, these nonlinear dynamics are approximated with a linear operator, enabling interpretability. Spectral analysis is then used to quantify the worst-case impact of domain shifts on the generalization error. Building on this analysis, a domain generalization method is proposed that reduces the OOD generalization error and improves the robustness to distribution shifts. Finally, the proposed analysis and domain generalization approach are validated on practical temporal pattern-learning tasks.
Paper Structure (36 sections, 2 theorems, 92 equations, 9 figures)

This paper contains 36 sections, 2 theorems, 92 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Motivation and Prior Work
  3. Contributions
  4. Background Concepts and Formal Definitions
  5. Definition of Domain Shift in LSTM
  6. Definition of Generalization Error
  7. Koopman Operator
  8. Dynamic Mode Decomposition with Control
  9. $\ell_2$ Norm, $Z$-transform, $\text{H}_\infty$ Norm
  10. Linear Inequality Matrix (LMI) Optimization
  11. Interpretable Model for LSTM Networks and Modeling Domain Shifts
  12. Modeling Dynamical Behavior of an LSTM Network
  13. Modeling Domain Shift
  14. Using Koopman Operator and DMDc to Approximate an LSTM Network Model
  15. Generalization Analysis of LSTM Networks
  16. ...and 21 more sections

Key Result

Theorem 1

For a given trained LSTM, suppose that the domain shift $\{\mathbf d_t\}_{t=0}^T$ satisfies $\|\mathbf d\|_{\ell_2}\le \alpha$. Let the linear map from $\mathbf d_z$ to $\Delta\mathbf h_z^N$ is the $\mathbf T^{dh}_z$, then

Figures (9)

  • Figure 1: LSTM model
  • Figure 2: Deep LSTM model
  • Figure 3: Impact of domain shift on dynamical behavior of an LSTM model
  • Figure 4: Domain shift rejection architecture of a trained LSTM
  • Figure 5: Generalization analysis under different domain-shift scenarios: (a) empirical bound versus theoretical upper bound; (b) RMSE increase.
  • ...and 4 more figures

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • proof
  • proof
  • proof
  • proof