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

TL;DR

This paper introduces a novel framework combining Koopman operator theory and spectral analysis to interpret and improve the domain generalization of recurrent neural networks, especially for sequential data with temporal correlations.

Contribution

It proposes a new method to analyze interpretability and out-of-domain generalization of RNNs using Koopman theory and spectral analysis, addressing limitations of existing approaches.

Findings

The method effectively models RNN dynamics as linear systems for interpretability.

Spectral analysis quantifies the impact of domain shifts on generalization error.

The proposed approach improves robustness to distribution shifts in temporal tasks.

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.