Continuous Temporal Domain Generalization

TL;DR

This work formalizes the concept of Continuous Temporal Domain Generalization (CTDG), where domain data are derived from continuous times and are collected at arbitrary times, and proposes a Koopman operator-driven continuous temporal domain generalization (Koodos) framework.

Abstract

Temporal Domain Generalization (TDG) addresses the challenge of training predictive models under temporally varying data distributions. Traditional TDG approaches typically focus on domain data collected at fixed, discrete time intervals, which limits their capability to capture the inherent dynamics within continuous-evolving and irregularly-observed temporal domains. To overcome this, this work formalizes the concept of Continuous Temporal Domain Generalization (CTDG), where domain data are derived from continuous times and are collected at arbitrary times. CTDG tackles critical challenges including: 1) Characterizing the continuous dynamics of both data and models, 2) Learning complex high-dimensional nonlinear dynamics, and 3) Optimizing and controlling the generalization across continuous temporal domains. To address them, we propose a Koopman operator-driven continuous temporal domain generalization (Koodos) framework. We formulate the problem within a continuous dynamic system and leverage the Koopman theory to learn the underlying dynamics; the framework is further enhanced with a comprehensive optimization strategy equipped with analysis and control driven by prior knowledge of the dynamics patterns. Extensive experiments demonstrate the effectiveness and efficiency of our approach. The code can be found at: https://github.com/Zekun-Cai/Koodos.

Figures (9)

• Figure 1: An example of continuous temporal domain generalization. Consider training classification models for public opinion prediction via tweets, where the training domains are only available at specific political events (e.g., presidential debates), we wish to generalize the model to any future based on the underlying data distribution drift within the time-irregularly distributed training domains.
• Figure 2: Macro-flows and micro-constraints in the proposed model framework.
• Figure 3: Visualization of decision boundary of the 2-Moons dataset (purple and yellow show data regions, red line shows the decision boundary). Top to bottom compares two baseline methods with ours; left to right shows partial test domains (all test domains are marked with red points on the timeline). All models are learned using data before the last train domain.
• Figure 4: Interpolated and extrapolated predictive model trajectories. Left: Koodos captures the essence of generalization through the harmonious synchronization of model and data dynamics; Middle: DRAIN, as a probabilistic model, fails to capture continuous dynamics, which is presented as jumps from one random state to another. Right: DeepODE demonstrates a certain degree of continuity, but the dynamics are incorrect.
• Figure 5: Eigenvalue distribution of the Koopman operator. Left: $\mathcal{K}$ as learnable; Right: $\mathcal{K}=B-B^T$ with $B$ as learnable.

Theorems & Definitions (5)

• Theorem 1
• proof
• Theorem 2: Superiority of continuous-time methods over discrete-time methods (informal)
• Theorem 3: Formal version of Theorem \ref{['thm: continuous better than discrete']}
• proof