Generalizing across Temporal Domains with Koopman Operators

In the field of domain generalization, the task of constructing a predictive model capable of generalizing to a target domain without access to target data remains challenging. This problem becomes further complicated when considering evolving dynamics between domains. While various approaches have been proposed to address this issue, a comprehensive understanding of the underlying generalization theory is still lacking. In this study, we contribute novel theoretic results that aligning conditional distribution leads to the reduction of generalization bounds. Our analysis serves as a key motivation for solving the Temporal Domain Generalization (TDG) problem through the application of Koopman Neural Operators, resulting in Temporal Koopman Networks (TKNets). By employing Koopman Operators, we effectively address the time-evolving distributions encountered in TDG using the principles of Koopman theory, where measurement functions are sought to establish linear transition relations between evolving domains. Through empirical evaluations conducted on synthetic and real-world datasets, we validate the effectiveness of our proposed approach.

Hessian Aware Low-Rank Weight Perturbation for Continual Learning

Continual learning aims to learn a series of tasks sequentially without forgetting the knowledge acquired from the previous ones. In this work, we propose the Hessian Aware Low-Rank Perturbation algorithm for continual learning. By modeling the parameter transitions along the sequential tasks with the weight matrix transformation, we propose to apply the low-rank approximation on the task-adaptive parameters in each layer of the neural networks. Specifically, we theoretically demonstrate the quantitative relationship between the Hessian and the proposed low-rank approximation. The approximation ranks are then globally determined according to the marginal increment of the empirical loss estimated by the layer-specific gradient and low-rank approximation error. Furthermore, we control the model capacity by pruning less important parameters to diminish the parameter growth. We conduct extensive experiments on various benchmarks, including a dataset with large-scale tasks, and compare our method against some recent state-of-the-art methods to demonstrate the effectiveness and scalability of our proposed method. Empirical results show that our method performs better on different benchmarks, especially in achieving task order robustness and handling the forgetting issue. A demo code can be found at https://github.com/lijiaqi/HALRP.