LoRanPAC: Low-rank Random Features and Pre-trained Models for Bridging Theory and Practice in Continual Learning

TL;DR

LoRanPAC is a stable, theoretically grounded continual learning method that leverages low-rank approximations of pre-trained features to achieve high performance across multiple tasks.

Contribution

It introduces a novel low-rank feature truncation approach with theoretical guarantees, bridging the gap between empirical performance and theoretical understanding in continual learning.

Findings

Outperforms state-of-the-art CL methods on multiple datasets

Handles hundreds of tasks with stability and high accuracy

Provides theoretical guarantees for training and generalization errors

Abstract

The goal of continual learning (CL) is to train a model that can solve multiple tasks presented sequentially. Recent CL approaches have achieved strong performance by leveraging large pre-trained models that generalize well to downstream tasks. However, such methods lack theoretical guarantees, making them prone to unexpected failures. Conversely, principled CL approaches often fail to achieve competitive performance. In this work, we aim to bridge this gap between theory and practice by designing a simple CL method that is theoretically sound and highly performant. Specifically, we lift pre-trained features into a higher dimensional space and formulate an over-parametrized minimum-norm least-squares problem. We find that the lifted features are highly ill-conditioned, potentially leading to large training errors (numerical instability) and increased generalization errors. We address these challenges by continually truncating the singular value decomposition of the lifted features. Our approach, termed LoRanPAC, is stable with respect to the choice of hyperparameters, can handle hundreds of tasks, and outperforms state-of-the-art CL methods on multiple datasets. Importantly, our method satisfies a recurrence relation throughout its continual learning process, which allows us to prove it maintains small training and test errors by appropriately truncating a fraction of SVD factors. This results in a stable continual learning method with strong empirical performance and theoretical guarantees. Code available: https://github.com/liangzu/loranpac.