# Continuous Subspace Optimization for Continual Learning

arXiv: https://arxiv.org/abs/2505.11816 · HTML: https://mlsanity.com/paper/2505.11816/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 54 references that are in the map.
- First posted on arXiv 17 May 2025.

## Builds on (most cited first, 30 of 39)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Emerging Properties in Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.14294/) (2021)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [The Power of Scale for Parameter-Efficient Prompt Tuning](https://mlsanity.com/paper/2104.08691/) (2021)
- [Learning without Forgetting](https://mlsanity.com/paper/1606.09282/) (2016)
- [iCaRL: Incremental Classifier and Representation Learning](https://mlsanity.com/paper/1611.07725/) (2016)
- [Gradient Episodic Memory for Continual Learning](https://mlsanity.com/paper/1706.08840/) (2017)
- [The Many Faces of Robustness: A Critical Analysis of Out-of-Distribution Generalization](https://mlsanity.com/paper/2006.16241/) (2020)
- [Moment Matching for Multi-Source Domain Adaptation](https://mlsanity.com/paper/1812.01754/) (2018)
- [Efficient Lifelong Learning with A-GEM](https://mlsanity.com/paper/1812.00420/) (2018)
- [Learning to Prompt for Continual Learning](https://mlsanity.com/paper/2112.08654/) (2021)
- [A Comprehensive Survey of Continual Learning: Theory, Method and Application](https://mlsanity.com/paper/2302.00487/) (2023)
- [Lifelong Learning with Dynamically Expandable Networks](https://mlsanity.com/paper/1708.01547/) (2017)
- [DualPrompt: Complementary Prompting for Rehearsal-free Continual Learning](https://mlsanity.com/paper/2204.04799/) (2022)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [CODA-Prompt: COntinual Decomposed Attention-based Prompting for Rehearsal-Free Continual Learning](https://mlsanity.com/paper/2211.13218/) (2022)
- [Class-incremental learning: survey and performance evaluation on image classification](https://mlsanity.com/paper/2010.15277/) (2020)
- [Orthogonal Gradient Descent for Continual Learning](https://mlsanity.com/paper/1910.07104/) (2019)
- [Gradient Projection Memory for Continual Learning](https://mlsanity.com/paper/2103.09762/) (2021)
- [Learn to Grow: A Continual Structure Learning Framework for Overcoming Catastrophic Forgetting](https://mlsanity.com/paper/1904.00310/) (2019)
- [Mnemonics Training: Multi-Class Incremental Learning without Forgetting](https://mlsanity.com/paper/2002.10211/) (2020)
- [Class-Incremental Learning: A Survey](https://mlsanity.com/paper/2302.03648/) (2023)
- [InfLoRA: Interference-Free Low-Rank Adaptation for Continual Learning](https://mlsanity.com/paper/2404.00228/) (2024)
- [UniPELT: A Unified Framework for Parameter-Efficient Language Model Tuning](https://mlsanity.com/paper/2110.07577/) (2021)
- [Continual Learning with Pre-Trained Models: A Survey](https://mlsanity.com/paper/2401.16386/) (2024)
- [Training Networks in Null Space of Feature Covariance for Continual Learning](https://mlsanity.com/paper/2103.07113/) (2021)
- [A Unified Continual Learning Framework with General Parameter-Efficient Tuning](https://mlsanity.com/paper/2303.10070/) (2023)

## Built on by (most cited first, 9 of 9)

- [Task-Driven Subspace Decomposition for Knowledge Sharing and Isolation in LoRA-based Continual Learning](https://mlsanity.com/paper/2603.00191/) (2026)
- [FOREVER: Forgetting Curve-Inspired Memory Replay for Language Model Continual Learning](https://mlsanity.com/paper/2601.03938/) (2026)
- [Bilinear Optimization Divergence: Diagnosing Factor-Constrained LoRA Continual Learning](https://mlsanity.com/paper/2609.23594/) (2026)
- [Hidden Failure Modes of Gradient Modification under Adam in Continual Learning, and Adaptive Decoupled Moment Routing as a Repair](https://mlsanity.com/paper/2604.22407/) (2026)
- [Can Tabular Foundation Models Guide Exploration in Robot Policy Learning?](https://mlsanity.com/paper/2604.27667/) (2026)
- [KeepLoRA++: Continual Learning with Layer-Scaled Residual Gradient Adaptation](https://mlsanity.com/paper/2606.16256/) (2026)
- [DeCoFlow: Structural Decomposition of Normalizing Flows for Continual Anomaly Detection](https://mlsanity.com/paper/2606.26687/) (2026)
- [Breaking the Model Forgetting Cycle in Long-Incremental 3D Object Detection](https://mlsanity.com/paper/2607.14560/) (2026)
- [MuLoRA: Spectrally Balanced Low-Rank Adaptation for Continual Learning](https://mlsanity.com/paper/2610.02283/) (2026)

## Abstract (arXiv)

Continual learning aims to learn multiple tasks sequentially while preserving prior knowledge, but faces the challenge of catastrophic forgetting when adapting to new tasks. Recently, approaches leveraging pre-trained models have gained increasing popularity in mitigating this issue, due to the strong generalization ability of foundation models. To adjust pre-trained models for new tasks, existing methods usually employ low-rank adaptation, which restricts parameter updates to a fixed low-rank subspace. However, constraining the optimization space inherently compromises the model's learning capacity, resulting in inferior performance. To address this limitation, we propose Continuous Subspace Optimization for Continual Learning (CoSO) to fine-tune the model in a series of subspaces rather than a single one. These sequential subspaces are dynamically determined through the singular value decomposition of the gradients. CoSO updates the model by projecting gradients onto these subspaces, ensuring memory-efficient optimization. To mitigate forgetting, the optimization subspace of each task is constrained to be orthogonal to the historical task subspace. During task learning, CoSO maintains a task-specific component that captures the critical update directions for the current task. Upon completing a task, this component is used to update the historical task subspace, laying the groundwork for subsequent learning. Extensive experiments on multiple datasets demonstrate that CoSO significantly outperforms state-of-the-art methods, especially in challenging scenarios with long task sequences.