# CRMA: A Spectrally-Bounded Backbone for Modular Continual Fine-Tuning of LLMs

arXiv: https://arxiv.org/abs/2606.00382 · HTML: https://mlsanity.com/paper/2606.00382/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 38 of its 49 references that are in the map.
- First posted on arXiv 29 May 2026.

## Builds on (most cited first, 30 of 38)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [Prefix-Tuning: Optimizing Continuous Prompts for Generation](https://mlsanity.com/paper/2101.00190/) (2021)
- [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)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Gradient Episodic Memory for Continual Learning](https://mlsanity.com/paper/1706.08840/) (2017)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Progressive Neural Networks](https://mlsanity.com/paper/1606.04671/) (2016)
- [PackNet: Adding Multiple Tasks to a Single Network by Iterative Pruning](https://mlsanity.com/paper/1711.05769/) (2017)
- [A Comprehensive Survey of Continual Learning: Theory, Method and Application](https://mlsanity.com/paper/2302.00487/) (2023)
- [TIES-Merging: Resolving Interference When Merging Models](https://mlsanity.com/paper/2306.01708/) (2023)
- [AdapterFusion: Non-Destructive Task Composition for Transfer Learning](https://mlsanity.com/paper/2005.00247/) (2020)
- [Language Models are Super Mario: Absorbing Abilities from Homologous Models as a Free Lunch](https://mlsanity.com/paper/2311.03099/) (2023)
- [Overcoming catastrophic forgetting with hard attention to the task](https://mlsanity.com/paper/1801.01423/) (2018)
- [Three scenarios for continual learning](https://mlsanity.com/paper/1904.07734/) (2019)
- [LoRA+: Efficient Low Rank Adaptation of Large Models](https://mlsanity.com/paper/2402.12354/) (2024)
- [LoraHub: Efficient Cross-Task Generalization via Dynamic LoRA Composition](https://mlsanity.com/paper/2307.13269/) (2023)
- [Gradient Projection Memory for Continual Learning](https://mlsanity.com/paper/2103.09762/) (2021)
- [LoRA Learns Less and Forgets Less](https://mlsanity.com/paper/2405.09673/) (2024)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [PiSSA: Principal Singular Values and Singular Vectors Adaptation of Large Language Models](https://mlsanity.com/paper/2404.02948/) (2024)
- [AdaMerging: Adaptive Model Merging for Multi-Task Learning](https://mlsanity.com/paper/2310.02575/) (2023)
- [Sparse Sinkhorn Attention](https://mlsanity.com/paper/2002.11296/) (2020)
- [InfLoRA: Interference-Free Low-Rank Adaptation for Continual Learning](https://mlsanity.com/paper/2404.00228/) (2024)
- [Mixture of LoRA Experts](https://mlsanity.com/paper/2404.13628/) (2024)
- [Learning Latent Permutations with Gumbel-Sinkhorn Networks](https://mlsanity.com/paper/1802.08665/) (2018)
- [Training Networks in Null Space of Feature Covariance for Continual Learning](https://mlsanity.com/paper/2103.07113/) (2021)
- [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021)

## Abstract (arXiv)

Sequential fine-tuning of large language models forces a choice: let the shared substrate keep learning and accept catastrophic forgetting, or freeze it after task one and foreclose cross-task refinement. Per-task adapter methods (LoRAHub, AdapterFusion, PackNet, Progressive Networks) take the second path. We introduce CRMA (Constrained Residual Mixing Adapter), a residual adapter whose internal mixing matrix M is doubly-stochastic at every forward pass via Sinkhorn normalization, so by Birkhoff's theorem ||M||_2 <= 1 holds by construction -- a structural bound, not a penalty. CRMA's spectrally bounded backbone provides a continuously trained shared substrate that earlier modular methods could not, while preserving their forgetting guarantees. On Mistral-7B across 5 sequential domains and 3 seeds, modular per-task LoRA on a CRMA backbone reduces loss-relative drift from +42.96% +/- 5.5 (naive sequential fine-tuning) to -0.17% +/- 0.17, with disjoint per-seed ranges, and improves prior-task holdout loss by 1.99% +/- 0.54 over a matched frozen-substrate baseline. Three independent experimental setups (Mistral-7B 4-domain controlled ablation, TinyLlama 3-domain contamination-controlled replication, Mistral-7B cross-domain probes at 7B) all show positive backward transfer -- without replay buffers, without growing per-task memory, and without distillation. An inference-time ablation on Gemma-2-9B confirms CRMA mediates access to sequentially trained knowledge: 98/100 vs. 38/100 on the same weights and same questions with only CRMA injection toggled. 867 logged training steps verify ||M||_2 = 1.0 within float32 precision (max deviation 1.2 x 10^-7). The forgetting-prevention effect holds across 1.1B-9.2B parameters and four architecture families.