Bilinear Optimization Divergence: Diagnosing Factor-Constrained LoRA Continual Learning

Low-rank adaptation (LoRA) makes finetuning large language models cheaper by adding to each weight matrix a trainable low-rank update parameterized as the product of two matrices. These matrices are usually trained with Adam, which treats them as a single flat vector of parameters and ignores both the matrix and product structure of LoRA. Applying a matrix-aware optimizer such as Muon to each factor does not consistently improve over Adam, and neither do the product-aware Muon variants proposed in concurrent works. To realize consistent gains, we introduce PoLoRA, a Preconditioned Orthogonalized LoRA optimizer built from three ingredients: a product-aware spectral update direction, curvature preconditioning derived from controlling the per-sample loss change, and a magnitude rule that controls the sizes of both the factor and merged updates. We evaluate PoLoRA on instruction-tuning datasets for code and math across models from 1B to 8B parameters, and find that it reaches the final held-out loss achieved by tuned Adam in 1.2-1.7 times fewer steps, while adding at most 3% per-step overhead. Compared to Adam, PoLoRA is also less sensitive to the learning rate, and its optimal learning rate is stable across ranks.

Libraries of Low-Rank Adaptation (LoRA) adapters are becoming a practical by-product of parameter-efficient adaptation. Once such adapters accumulate, a natural question is no longer how to train one adapter for one task, but how to reuse an open pool of adapters for a new task given only a small support set. Prior work has shown that LoRA modules can be composed at the task level and dynamically selected at the instance level. However, open-pool LoRA reuse is not automatic: retrieving relevant adapters does not guarantee that their parameter updates are compatible, and composing adapters does not guarantee reliable outputs. We introduce the Sparse-Composition Agreement Layer (SCALE), a post-retrieval audit and composition framework for open-pool LoRA reuse. SCALE contains a deployable 1.0* merge path, Layer-Adaptive Sparse Residual Composition (LASRC), and a higher-cost reliability-analysis layer for multi-view disagreement. LASRC addresses merge interference by preserving a linear anchor while residualizing block-wise adapter update directions. The reliability layer treats disagreement among sparse composition views as an observable uncertainty signal and compares agreement, support-loss proxy selection, and oracle headroom under explicit path cost. In matched FLAN-T5-Large, BIG-Bench Hard (BBH), and 97-LoRA experiments, LASRC gives a directional single-view gain under fixed retrieval, while SCALE-support is reported as a query-label-free 3.0* reliability-analysis variant rather than as a calibrated or throughput-equivalent selector. Protocol-distinct BBH-8 validation shows the same qualitative trend on three decoder-only backbones. Detailed scores, paired audits, and path-cost records are reported in the experimental section.

While finetuning effectively adapts foundation models to specialized downstream tasks, it can degrade nontarget capabilities acquired during pretraining. Existing forgetting aware methods typically seek safer updates through specialized initialization or fixed constraints, but do not regulate the adaptation preservation trade-off during training. We propose Foundation Preserving LoRA (FoLoRA), a forgetting aware optimization framework. Guided by a first order preservation condition, FoLoRA defines a forgetting penalty over pretraining-proxy activations and a task utility over downstream task activations. It then scores update directions by task utility per unit forgetting penalty via a generalized Rayleigh quotient. The resulting spectral coordinate system enables direction wise gated Adam updates, attenuating low utility to penalty directions during training. To estimate the forgetting penalty, FoLoRA constructs pretraining proxy calibration data by sampling from the pretrained model rather than relying on a single proxy dataset. Experiments on math, code, and instruction following adaptation show that FoLoRA achieves the strongest preservation adaptation balance over baselines, improving target task performance with best aggregate preservation of non target capabilities.