Hidden Failure Modes of Gradient Modification under Adam in Continual Learning, and Adaptive Decoupled Moment Routing as a Repair

Continual learning with Low-Rank Adapters (LoRA) typically mitigates forgetting by penalizing the overlap between a new update and the accumulated past weights, which discourages certain update directions without controlling how an update distributes its energy over the ones that remain. We ask whether that restriction has to be task-aware, or whether a generic one supplied by the optimizer is enough. We train a plain incremental LoRA (IncLoRA) with Muon, which orthogonalizes each update, and compare it against O-LoRA and ELLA over five seeds and three task orders on the Standard CL Benchmark and three seeds on TRACE. IncLoRA+Muon reaches the accuracy band of the dedicated methods on Standard CL and improves on every AdamW configuration on TRACE. One update-constraining mechanism is enough, whether it comes from the loss or from the optimizer; on Standard CL a second one does not help, and for the most restrictive method it costs 8.4 points of accuracy and the plasticity to fit each task. What separates the two optimizers is not the size of the update, which under Muon is 0.91 to 2.06 times that under AdamW, but how it is distributed. AdamW confines it to between 1.4 and 1.8 effective singular directions, Muon spreads it over 7.0, and the two do not overlap in any tracked run. Part of the advantage usually attributed to dedicated CL methods may therefore be explained by the geometry of the optimizer's updates.

LoRA is widely adopted for continual fine-tuning of Large Language Models due to its parameter efficiency, modularity across tasks, and compatibility with replay strategies. However, LoRA-based continual learning remains vulnerable to catastrophic forgetting, whose severity depends on how successive task gradients interact: when consecutive task gradients conflict, standard adapter initializations channel updates into subspaces that overwrite previously learned directions. We propose SLICE, a gradient-surgery-based initialization for LoRA adapters in continual learning. SLICE accumulates gradients from both the current task and a replay buffer of prior tasks, reconciles them through a projection operator, and decomposes the result via truncated SVD to initialize the adapter weights. We evaluate SLICE on the TRACE benchmark and sequences of Super-NI tasks, including a set of adversarial Super-NI sequences that we construct by mining task pairs with maximally opposing gradients. Compared to vanilla LoRA, LoRA-GA, and LoRAM, SLICE consistently achieves a better stability-plasticity trade-off, improving Average Performance, Final Performance and Forgetting metrics while preserving General Performance and In Context Performance across both standard and adversarial continual learning sequences.

Continual learning agents with finite capacity must balance acquiring new knowledge with retaining the old. This requires controlled forgetting of knowledge that is no longer needed, freeing up capacity to learn. Weight decay, viewed as a mechanism for forgetting, can serve this role by gradually discarding information stored in the weights. However, a fixed scalar weight decay drives this forgetting uniformly over time and uniformly across all parameters, even when some encode stable knowledge while others track rapidly changing targets. We introduce Forgetting through Adaptive Decay (FADE), which adapts per-parameter weight decay rates online via approximate meta-gradient descent. We derive FADE for the online linear setting and apply it to the final layer of neural networks. Our empirical analysis shows that FADE automatically discovers distinct decay rates for different parameters, complements step-size adaptation, and consistently improves over fixed weight decay across online tracking and streaming classification problems.