Linear attention architectures enable efficient long-context modeling by compressing sequence history into a fixed-size memory state. However, dense key value updates can overload the state as the context grows, causing memory collisions and degraded retrieval. Existing methods rely on memory decay, rank reduction, or heuristic routing to reduce interference, but they largely preserve dense online updates and do not explicitly control the structural complexity of memory. We introduce BregNet, a linear sequence architecture that formulates associative memory updates as a regularized online regression problem. Based on Linearized Bregman Iteration (LBI), BregNet decouples memory into a continuous dual accumulator and a structurally sparse primal state. The dual state records incoming information, while the sparse primal state keeps only important associations and removes noise through soft shrinkage. Experiments show that BregNet matches strong linear baselines in language modeling even with high memory sparsity. In this way, BregNet prevents the memory state from being overloaded by dense and interfering updates.

The heavy-tailed distribution of output lengths in Large Language Model (LLM) serving poses major challenges for resource provisioning and cluster scheduling. Although output-length prediction can mitigate these issues, existing approaches have key drawbacks: external proxy models add substantial latency and often have limited fidelity, whereas internal state-based methods are efficient but rely on shallow probes of current model states. We identify a structural connection between speculative decoding (SD) and length prediction: latent representations produced by the draft decoder in advanced frameworks (e.g., EAGLE-3) encode signals that are predictive of generation length. Building on this insight, we introduce **OUTLETS** (**Out**put-**Le**ng**t**h Prediction from **S**peculative Decoding Backbones), which repurposes the speculative backbone as a trajectory-aware length predictor. When its draft representations are already computed for speculative decoding, OUTLETS adds only a lightweight regression head and achieves lower MAE than the evaluated methods. Under saturated disaggregated serving, OUTLETS predictions enable standard scheduling policies to prioritize shorter requests and distribute requests more evenly across decoding instances, reducing short-request P99 latency by **34.8%**.