The rapid evolution of Time Series Foundation Models (TSFMs) has advanced zero-shot forecasting across diverse domains. Inspired by the current form of Large Language Models, future TSFMs may be offered as commercialized, closed-source API services. However, many existing online adaptation methods still rely on white-box access for parameter fine-tuning or gradient backpropagation. This paradigm mismatch raises a question: In black-box online adaptation for TSFMs, what should we learn? We answer this with an insight: the predictive errors of the base model are conditioned on both the input and output of the base model (i.e., the context of errors). To validate this insight, we propose ORCA (Online Residual Contextual Adaptation). We conduct extensive experiments across 5 state-of-the-art TSFMs and 8 datasets to demonstrate the effectiveness of our approach. Furthermore, through ablation studies, we quantitatively analyze the impact of different adapter learning hypotheses on the final adaptation performance in black-box online adaptation. Code available at https://github.com/Fifthky/ORCA.

翻译：时间序列基础模型（TSFMs）的快速发展推动了跨领域零样本预测的进步。受当前大型语言模型形态的启发，未来的TSFMs可能作为商业化、闭源的API服务提供。然而，许多现有的在线适应方法仍依赖白箱访问进行参数微调或梯度反向传播。这种范式错位引发了一个问题：在TSFMs的黑箱在线适应中，我们应该学习什么？我们用一个洞见来回答：基础模型的预测误差同时受其输入和输出（即误差语境）条件约束。为验证这一洞见，我们提出ORCA（在线残差语境适应）。我们在5个最先进TSFMs和8个数据集上进行广泛实验，证明了我们方法的有效性。此外，通过消融研究，我们定量分析了不同适配器学习假设对黑箱在线适应最终性能的影响。代码见https://github.com/Fifthky/ORCA。