Learning the Context of Errors: Black-Box Online Adaptation of Time Series Foundation Models

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://anonymous.4open.science/r/ORCA1/.