Sections S3 to S5 contain the proofs for all results stated in the matching sections from the main article. References to contents in the supplementary material are prefixed with an "S", e.g., Lemma S3.20, eq. References to contents in the main article do not have any prefix, e.g., Theorem 1.3 and Section 1.4. We introduce notations in addition to those already defined in in the main article's Section 1.3. L always denotes the WW-hinge loss (Definition 1.4) and l always denotes the ordered partition loss (Definition 2.2).

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Clouds in satellite imagery pose a significant challenge for downstream applications. A major challenge in current cloud removal research is the absence of a comprehensive benchmark and a sufficiently large and diverse training dataset. To address this problem, we introduce the largest public dataset -- AllClear for cloud removal, featuring 23,742 globally distributed regions of interest (ROIs) with diverse land-use patterns, comprising 4 million images in total. Each ROI includes complete temporal captures from the year 2022, with (1) multi-spectral optical imagery from Sentinel-2 and Landsat 8/9, (2) synthetic aperture radar (SAR) imagery from Sentinel-1, and (3) auxiliary remote sensing products such as cloud masks and land cover maps. We validate the effectiveness of our dataset by benchmarking performance, demonstrating the scaling law -- the PSNR rises from 28.47 to 33.87 with 30 more data, and conducting ablation studies on the temporal length and the importance of individual modalities. This dataset aims to provide comprehensive coverage of the Earth's surface and promote better cloud removal results.

Monitoring, estimating, and explaining performance of deployed ML models is a growing area with significant economic and social impact. In this paper, we propose SJS, a new data distribution shift model to consider when both labels and features shift after model deployment. We show how SJS generalizes existing data shift models, and further propose SEES, a generic framework that efficiently explains and estimates an ML model's performance under SJS. This may serve as a monitoring tool to help ML practitioners recognize performance changes, discover potential fairness issues and take appropriate business decisions (e.g., switching to other models or debugging the existing ones). One limitation in general is adaption to continuously changing data streams.