Real-world data often exhibit imbalanced label distributions. Existing studies on data imbalance focus on single-domain settings, i.e., samples are from the same data distribution. However, natural data can originate from distinct domains, where a minority class in one domain could have abundant instances from other domains. We formalize the task of Multi-Domain Long-Tailed Recognition (MDLT), which learns from multi-domain imbalanced data, addresses label imbalance, domain shift, and divergent label distributions across domains, and generalizes to all domain-class pairs. We first develop the domain-class transferability graph, and show that such transferability governs the success of learning in MDLT. We then propose BoDA, a theoretically grounded learning strategy that tracks the upper bound of transferability statistics, and ensures balanced alignment and calibration across imbalanced domain-class distributions. We curate five MDLT benchmarks based on widely-used multi-domain datasets, and compare BoDA to twenty algorithms that span different learning strategies. Extensive and rigorous experiments verify the superior performance of BoDA. Further, as a byproduct, BoDA establishes new state-of-the-art on Domain Generalization benchmarks, highlighting the importance of addressing data imbalance across domains, which can be crucial for improving generalization to unseen domains. Code and data are available at: https://github.com/YyzHarry/multi-domain-imbalance.
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