In this work, we introduce a novel paradigm for generalized In-Context Learning (ICL), termed Indirect In-Context Learning. In Indirect ICL, we explore demonstration selection strategies tailored for two distinct real-world scenarios: Mixture of Tasks and Noisy ICL. We systematically evaluate the effectiveness of Influence Functions (IFs) as a selection tool for these settings, highlighting the potential of IFs to better capture the informativeness of examples within the demonstration pool. For the Mixture of Tasks setting, demonstrations are drawn from 28 diverse tasks, including MMLU, BigBench, StrategyQA, and CommonsenseQA. We demonstrate that combining BertScore-Recall (BSR) with an IF surrogate model can further improve performance, leading to average absolute accuracy gains of 0.37\% and 1.45\% for 3-shot and 5-shot setups when compared to traditional ICL metrics. In the Noisy ICL setting, we examine scenarios where demonstrations might be mislabeled or have adversarial noise. Our experiments show that reweighting traditional ICL selectors (BSR and Cosine Similarity) with IF-based selectors boosts accuracy by an average of 2.90\% for Cosine Similarity and 2.94\% for BSR on noisy GLUE benchmarks. For the adversarial sub-setting, we show the utility of using IFs for task-agnostic demonstration selection for backdoor attack mitigation. Showing a 32.89\% reduction in Attack Success Rate compared to task-aware methods. In sum, we propose a robust framework for demonstration selection that generalizes beyond traditional ICL, offering valuable insights into the role of IFs for Indirect ICL.

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