Large-scale black-box models have become ubiquitous across numerous applications. Understanding the influence of individual training data sources on predictions made by these models is crucial for improving their trustworthiness. Current influence estimation techniques involve computing gradients for every training point or repeated training on different subsets. These approaches face obvious computational challenges when scaled up to large datasets and models. In this paper, we introduce and explore the Mirrored Influence Hypothesis, highlighting a reciprocal nature of influence between training and test data. Specifically, it suggests that evaluating the influence of training data on test predictions can be reformulated as an equivalent, yet inverse problem: assessing how the predictions for training samples would be altered if the model were trained on specific test samples. Through both empirical and theoretical validations, we demonstrate the wide applicability of our hypothesis. Inspired by this, we introduce a new method for estimating the influence of training data, which requires calculating gradients for specific test samples, paired with a forward pass for each training point. This approach can capitalize on the common asymmetry in scenarios where the number of test samples under concurrent examination is much smaller than the scale of the training dataset, thus gaining a significant improvement in efficiency compared to existing approaches. We demonstrate the applicability of our method across a range of scenarios, including data attribution in diffusion models, data leakage detection, analysis of memorization, mislabeled data detection, and tracing behavior in language models. Our code will be made available at https://github.com/ruoxi-jia-group/Forward-INF.

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