Capturing the Temporal Dependence of Training Data Influence

Traditional data influence estimation methods, like influence function, assume that learning algorithms are permutation-invariant with respect to training data. However, modern training paradigms—especially for foundation models using stochastic algorithms and non-convergent, multi-stage curricula—are sensitive to data ordering, thus violating this assumption. This mismatch renders influence functions inadequate for answering some critical questions in current machine learning: How can we differentiate the influence of the same data contributing at different stages of training? More generally, how can we capture the dependence of data influence on the optimization trajectory during training? To address this gap, we formalize the concept of \emph{trajectory-specific leave-one-out (LOO) influence}, which quantifies the impact of removing a data point from a specific iteration during training, accounting for the exact sequence of data encountered and the model's optimization trajectory. However, exactly evaluating the trajectory-specific LOO presents a significant computational challenge. To address this, we propose \emph{data value embedding}, a novel technique enabling efficient approximation of trajectory-specific LOO. Specifically, we compute a training data embedding that encapsulates the cumulative interactions between data and the evolving model parameters. The LOO can then be efficiently approximated through a simple dot-product between the data value embedding and the gradient of the given test data. As data value embedding captures training data ordering, it offers valuable insights into model training dynamics. In particular, we uncover distinct phases of data influence, revealing that data points in the early and late stages of training exert a greater impact on the final model. These insights translate into actionable strategies for managing the computational overhead of data selection by strategically timing the selection process, potentially opening new avenues in data curation research.