The Relevancy Metric: Understanding the Impact of Training Data

We thank the reviewer for their feedback and aim to address the concerns raised below.

1. We appreciate the reviewer’s thoughtful comment regarding the reliability of the relevancy metric, especially in extreme cases and its reliance on loss information. While the relevancy metric does rely on loss dynamics, its correlation-based formulation mitigates the instability caused by extreme loss values, as it focuses on the relative changes in loss across training rather than absolute values. However, we agree that specific cases, such as using discontinuous loss functions like 0-1 loss, could lead to instability or unintended behavior. In such cases, relevancy’s effectiveness may be diminished, as the metric depends on continuous and meaningful loss gradients to capture train-test relationships.

   To provide more clarity, the metric is most effective when the loss function is smooth and reflects generalization behaviors, such as cross-entropy or mean squared error, which are commonly used in modern deep learning tasks. In contrast, tasks with discontinuous loss functions or highly adversarial data distributions might require additional adaptations or complementary methods to ensure reliability.

   We have included a section in the appendix (Appendix A, highlighted in red) to outline these limitations and the conditions where relevancy performs best. Future work could explore extending the metric to account for these edge cases, enhancing its robustness for diverse datasets and loss functions.

2. We tested and showcased the utility of the relevancy metric across model architectures (in Section 62.) and across datasets (in Section 6.1).

3. We used a separate validation set to select the coreset as explained in Section 5.

4. Yes, a training sample with a negative relevancy score for a testing point suggests that the training sample negatively impacts the test sample's prediction. This occurs when the features learned from the training sample during training conflict with or diverge from the features of the testing sample during inference.

   For example, a training sample of a white tiger might negatively impact a test sample of a regular (orange) tiger. During training, the white tiger sample could influence the model by associating the "tiger" class with white and black coloration (simplified for explanation). During inference, this learned association may conflict with the features of the regular tiger test sample, which lacks white and black coloration, reducing the model's confidence or accuracy for the test sample.

We thank the reviewer for their feedback and aim to address the concerns raised below.

1. We thank the reviewer for pointing out relevant prior work on training data attributions (TDA), particularly [1] and [2]. These studies, including the datamodeling approach by Ilyas et al. [1] and the linear datamodeling score (LDS) proposed in Park et al. [2], are indeed closely related to our goal of understanding the influence of training data on model behavior. We acknowledge that these works were not explicitly discussed in our paper and appreciate the opportunity to address this oversight.

   Both datamodels and LDS use counterfactual approaches to evaluate training data influence, offering valuable insights into training-test relationships. However, these methods often involve additional computational complexity and require specific assumptions (e.g., model linearity or surrogate models). In contrast, our relevancy metric provides a lightweight and scalable alternative by directly leveraging the learning dynamics during training, making it broadly applicable without requiring explicit counterfactual constructions.

   We have incorporated such a discussion on datamodels and LDS in the Section 2 (highlighted in red in the revised manuscript) to highlight these connections and position relevancy as a complementary approach to existing TDA methods.

2. We appreciate the reviewer’s observation regarding the distinction between correlation and causation in relevancy. To clarify, relevancy is designed to measure the relationship between training samples and test examples, capturing correlations in learning dynamics rather than strict causal inference. While this does not establish direct causality, it provides actionable insights into how training samples influence generalization.

   In the scenario with tr1 and tr2, relevancy would capture the relationship between tr1 and te indirectly via tr2, reflecting their shared optimization dynamics. This aligns with the metric’s goal of identifying samples that play a significant role in shaping the model’s predictions, even indirectly.

   Relevancy is a practical and computationally efficient tool for understanding train-test relationships and complements more complex methods focused on causality. We acknowledge the distinction and see future work as an opportunity to disentangle these effects further.

3. Influence scores measure the change in prediction probability when a sample is included in the training dataset to when it is not. Since many samples assist in making the model successfully predict on an inference sample, exclusion of a sample may not have a huge effect. This bimodal distribution of influence scores where most values are 0 or close to 1 has been extensively studied in literature. It is hard to utilize this information to generate coresets from this information as most values are 0. However, our metric assigns a non-zero value to most train-test pairs and is a more continuous distribution. Hence, we can leverage this more successfully in applications such as generating coresets. This is why we say the relevancy metric provides more of an insight (meaningful relationship) than influence scores.

4. The reviewer is right that due to the stochastic nature of training, each run may provide slightly different loss trajectories. We take this into account and show a detailed study on how the randomness of the training (seed) or the model architectures do not affect the relevancy metric in Section 6.2.

4. Experimental Validation

• a) The relevancy metric theoretically assigns near-zero scores to pure noise inputs due to their lack of correlation with meaningful training data. Our existing results demonstrate robustness in identifying mislabeled and atypical samples, supporting its reliability in real-world settings. Synthetic experiments to further validate this behavior are explained in part c of this response.

• b) Relevancy outperforms other coreset selection methods by directly quantifying the evolving impact of training samples on test generalization through loss dynamics. In contrast, other methods often rely on static heuristics, which may not effectively capture the dynamic train-test relationship.

• c) High-Noise Regime: We appreciate the reviewer’s suggestion to evaluate the relevancy score with synthetic experiments. We introduced synthetically modified noisy samples (10% corruption) into the training data and analyzed their $Rel_{Mem}$ scores to address this. We have included the full experimental details and analysis in appendix D (highlighted in red) to demonstrate the robustness and reliability of the relevancy metric. The histogram (Figure 13 in the revised manuscript) shows these noisy samples exhibit significantly higher $Rel_{Mem}$ scores (marked as red crosses), making them distinguishable from regular samples. This validates the metric’s ability to identify noisy and atypical samples effectively. Furthermore, the design of the relevancy metric ensures that it captures meaningful correlations between training and test samples, with noise inputs expected to yield higher scores due to their inherent tendency to be memorized during training. This behavior is consistent with prior results in our paper, where the metric successfully identifies mislabeled and memorized samples in real-world datasets. While we agree that further exploration of edge cases could strengthen the evaluation, we believe this experiment, combined with existing results, sufficiently demonstrates the robustness and practical utility of the metric.

• d) Relevancy has shown consistency across different model backbones, as demonstrated in Section 6.2. The metric is task and loss-dependent and differs for each task. While influence functions could provide a useful comparison, their computational expense (due to multiple retrainings) limits their practicality in such evaluations.

References:

[1] Mariya Toneva, Alessandro Sordoni, Remi Tachet des Combes, Adam Trischler, Yoshua Bengio, and Geoffrey J Gordon. An empirical study of example forgetting during deep neural network learning. In International Conference on Learning Representations, 2018.

[2] Karttikeya Mangalam and Vinay Uday Prabhu. Do deep neural networks learn shallow learnable examples first? In ICML 2019 Workshop on Identifying and Understanding Deep Learning Phenomena, 2019

[3] Ziheng Jiang, Chiyuan Zhang, Kunal Talwar, and Michael C Mozer. Characterizing structural regularities of labeled data in overparameterized models. In Marina Meila and Tong Zhang (eds.), Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pp. 5034–5044. PMLR, 18–24 Jul 2021.

[4] Isha Garg, Deepak Ravikumar, and Kaushik Roy. Memorization through the lens of curvature of loss function around samples. In Proceedings of the 41st International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, pp. 15083–15101. PMLR, 21–27 Jul 2024

We thank the reviewer for their detailed feedback and constructive criticism. Below, we address the concerns and clarify the proposed relevancy metric.

The relevancy metric evaluates the influence of a training sample on a sample of interest (e.g., test or other training samples) for a specific task. It quantifies this impact dynamically by correlating the loss changes of the training sample with those of the sample of interest across training iterations. Unlike influence functions, which measure the static effects of perturbing or removing samples, the relevancy metric captures the evolving contribution of training samples to both memorization and generalization during learning. This dynamic perspective provides actionable insights into train-sample interactions, enabling tasks such as coreset selection, noisy data detection, and understanding model behavior.

1. Impact of Individual Training Samples and Formal Definition: The relevancy metric formally quantifies the dynamic impact of training samples by analyzing the correlation between their loss changes and those of the sample of interest across training iterations. This approach reveals how training samples influence the model during training, providing richer insights than static methods like influence functions. For example, samples with higher relevancy positively contribute to generalization, while low-relevancy samples indicate noise or atypical patterns. This is further demonstrated in tasks such as coreset selection and mislabeled sample detection.

2. Concerns Regarding Correlation-Based Definitions

1. Observing learning dynamics is crucial for understanding sample-level interactions, as supported by prior works [1-4]. Relevancy scores remain consistent across models with identical tasks, irrespective of differences in learning hyperparameters, architectures, or initialization seeds. We demonstrate this in Section 6.2.

2. The metric is task-specific and reflects loss dynamics within the training trajectory for a given task. Extending the model to a new task (e.g., after epoch T+1) requires recomputation, similar to other metrics in this domain.

3. While relevancy considers changes in loss values during training, it inherently captures the relationship between losses of training and test samples rather than being directly influenced by the magnitude of individual losses. This ensures that outliers with disproportionately large loss values do not dominate the metric. Additionally, the relevancy's averaging and correlation-based nature mitigates such outliers' impact.

4. Since relevancy is tied to task-specific loss functions, its computation is naturally dependent on the chosen loss. Differences in loss transformations (e.g., cross-entropy vs. perplexity) could affect order preservation. Future work could explore principled loss normalization techniques to address this.

3. Limitations of the Metric: We acknowledge the reviewer’s concern regarding the memory-intensive nature of storing multiple checkpoints. We will include this limitation in the paper. However, relevancy can be computed on smaller models and reused for tasks like coreset generation in larger models, making it computationally efficient compared to retraining-based metrics. As shown in Table 1, relevancy significantly reduces computational overhead compared to influence functions.

We thank the reviewer for their thoughtful feedback and positive assessment of our paper. We address each point raised in detail below:

1. Checkpoint Requirements and Impact of Trajectory Length (T): We acknowledge the concern regarding the practicality of frequent checkpoints and address this limitation in Appendix C, where we provide an analysis on selecting fewer (down sampling) checkpoints. Specifically, we demonstrate that the relevancy score remains consistent when we reduce the frequency of checkpoints (e.g., by skipping epochs). However, we also observe that as the down sampling rate increases, the relevancy score tends to approach 1, reducing the metric's sensitivity to sample-specific learning dynamics. To balance computational efficiency with accuracy, we suggest using every second or third checkpoint, which retains sufficient granularity for differentiating learning patterns while reducing the computational burden. The trajectory length T similarly impacts the metric, with shorter trajectories providing less detailed information but offering a feasible option for lower-resource environments.

2. Generalization Across Tasks Beyond Classification: The current work focuses on supervised classification tasks to establish the foundational utility of the relevancy metric. While we agree that exploring its application in regression or sequential tasks could significantly enhance its impact, extending the metric to such tasks involves additional considerations that are beyond the current paper’s scope. We are excited by the potential to adapt and test the relevancy metric in these domains as part of future research. This extension would require validating the metric's adaptability to different learning dynamics characteristic of non-classification tasks, which we believe will further strengthen the generalizability and utility of our approach.