Scaling Laws for the Value of Individual Data Points in Machine Learning (2405.20456v1)

Abstract: Recent works have shown that machine learning models improve at a predictable rate with the total amount of training data, leading to scaling laws that describe the relationship between error and dataset size. These scaling laws can help design a model's training dataset, but they typically take an aggregate view of the data by only considering the dataset's size. We introduce a new perspective by investigating scaling behavior for the value of individual data points: we find that a data point's contribution to model's performance shrinks predictably with the size of the dataset in a log-linear manner. Interestingly, there is significant variability in the scaling exponent among different data points, indicating that certain points are more valuable in small datasets while others are relatively more useful as a part of large datasets. We provide learning theory to support our scaling law, and we observe empirically that it holds across diverse model classes. We further propose a maximum likelihood estimator and an amortized estimator to efficiently learn the individualized scaling behaviors from a small number of noisy observations per data point. Using our estimators, we provide insights into factors that influence the scaling behavior of different data points. Finally, we demonstrate applications of the individualized scaling laws to data valuation and data subset selection. Overall, our work represents a first step towards understanding and utilizing scaling properties for the value of individual data points.

Scaling Laws for the Value of Individual Data Points in Machine Learning

The paper "Scaling Laws for the Value of Individual Data Points in Machine Learning" by Covert, Ji, Hashimoto, and Zou from Stanford University introduces a novel perspective on understanding how individual data points affect machine learning model performance as the size of the dataset varies. This research builds on prior work that has established scaling laws describing the relationship between a model's error and the dataset size in aggregate terms. The authors extend this by proposing individualized scaling laws that detail how the marginal contribution of a single data point changes with the dataset size.

Key Contributions and Findings

1. Individualized Scaling Law: The authors posit that the marginal contribution of a data point to the performance of a model follows a predictable pattern that can be described by a simple parametric form: $\psi_k(z) \approx \frac{c(z)}{k^{\alpha(z)}}$, where $\psi_k(z)$ is the expected marginal contribution of data point $z$ for a dataset of size $k$. This relation indicates that the contribution of a data point diminishes in a log-linear manner as the size of the dataset increases.
2. Empirical Validation: Through extensive experimentation across diverse datasets (e.g., IMDB, MiniBooNE, CIFAR-10) and model types (logistic regression, MLPs, SVMs), the authors validate that the proposed scaling law holds true. The results consistently show a strong match between predicted and observed marginal contributions, achieving high $R^2$ scores, often greater than 0.9, across different settings.
3. Estimation Techniques: To efficiently estimate the individualized scaling laws, the paper introduces two estimation techniques:
   • Maximum Likelihood Estimator (MLE): This approach leverages the Gaussian distribution properties to fit the scaling parameters $c(z)$ and $\alpha(z)$ using a small number of marginal contribution samples per data point.
   • Amortized Estimation: By sharing parameter information across data points and using a neural network to predict the scaling parameters, this method enhances efficiency and reduces the number of required samples.
4. Theoretical Support: The paper provides mathematical analysis for linear regression and general M-estimators, demonstrating theoretically that the marginal contributions for simple models follow a scaling law. Factors influencing the parameters $c(z)$ and $\alpha(z)$ are also discussed, such as the relative noise level and leverage score of the data point.
5. Applications to Data Valuation and Subset Selection:
   • Data Valuation: The scaling laws can be applied to estimate data valuation scores, which are typically computed based on the marginal contributions across different dataset sizes. The proposed methods show improved or comparable accuracy with fewer samples compared to traditional Monte Carlo estimators.
   • Data Subset Selection: The variability in scaling exponents among data points suggests different points may be more valuable depending on the current dataset size. The paper demonstrates that using the scaling law to guide the selection of additional data points can result in better model performance improvements compared to other methods.

Implications and Future Research

The implications of this research are both practical and theoretical:

• Practical Implications: Understanding individualized scaling laws can guide the design and curation of training datasets. By identifying high-value data points and understanding how their contributions change with the dataset size, practitioners can make informed decisions about which data to prioritize or discard, particularly as datasets grow.
• Theoretical Implications: This work provides a deeper understanding of learning dynamics at a granular level. The variability in scaling exponents among data points reveals complexities in how data points interact with machine learning models, prompting further investigation.

Future Developments

Several future directions are foreseeable from this research:

• Extensions to Larger Scale Models: The insights and methods developed could be extended to more complex and large-scale models, such as deep neural networks used in natural language processing and vision.
• Dynamic Dataset Selection: Integrating individualized scaling laws into dynamic dataset selection algorithms could create more adaptive and efficient training regimes.
• Higher-Order Interactions: Studying the interactions between subsets of data points, rather than individual points, could provide more nuanced tools for dataset optimization.
• Practical Implementations: Developing user-friendly tools and libraries that implement these individualized scaling law estimations could democratize access to these advanced data curation techniques.

In conclusion, the paper makes significant strides in bridging the gap between aggregate data scaling laws and individual data point contributions. This approach opens new avenues for optimizing machine learning models by finely understanding and utilizing the contributions of specific training examples.