Small Coresets via Determinantal Point Processes: Achieving Better Accuracy Guarantees than Independent Sampling for Machine Learning

Core Concepts

DPP-based coresets can provably outperform independently drawn coresets in machine learning tasks by achieving better accuracy guarantees with a smaller coreset size.

Abstract

• Bibliographic Information: Bardenet, R., Ghosh, S., Simon-Onfroy, H., & Tran, H.-S. (2024). Small coresets via negative dependence: DPPs, linear statistics, and concentration. arXiv preprint arXiv:2411.00611v1.
• Research Objective: This paper investigates the use of Determinantal Point Processes (DPPs) for constructing smaller and more efficient coresets in machine learning, aiming to theoretically prove their advantage over traditional independent sampling methods.
• Methodology: The authors leverage the inherent negative dependence property of DPPs to reduce the variance of linear statistics, which are directly related to coreset loss. They derive novel concentration inequalities for linear statistics of DPPs, applicable to both symmetric and non-symmetric kernels, as well as vector-valued statistics. These inequalities are then used to establish theoretical guarantees for DPP-based coresets, demonstrating their superior accuracy and size advantages.
• Key Findings: The paper proves that DPP-based coresets can achieve a cardinality of O(ε−2/(1+δ)), where δ is a positive value related to the variance reduction achieved by the DPP, while independent sampling methods are limited to O(ε−2). This demonstrates that DPPs require significantly fewer samples to achieve the same level of accuracy. The authors also provide empirical evidence supporting their theoretical findings through experiments on k-means clustering with various datasets.
• Main Conclusions: This work establishes the superiority of DPPs for coreset construction in machine learning, offering both theoretical guarantees and empirical validation. The use of DPPs allows for smaller coresets without compromising accuracy, leading to more efficient training and potentially improved generalization in various machine learning tasks.
• Significance: This research significantly contributes to the field of coreset construction by introducing a novel and theoretically grounded approach based on DPPs. It paves the way for more efficient large-scale machine learning by enabling the use of smaller, more representative subsets of data.
• Limitations and Future Research: The paper acknowledges limitations regarding the computational complexity of DPP sampling and the dependence of the improvement rate on data dimensionality. Future research directions include exploring faster DPP sampling techniques, investigating dimension-independent performance guarantees, and extending the theoretical framework to other machine learning algorithms beyond k-means clustering.

Stats

The paper demonstrates the effectiveness of DPP-based coresets on the k-means problem, achieving a faster decay rate of QS(0.9) (quantile function of the supremum of relative error) compared to independent sampling methods.
In experiments with a synthetic uniform dataset, the DPP-based methods and Gaussian m-DPP achieved a faster decay rate (around m−3/4) than the m−1/2 rate of independent sampling.
The stratified sampling baseline, suitable for uniformly-spread datasets, achieved the best performance with a m−1 rate.
For a trimodal dataset, DPP-based samplers still outperformed independent ones, showcasing their effectiveness in capturing data distribution characteristics.
Experiments with the MNIST dataset (projected to 4 dimensions) showed a faster decay for DPP-based methods, but the advantage decreased as dimensionality increased.

Quotes

"DPPs can provably outperform independently drawn coresets."
"DPP-based coresets actually can achieve cardinality m = O(ε−2/(1+δ))."
"sampling with this DPP gives |LS(f)/L(f)−1| ≤m−(1/2+1/(2d))−, ∀f ∈F, where (1/2+1/(2d))− denotes any positive number strictly smaller than 1/2+1/(2d). Meanwhile, for i.i.d. sampling, the accuracy rate ε is at best m−1/2."

Key Insights Distilled From

Small coresets via negative dependence: DPPs, linear statistics, and concentration

by Rémi... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00611.pdf

Deeper Inquiries

How can the theoretical framework presented in this paper be extended to other machine learning algorithms beyond k-means clustering, such as support vector machines or deep neural networks?

Extending the DPP-based coreset framework to algorithms like SVMs and DNNs presents exciting challenges and opportunities:
1. Identifying Suitable Linear Statistics:

SVMs: The coreset construction relies on the existence of a low-dimensional linear statistic that captures the essence of the loss function. For SVMs, the hinge loss is not directly a linear statistic. However, we can explore:

Dual Formulation: The dual SVM problem involves maximizing a quadratic function subject to linear constraints. Coresets for quadratic programming could be investigated.
Approximations: Smooth approximations to the hinge loss, potentially amenable to linear statistic representations, could be considered.

DNNs: The highly non-linear and compositional nature of DNNs makes direct application of linear statistics challenging. Possible avenues include:

Layer-wise Coresets: Constructing coresets for individual layers or groups of layers, exploiting potential linearity within those substructures.
Gradient-Based Coresets: Focusing on coresets that accurately represent the gradients of the loss function with respect to the network parameters.
2. Handling Non-Linearity:

Kernel Methods: For algorithms like SVMs that employ kernels, the DPP kernel selection should align with the kernel used in the learning algorithm. This might involve designing DPP kernels that capture the similarity structure induced by the SVM kernel.
DNN Feature Space: For DNNs, one could explore constructing DPPs in the high-dimensional feature space learned by the network. This would require efficient approximations to handle the dimensionality.
3. Theoretical Analysis:

Generalization Bounds: Extending the coreset guarantees to provide generalization bounds for SVMs and DNNs would require careful analysis of the interplay between the DPP sampling, the algorithm's loss function, and the data distribution.
Computational Tractability: Maintaining computational efficiency is crucial. Approximations and efficient sampling techniques for DPPs in high-dimensional or complex feature spaces would be essential.

Could the reliance on the assumption of data being generated i.i.d. from a specific distribution be relaxed to accommodate more complex real-world datasets with unknown or intricate generative processes?

Relaxing the i.i.d. assumption is crucial for broader applicability. Here are some potential directions:
1. Beyond Parametric Assumptions:

Non-Parametric Density Estimation: Instead of assuming a specific parametric form for the data distribution (like in the discretized OPE), explore non-parametric density estimators like kernel density estimation with adaptive bandwidth selection or nearest-neighbor methods.
Mixture Models: Model the data as arising from a mixture of simpler distributions. DPP constructions for mixture models could be investigated.
2. Leveraging Data Structure:

Local DPPs: Instead of a single global DPP, consider a collection of local DPPs, each capturing the structure of a subset of the data. This could adapt to varying densities and local correlations.
Graph-Based DPPs: Represent the data as a graph, where edges encode similarity. Construct DPPs on this graph to capture dependencies beyond simple pairwise similarities.
3. Distribution-Free Approaches:

Empirical Risk Minimization: Focus on coreset constructions that directly minimize the empirical risk on the observed data, without explicit reliance on a generative model.
Adversarial Learning: Train a DPP-based coreset sampler in an adversarial setting, where an adversary tries to find queries for which the coreset performs poorly. This could lead to more robust coresets.

What are the potential implications of using DPP-based coresets on the fairness and bias of machine learning models, considering that DPPs inherently promote diversity in the selected data points?

The use of DPP-based coresets in fairness and bias presents a nuanced picture:
Potential Benefits:

Mitigating Undersampling Bias: DPPs can help ensure representation from minority groups or data points that might be otherwise under-sampled in traditional coreset methods. This could lead to models that are more robust and generalize better to diverse populations.
Promoting Feature Diversity: By encouraging diversity in the selected features, DPPs might help uncover and mitigate biases encoded in specific feature combinations.
Potential Concerns:

Amplifying Existing Biases: If the DPP kernel is not carefully designed, it might inadvertently amplify existing biases in the data. For instance, if the kernel captures spurious correlations related to sensitive attributes, the resulting coreset might exacerbate unfairness.
Trade-off with Accuracy: In some cases, promoting diversity in the coreset might come at the cost of slight reductions in overall accuracy. Balancing fairness and accuracy would require careful consideration.
Mitigation Strategies and Future Research:

Fairness-Aware Kernel Design: Develop DPP kernels that explicitly incorporate fairness constraints or are designed to be invariant to sensitive attributes.
Regularization Techniques: Explore regularization methods during DPP training to penalize coresets that exhibit high bias or unfairness.
Evaluation Metrics: Establish comprehensive evaluation metrics that quantify both the fairness and accuracy of models trained on DPP-based coresets.
Key Takeaway:
DPP-based coresets offer a promising avenue for fairness-aware machine learning. However, careful consideration of kernel design, potential bias amplification, and the trade-off with accuracy is crucial. Further research is needed to develop robust and fair DPP-based coreset methods.