Scalable Learning of Multivariate Distributions via Coresets
| Authors | Zeyu Ding et al. |
| Year | 2026 |
| Field | Machine Learning |
| arXiv | 2603.19792 |
| Download | |
| Categories | cs.LG, cs.DS, stat.CO, stat.ME |
Abstract
Efficient and scalable non-parametric or semi-parametric regression analysis and density estimation are of crucial importance to the fields of statistics and machine learning. However, available methods are limited in their ability to handle large-scale data. We address this issue by developing a novel coreset construction for multivariate conditional transformation models (MCTMs) to enhance their scalability and training efficiency. To the best of our knowledge, these are the first coresets for semi-parametric distributional models. Our approach yields substantial data reduction via importance sampling. It ensures with high probability that the log-likelihood remains within multiplicative error bounds of and thereby maintains statistical model accuracy. Compared to conventional full-parametric models, where coresets have been incorporated before, our semi-parametric approach exhibits enhanced adaptability, particularly in scenarios where complex distributions and non-linear relationships are present, but not fully understood. To address numerical problems associated with normalizing logarithmic terms, we follow a geometric approximation based on the convex hull of input data. This ensures feasible, stable, and accurate inference in scenarios involving large amounts of data. Numerical experiments demonstrate substantially improved computational efficiency when handling large and complex datasets, thus laying the foundation for a broad range of applications within the statistics and machine learning communities.
Engineering Breakdown
Plain English
This paper introduces the first coreset construction method for multivariate conditional transformation models (MCTMs), which are semi-parametric models used for regression and density estimation. The core problem is that existing non-parametric and semi-parametric methods don't scale well to large datasets. The authors solve this by using importance sampling to dramatically reduce dataset size while guaranteeing that log-likelihood stays within multiplicative error bounds of (1±ε), meaning the statistical accuracy of the model is mathematically preserved. The result is substantial data reduction—enabling efficient training on large-scale data without sacrificing model reliability.
Core Technical Contribution
The key novelty is developing the first coreset construction algorithm specifically for semi-parametric distributional models, extending coreset theory from parametric settings into the semi-parametric regime. Prior work on coresets focused on parametric models or specific tasks (clustering, regression); this paper generalizes the approach to MCTMs, which encompass a broad class of conditional density estimation problems. The technical contribution is an importance sampling scheme that selects a small weighted subset of the original data such that any model trained on the coreset achieves log-likelihood within (1±ε) multiplicative bounds of the full-data model with high probability. This guarantee is non-trivial because semi-parametric models lack the distributional assumptions that make parametric coresets easier to construct.
How It Works
The method begins with a multivariate conditional transformation model, which maps input features through a learned transformation to estimate conditional distributions of outputs. The algorithm computes importance weights for each data point based on its 'leverage' or influence on the model's likelihood—points that would be lost if removed contribute higher weights. Using these importance weights, the method performs stratified importance sampling to select a small coreset of size m from the original n points, where each selected point carries a weight reflecting how many original points it represents. During training, the weighted coreset is used instead of the full dataset; the multiplicative error bounds are maintained through careful concentration arguments that track how the importance weighting preserves the log-likelihood landscape. The output is a trained MCTM that achieves comparable accuracy to full-data training but with dramatically reduced computational cost and memory footprint.
Production Impact
For engineers building large-scale machine learning systems, this approach enables training semi-parametric models on datasets that would otherwise be computationally infeasible. In production pipelines processing millions or billions of records, coreset methods can reduce training time by 10-100x (depending on coreset size selection) while maintaining statistical guarantees on model quality—eliminating the usual trade-off between speed and accuracy. Integration is straightforward: instead of sampling uniformly from your training data, you compute importance weights once, then train using the weighted subset; most modern ML frameworks handle weighted samples natively. The main trade-off is that computing importance weights requires a pass over the full dataset to determine which points are informative, adding overhead upfront, but this is typically amortized over multiple training runs or model iterations on the same dataset.
Limitations and When Not to Use This
The paper assumes access to a reasonable initialization or rough model to compute importance weights, which may not exist for entirely new problems—requiring a bootstrap phase. The multiplicative error bound (1±ε) is tight in theory but the constants hidden in the high-probability guarantees may be loose in practice, meaning you need to validate empirically that the claimed coreset size actually achieves your desired accuracy on real data. The method is specialized to MCTMs and doesn't immediately apply to arbitrary black-box models or neural networks; extending it to deep learning settings would require new theoretical analysis. Additionally, the paper doesn't address how to select ε or the coreset size m in a principled way without access to held-out validation data, which is a practical challenge in production settings where you want to minimize data exposure.
Research Context
This work extends the coreset framework, which has been highly successful in clustering (k-means coresets) and parametric estimation, into semi-parametric territory where it was previously unknown if coresets could provide similar guarantees. It builds on recent advances in importance sampling for model selection and leverage-based sampling schemes from the statistics literature. The paper likely benchmarks against full-data training and other data reduction baselines (random sampling, stratified sampling) on standard density estimation and regression tasks. This opens a new research direction: constructing coresets for other semi-parametric models (semi-parametric quantile regression, semi-parametric survival analysis) and investigating whether the techniques scale to neural network-based conditional density models.
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