Learning and Generalization with Mixture Data

• URL: http://arxiv.org/abs/2504.20651v1
• Date: Tue, 29 Apr 2025 11:21:15 GMT
• Title: Learning and Generalization with Mixture Data
• Authors: Harsh Vardhan, Avishek Ghosh, Arya Mazumdar,
• Abstract summary: We study generalization performance and statistical rates when data is sampled from a mixture distribution.<n>In particular, we study the generalization performance under the classical PAC framework.
• Score: 20.282908756947936
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In many, if not most, machine learning applications the training data is naturally heterogeneous (e.g. federated learning, adversarial attacks and domain adaptation in neural net training). Data heterogeneity is identified as one of the major challenges in modern day large-scale learning. A classical way to represent heterogeneous data is via a mixture model. In this paper, we study generalization performance and statistical rates when data is sampled from a mixture distribution. We first characterize the heterogeneity of the mixture in terms of the pairwise total variation distance of the sub-population distributions. Thereafter, as a central theme of this paper, we characterize the range where the mixture may be treated as a single (homogeneous) distribution for learning. In particular, we study the generalization performance under the classical PAC framework and the statistical error rates for parametric (linear regression, mixture of hyperplanes) as well as non-parametric (Lipschitz, convex and H\"older-smooth) regression problems. In order to do this, we obtain Rademacher complexity and (local) Gaussian complexity bounds with mixture data, and apply them to get the generalization and convergence rates respectively. We observe that as the (regression) function classes get more complex, the requirement on the pairwise total variation distance gets stringent, which matches our intuition. We also do a finer analysis for the case of mixed linear regression and provide a tight bound on the generalization error in terms of heterogeneity.

Related papers

• Conformal inference for regression on Riemannian Manifolds [49.7719149179179]
  We investigate prediction sets for regression scenarios when the response variable, denoted by $Y$, resides in a manifold, and the covariable, denoted by X, lies in Euclidean space. We prove the almost sure convergence of the empirical version of these regions on the manifold to their population counterparts.
  arXiv  Detail & Related papers  (2023-10-12T10:56:25Z)
• High-Dimensional Undirected Graphical Models for Arbitrary Mixed Data [2.2871867623460207]
  In many applications data span variables of different types, whose principled joint analysis is nontrivial. Recent advances have shown how the binary-continuous case can be tackled, but the general mixed variable type regime remains challenging. We propose flexible and scalable methodology for data with variables of entirely general mixed type.
  arXiv  Detail & Related papers  (2022-11-21T18:21:31Z)
• C-Mixup: Improving Generalization in Regression [71.10418219781575]
  Mixup algorithm improves generalization by linearly interpolating a pair of examples and their corresponding labels. We propose C-Mixup, which adjusts the sampling probability based on the similarity of the labels. C-Mixup achieves 6.56%, 4.76%, 5.82% improvements in in-distribution generalization, task generalization, and out-of-distribution robustness, respectively.
  arXiv  Detail & Related papers  (2022-10-11T20:39:38Z)
• On the Strong Correlation Between Model Invariance and Generalization [54.812786542023325]
  Generalization captures a model's ability to classify unseen data. Invariance measures consistency of model predictions on transformations of the data. From a dataset-centric view, we find a certain model's accuracy and invariance linearly correlated on different test sets.
  arXiv  Detail & Related papers  (2022-07-14T17:08:25Z)
• Gaussian Universality of Linear Classifiers with Random Labels in High-Dimension [24.503842578208268]
  We prove that data coming from a range of generative models in high-dimensions have the same minimum training loss as Gaussian data with corresponding data covariance. In particular, our theorem covers data created by an arbitrary mixture of homogeneous Gaussian clouds, as well as multi-modal generative neural networks.
  arXiv  Detail & Related papers  (2022-05-26T12:25:24Z)
• Fluctuations, Bias, Variance & Ensemble of Learners: Exact Asymptotics for Convex Losses in High-Dimension [25.711297863946193]
  We develop a theory for the study of fluctuations in an ensemble of generalised linear models trained on different, but correlated, features. We provide a complete description of the joint distribution of the empirical risk minimiser for generic convex loss and regularisation in the high-dimensional limit.
  arXiv  Detail & Related papers  (2022-01-31T17:44:58Z)
• A Robust and Flexible EM Algorithm for Mixtures of Elliptical Distributions with Missing Data [71.9573352891936]
  This paper tackles the problem of missing data imputation for noisy and non-Gaussian data. A new EM algorithm is investigated for mixtures of elliptical distributions with the property of handling potential missing data. Experimental results on synthetic data demonstrate that the proposed algorithm is robust to outliers and can be used with non-Gaussian data.
  arXiv  Detail & Related papers  (2022-01-28T10:01:37Z)
• Uniform Consistency in Nonparametric Mixture Models [12.382836502781258]
  We study uniform consistency in nonparametric mixture models and mixed regression models. In the case of mixed regression, we prove $L1$ convergence of the regression functions while allowing for the component regression functions to intersect arbitrarily often.
  arXiv  Detail & Related papers  (2021-08-31T17:53:52Z)
• Learning Gaussian Mixtures with Generalised Linear Models: Precise Asymptotics in High-dimensions [79.35722941720734]
  Generalised linear models for multi-class classification problems are one of the fundamental building blocks of modern machine learning tasks. We prove exacts characterising the estimator in high-dimensions via empirical risk minimisation. We discuss how our theory can be applied beyond the scope of synthetic data.
  arXiv  Detail & Related papers  (2021-06-07T16:53:56Z)
• What causes the test error? Going beyond bias-variance via ANOVA [21.359033212191218]
  Modern machine learning methods are often overparametrized, allowing adaptation to the data at a fine level. Recent work aimed to understand in greater depth why overparametrization is helpful for generalization. We propose using the analysis of variance (ANOVA) to decompose the variance in the test error in a symmetric way.
  arXiv  Detail & Related papers  (2020-10-11T05:21:13Z)
• Good Classifiers are Abundant in the Interpolating Regime [64.72044662855612]
  We develop a methodology to compute precisely the full distribution of test errors among interpolating classifiers. We find that test errors tend to concentrate around a small typical value $varepsilon*$, which deviates substantially from the test error of worst-case interpolating model. Our results show that the usual style of analysis in statistical learning theory may not be fine-grained enough to capture the good generalization performance observed in practice.
  arXiv  Detail & Related papers  (2020-06-22T21:12:31Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.