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Quantifying the Prediction Uncertainty of Machine Learning Models for Individual Data

Koby Bibas

TL;DR

The thesis develops an information-theoretic framework for quantifying prediction uncertainty of machine learning models on individual data using predictive normalized maximum likelihood (pNML). It derives explicit pNML solutions for ordinary and over-parameterized linear regression, introduces a luckiness-enhanced variant (LpNML), and analyzes learnable subspaces where pNML performs well, even without distributional assumptions. Extending to neural networks, it provides analytical pNML formulations for single-layer and last-layer adaptations, enabling confidence estimates and improved OOD detection, while also proposing Adversarial pNML as a defense and a Deep Individual Active Learning (DIAL) strategy. Across experiments on UCI, MNIST, CIFAR, and ImageNet, the approach yields accurate uncertainty quantification, better OOD detection, targeted adversarial defense gains, and sample-efficient active learning, often outperforming strong baselines. Overall, the work offers distribution-free, point-wise uncertainty measures with practical benefits for robustness, open-set recognition, and active data acquisition in modern ML systems.

Abstract

Machine learning models have exhibited exceptional results in various domains. The most prevalent approach for learning is the empirical risk minimizer (ERM), which adapts the model's weights to reduce the loss on a training set and subsequently leverages these weights to predict the label for new test data. Nonetheless, ERM makes the assumption that the test distribution is similar to the training distribution, which may not always hold in real-world situations. In contrast, the predictive normalized maximum likelihood (pNML) was proposed as a min-max solution for the individual setting where no assumptions are made on the distribution of the tested input. This study investigates pNML's learnability for linear regression and neural networks, and demonstrates that pNML can improve the performance and robustness of these models on various tasks. Moreover, the pNML provides an accurate confidence measure for its output, showcasing state-of-the-art results for out-of-distribution detection, resistance to adversarial attacks, and active learning.

Quantifying the Prediction Uncertainty of Machine Learning Models for Individual Data

TL;DR

The thesis develops an information-theoretic framework for quantifying prediction uncertainty of machine learning models on individual data using predictive normalized maximum likelihood (pNML). It derives explicit pNML solutions for ordinary and over-parameterized linear regression, introduces a luckiness-enhanced variant (LpNML), and analyzes learnable subspaces where pNML performs well, even without distributional assumptions. Extending to neural networks, it provides analytical pNML formulations for single-layer and last-layer adaptations, enabling confidence estimates and improved OOD detection, while also proposing Adversarial pNML as a defense and a Deep Individual Active Learning (DIAL) strategy. Across experiments on UCI, MNIST, CIFAR, and ImageNet, the approach yields accurate uncertainty quantification, better OOD detection, targeted adversarial defense gains, and sample-efficient active learning, often outperforming strong baselines. Overall, the work offers distribution-free, point-wise uncertainty measures with practical benefits for robustness, open-set recognition, and active data acquisition in modern ML systems.

Abstract

Machine learning models have exhibited exceptional results in various domains. The most prevalent approach for learning is the empirical risk minimizer (ERM), which adapts the model's weights to reduce the loss on a training set and subsequently leverages these weights to predict the label for new test data. Nonetheless, ERM makes the assumption that the test distribution is similar to the training distribution, which may not always hold in real-world situations. In contrast, the predictive normalized maximum likelihood (pNML) was proposed as a min-max solution for the individual setting where no assumptions are made on the distribution of the tested input. This study investigates pNML's learnability for linear regression and neural networks, and demonstrates that pNML can improve the performance and robustness of these models on various tasks. Moreover, the pNML provides an accurate confidence measure for its output, showcasing state-of-the-art results for out-of-distribution detection, resistance to adversarial attacks, and active learning.

Paper Structure

This paper contains 109 sections, 9 theorems, 164 equations, 26 figures, 22 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Ordinary Least Squares Regression
  3. Introduction
  4. Formal Problem Definition
  5. The pNML solution
  6. pNML with regularization
  7. The learnable space
  8. Simulation
  9. Concluding remarks
  10. Overparameterized Linear Regression
  11. Introduction
  12. Notation and preliminaries
  13. The norm of the minimum norm solution
  14. The pNML for over-parameterized linear regression
  15. The pNML regret upper bound
  16. ...and 94 more sections

Key Result

Theorem 3.3.1

Denote the projection of the test sample onto the orthogonal subspace of the training data empirical correlation matrix as the norm of the MN solution based on the training set $\mathcal{D}_N$ and the test sample $(x,y)$ is given by

Figures (26)

  • Figure 1: The mean pNML least squares estimator and its associated regret
  • Figure 2: The pNML regret for over-parameterized linear regression
  • Figure 3: The MN and pNML loss for samples with regrets lower than a threshold
  • Figure 4: Double-descent of the test log-loss and regret for UCI datasets
  • Figure 5: Ridge ERM and LpNML prediction histograms
  • ...and 21 more figures

Theorems & Definitions (18)

  • Theorem 3.3.1
  • proof
  • Lemma 3.4.1
  • proof
  • Lemma 3.4.2
  • proof
  • Theorem 3.4.3
  • proof
  • Theorem 4.3.1
  • proof
  • ...and 8 more