Theoretical characterization of uncertainty in high-dimensional linear classification

• URL: http://arxiv.org/abs/2202.03295v1
• Date: Mon, 7 Feb 2022 15:32:07 GMT
• Title: Theoretical characterization of uncertainty in high-dimensional linear classification
• Authors: Lucas Clart\'e, Bruno Loureiro, Florent Krzakala, Lenka Zdeborov\'a
• Abstract summary: We show that uncertainty for learning from limited number of samples of high-dimensional input data and labels can be obtained by the approximate message passing algorithm. We discuss how over-confidence can be mitigated by appropriately regularising, and show that cross-validating with respect to the loss leads to better calibration than with the 0/1 error.
• Score: 24.073221004661427
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Being able to reliably assess not only the accuracy but also the uncertainty of models' predictions is an important endeavour in modern machine learning. Even if the model generating the data and labels is known, computing the intrinsic uncertainty after learning the model from a limited number of samples amounts to sampling the corresponding posterior probability measure. Such sampling is computationally challenging in high-dimensional problems and theoretical results on heuristic uncertainty estimators in high-dimensions are thus scarce. In this manuscript, we characterise uncertainty for learning from limited number of samples of high-dimensional Gaussian input data and labels generated by the probit model. We prove that the Bayesian uncertainty (i.e. the posterior marginals) can be asymptotically obtained by the approximate message passing algorithm, bypassing the canonical but costly Monte Carlo sampling of the posterior. We then provide a closed-form formula for the joint statistics between the logistic classifier, the uncertainty of the statistically optimal Bayesian classifier and the ground-truth probit uncertainty. The formula allows us to investigate calibration of the logistic classifier learning from limited amount of samples. We discuss how over-confidence can be mitigated by appropriately regularising, and show that cross-validating with respect to the loss leads to better calibration than with the 0/1 error.

Related papers

• Calibrating Neural Simulation-Based Inference with Differentiable Coverage Probability [50.44439018155837]
  We propose to include a calibration term directly into the training objective of the neural model. By introducing a relaxation of the classical formulation of calibration error we enable end-to-end backpropagation. It is directly applicable to existing computational pipelines allowing reliable black-box posterior inference.
  arXiv  Detail & Related papers  (2023-10-20T10:20:45Z)
• A Data-Driven Measure of Relative Uncertainty for Misclassification Detection [25.947610541430013]
  We introduce a data-driven measure of uncertainty relative to an observer for misclassification detection. By learning patterns in the distribution of soft-predictions, our uncertainty measure can identify misclassified samples. We demonstrate empirical improvements over multiple image classification tasks, outperforming state-of-the-art misclassification detection methods.
  arXiv  Detail & Related papers  (2023-06-02T17:32:03Z)
• Convergence of uncertainty estimates in Ensemble and Bayesian sparse model discovery [4.446017969073817]
  We show empirical success in terms of accuracy and robustness to noise with bootstrapping-based sequential thresholding least-squares estimator. We show that this bootstrapping-based ensembling technique can perform a provably correct variable selection procedure with an exponential convergence rate of the error rate.
  arXiv  Detail & Related papers  (2023-01-30T04:07:59Z)
• The Implicit Delta Method [61.36121543728134]
  In this paper, we propose an alternative, the implicit delta method, which works by infinitesimally regularizing the training loss of uncertainty. We show that the change in the evaluation due to regularization is consistent for the variance of the evaluation estimator, even when the infinitesimal change is approximated by a finite difference.
  arXiv  Detail & Related papers  (2022-11-11T19:34:17Z)
• High-dimensional Measurement Error Models for Lipschitz Loss [2.6415509201394283]
  We develop high-dimensional measurement error models for a class of Lipschitz loss functions. Our estimator is designed to minimize the $L_1$ norm among all estimators belonging to suitable feasible sets. We derive theoretical guarantees in terms of finite sample statistical error bounds and sign consistency, even when the dimensionality increases exponentially with the sample size.
  arXiv  Detail & Related papers  (2022-10-26T20:06:05Z)
• Statistical and Computational Trade-offs in Variational Inference: A Case Study in Inferential Model Selection [27.817156428797567]
  Variational inference has emerged as a popular alternative to the classical Markov chain Monte Carlo. We study the statistical and computational trade-offs in variational inference via a case study in inferential model selection. We prove that, given a fixed computation budget, a lower-rank inferential model produces variational posteriors with a higher statistical approximation error.
  arXiv  Detail & Related papers  (2022-07-22T17:16:05Z)
• Evaluating State-of-the-Art Classification Models Against Bayes Optimality [106.50867011164584]
  We show that we can compute the exact Bayes error of generative models learned using normalizing flows. We use our approach to conduct a thorough investigation of state-of-the-art classification models.
  arXiv  Detail & Related papers  (2021-06-07T06:21:20Z)
• Scalable Marginal Likelihood Estimation for Model Selection in Deep Learning [78.83598532168256]
  Marginal-likelihood based model-selection is rarely used in deep learning due to estimation difficulties. Our work shows that marginal likelihoods can improve generalization and be useful when validation data is unavailable.
  arXiv  Detail & Related papers  (2021-04-11T09:50:24Z)
• Amortized Conditional Normalized Maximum Likelihood: Reliable Out of Distribution Uncertainty Estimation [99.92568326314667]
  We propose the amortized conditional normalized maximum likelihood (ACNML) method as a scalable general-purpose approach for uncertainty estimation. Our algorithm builds on the conditional normalized maximum likelihood (CNML) coding scheme, which has minimax optimal properties according to the minimum description length principle. We demonstrate that ACNML compares favorably to a number of prior techniques for uncertainty estimation in terms of calibration on out-of-distribution inputs.
  arXiv  Detail & Related papers  (2020-11-05T08:04:34Z)
• Unlabelled Data Improves Bayesian Uncertainty Calibration under Covariate Shift [100.52588638477862]
  We develop an approximate Bayesian inference scheme based on posterior regularisation. We demonstrate the utility of our method in the context of transferring prognostic models of prostate cancer across globally diverse populations.
  arXiv  Detail & Related papers  (2020-06-26T13:50:19Z)
• Quantifying the Uncertainty in Model Parameters Using Gaussian Process-Based Markov Chain Monte Carlo: An Application to Cardiac Electrophysiological Models [7.8316005711996235]
  Estimates of patient-specific model parameters are important for personalized modeling. Standard Markov Chain Monte Carlo sampling requires repeated model simulations that are computationally infeasible. A common solution is to replace the simulation model with a computationally-efficient surrogate for a faster sampling.
  arXiv  Detail & Related papers  (2020-06-02T23:48:15Z)

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.