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Adaptive, Robust and Scalable Bayesian Filtering for Online Learning

Gerardo Duran-Martin

TL;DR

This thesis reframes Bayesian filtering as a versatile toolkit for online learning, addressing adaptivity to non-stationarity, robustness to misspecification and outliers, and scalability to high-dimensional neural networks. It introduces the BONE framework for generalized-Bayes online learning in non-stationary environments, and the WoLF method for robust, loss-based updates with closed-form Kalman-like recursions. It also develops scalable online neural-network learning via subspace EKF (SSEKF), PULSE (projection-based last-layer learning), and LoFi (low-rank precision) to enable real-time training. The work includes theoretical robustness guarantees and extensive experiments across prequential prediction, online classification, contextual bandits, segmentation, and online deep learning, showing improved performance in dynamic, misspecified, and high-dimensional settings. Overall, the contributions offer a coherent framework and practical algorithms for robust, adaptive, and scalable Bayesian online learning in complex environments.

Abstract

In this thesis, we introduce Bayesian filtering as a principled framework for tackling diverse sequential machine learning problems, including online (continual) learning, prequential (one-step-ahead) forecasting, and contextual bandits. To this end, this thesis addresses key challenges in applying Bayesian filtering to these problems: adaptivity to non-stationary environments, robustness to model misspecification and outliers, and scalability to the high-dimensional parameter space of deep neural networks. We develop novel tools within the Bayesian filtering framework to address each of these challenges, including: (i) a modular framework that enables the development adaptive approaches for online learning; (ii) a novel, provably robust filter with similar computational cost to standard filters, that employs Generalised Bayes; and (iii) a set of tools for sequentially updating model parameters using approximate second-order optimisation methods that exploit the overparametrisation of high-dimensional parametric models such as neural networks. Theoretical analysis and empirical results demonstrate the improved performance of our methods in dynamic, high-dimensional, and misspecified models.

Adaptive, Robust and Scalable Bayesian Filtering for Online Learning

TL;DR

This thesis reframes Bayesian filtering as a versatile toolkit for online learning, addressing adaptivity to non-stationarity, robustness to misspecification and outliers, and scalability to high-dimensional neural networks. It introduces the BONE framework for generalized-Bayes online learning in non-stationary environments, and the WoLF method for robust, loss-based updates with closed-form Kalman-like recursions. It also develops scalable online neural-network learning via subspace EKF (SSEKF), PULSE (projection-based last-layer learning), and LoFi (low-rank precision) to enable real-time training. The work includes theoretical robustness guarantees and extensive experiments across prequential prediction, online classification, contextual bandits, segmentation, and online deep learning, showing improved performance in dynamic, misspecified, and high-dimensional settings. Overall, the contributions offer a coherent framework and practical algorithms for robust, adaptive, and scalable Bayesian online learning in complex environments.

Abstract

In this thesis, we introduce Bayesian filtering as a principled framework for tackling diverse sequential machine learning problems, including online (continual) learning, prequential (one-step-ahead) forecasting, and contextual bandits. To this end, this thesis addresses key challenges in applying Bayesian filtering to these problems: adaptivity to non-stationary environments, robustness to model misspecification and outliers, and scalability to the high-dimensional parameter space of deep neural networks. We develop novel tools within the Bayesian filtering framework to address each of these challenges, including: (i) a modular framework that enables the development adaptive approaches for online learning; (ii) a novel, provably robust filter with similar computational cost to standard filters, that employs Generalised Bayes; and (iii) a set of tools for sequentially updating model parameters using approximate second-order optimisation methods that exploit the overparametrisation of high-dimensional parametric models such as neural networks. Theoretical analysis and empirical results demonstrate the improved performance of our methods in dynamic, high-dimensional, and misspecified models.
Paper Structure (114 sections, 28 theorems, 232 equations, 39 figures, 8 tables, 16 algorithms)

This paper contains 114 sections, 28 theorems, 232 equations, 39 figures, 8 tables, 16 algorithms.

Table of Contents

  1. Introduction
  2. Recursive Bayesian online learning
  3. Recursive Bayesian inference
  4. Example tasks
  5. Unsupervised tasks
  6. Segmentation
  7. Filtering using state-space models (SSM)
  8. Segmentation using Switching state-space models (SSSM)
  9. Supervised tasks
  10. Prequential forecasting
  11. Online continual learning (OCL)
  12. Contextual bandits
  13. The multivariate Gaussian
  14. Linear and Gaussian measurement models
  15. Example: recursive and batch Ridge regression
  16. ...and 99 more sections

Key Result

Proposition 2.2

Let ${\bm{x}}\in\mathbb{R}^M$ and ${\bm{y}}\in\mathbb{R}^o$ be two random vectors such that $p({\bm{x}}) = {\cal N}({\bm{x}} \,\vert\, {\bm{m}}, \mathbf{P})$ and $p({\bm{y}} \,\vert\, {\bm{x}}) = {\cal N}({\bm{y}} \,\vert\, \mathbf{H}\,{\bm{x}} + {\bm{b}}, \mathbf{S})$. The joint density for $({\bm{

Figures (39)

  • Figure 1: One-dimensional projection of a noisy-two dimensional dynamical system. In the top panel, the gray arrows represent the underlying evolution of the system and the black dots show the sampled (but unknown) locations of the system. In the bottom panel, the red dots show the observed measurements projected onto a one-dimensional plane. The red vertical lines denote the projection from latent space to observation space.
  • Figure 2: (Left panel) Mean estimate of the parameters. The solid lines correspond to the recursive-Bayes estimate of the mean. The dashed lines correspond to the offline estimate of the mean using Ridge regression. (Right panel) RMSE on a held-out test set.
  • Figure 3: Decision boundaries for the logistic regression model as a function of the number of processed measurements.
  • Figure 4: The solid lines show the posterior mean estimate of model parameters and two standard deviations. The dashed lines show the batch estimate of the logistic regression parameters.
  • Figure 5: Decision boundaries for the classification problem using a neural network trained using the R-VGA. The crimson line shows the decision boundary at $0.5$. The dot coloured in cyan shows the observation seen at time $t$.
  • ...and 34 more figures

Theorems & Definitions (70)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4: Ridge regression
  • proof
  • Proposition 2.5: Multivariate recursive Bayesian linear regression
  • proof
  • Proposition 2.6
  • ...and 60 more