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Online Learning and Unlearning

Yaxi Hu, Bernhard Schölkopf, Amartya Sanyal

TL;DR

This work extends online learning to handle interleaved data deletions by formalizing an $(\alpha,\varepsilon)$-online unlearning guarantee and proposing two strategies: Passive Unlearning, which injects calibrated noise at deletion times without extra computation, and Active Unlearning, which uses a descent-to-delete step toward an offline ERM solution. Both approaches maintain sublinear regret close to standard Online Gradient Descent under convexity and smoothness assumptions, with stronger bounds in strongly convex settings. The passive method achieves provable unlearning with contractive updates and Gaussian perturbations, while the active method leverages offline unlearning to potentially reduce added noise at the cost of extra computation. The paper also discusses how these ideas compare to DP-based online methods and outlines open questions, such as lower bounds, more efficient active schemes, and broader unlearning-friendly online learners, highlighting practical relevance for privacy regulations and continuous-data environments.

Abstract

We formalize the problem of online learning-unlearning, where a model is updated sequentially in an online setting while accommodating unlearning requests between updates. After a data point is unlearned, all subsequent outputs must be statistically indistinguishable from those of a model trained without that point. We present two online learner-unlearner (OLU) algorithms, both built upon online gradient descent (OGD). The first, passive OLU, leverages OGD's contractive property and injects noise when unlearning occurs, incurring no additional computation. The second, active OLU, uses an offline unlearning algorithm that shifts the model toward a solution excluding the deleted data. Under standard convexity and smoothness assumptions, both methods achieve regret bounds comparable to those of standard OGD, demonstrating that one can maintain competitive regret bounds while providing unlearning guarantees.

Online Learning and Unlearning

TL;DR

This work extends online learning to handle interleaved data deletions by formalizing an -online unlearning guarantee and proposing two strategies: Passive Unlearning, which injects calibrated noise at deletion times without extra computation, and Active Unlearning, which uses a descent-to-delete step toward an offline ERM solution. Both approaches maintain sublinear regret close to standard Online Gradient Descent under convexity and smoothness assumptions, with stronger bounds in strongly convex settings. The passive method achieves provable unlearning with contractive updates and Gaussian perturbations, while the active method leverages offline unlearning to potentially reduce added noise at the cost of extra computation. The paper also discusses how these ideas compare to DP-based online methods and outlines open questions, such as lower bounds, more efficient active schemes, and broader unlearning-friendly online learners, highlighting practical relevance for privacy regulations and continuous-data environments.

Abstract

We formalize the problem of online learning-unlearning, where a model is updated sequentially in an online setting while accommodating unlearning requests between updates. After a data point is unlearned, all subsequent outputs must be statistically indistinguishable from those of a model trained without that point. We present two online learner-unlearner (OLU) algorithms, both built upon online gradient descent (OGD). The first, passive OLU, leverages OGD's contractive property and injects noise when unlearning occurs, incurring no additional computation. The second, active OLU, uses an offline unlearning algorithm that shifts the model toward a solution excluding the deleted data. Under standard convexity and smoothness assumptions, both methods achieve regret bounds comparable to those of standard OGD, demonstrating that one can maintain competitive regret bounds while providing unlearning guarantees.
Paper Structure (15 sections, 32 theorems, 126 equations, 1 figure, 2 tables, 3 algorithms)

This paper contains 15 sections, 32 theorems, 126 equations, 1 figure, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Setup
  3. Preliminaries
  4. Online Learning-Unlearning
  5. Passive Unlearning
  6. Unlearning guarantee
  7. Proof Sketch.
  8. Regret guarantee
  9. Active Unlearning
  10. Discussion and Open Problems
  11. Omitted Proofs for \ref{['sec:prelim']}
  12. Omitted Proofs for \ref{['sec:passive']}
  13. Unlearning guarantee of passive unlearning
  14. Regret guarantee of passive unlearning
  15. Omitted Proofs for \ref{['sec:active-unlearner']}

Key Result

Proposition 1

For $k \in \mathbb{N}$, any $(\alpha, \varepsilon)$-RDP online learning algorithm is an $(\alpha/k, k^{1.6}\varepsilon)$- for any deletion set $U$ and deletion-time set $\mathcal{T}$ of size $k$, if $\alpha \geq 2k$.

Figures (1)

  • Figure 1: Visualization of the output sequence of algorithm ${\mathcal{A}}$ up to the first deletion $\tau[1]$

Theorems & Definitions (55)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • ...and 45 more