# From Order to Distribution: An Exact Operator Framework for Forgetting in Continual Learning

arXiv: https://arxiv.org/abs/2604.13460 · HTML: https://mlsanity.com/paper/2604.13460/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 35 of its 46 references that are in the map.
- First posted on arXiv 15 Apr 2026.

## Builds on (most cited first, 30 of 35)

- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Learning without Forgetting](https://mlsanity.com/paper/1606.09282/) (2016)
- [Gradient Episodic Memory for Continual Learning](https://mlsanity.com/paper/1706.08840/) (2017)
- [Memory Aware Synapses: Learning what (not) to forget](https://mlsanity.com/paper/1711.09601/) (2017)
- [Dark Experience for General Continual Learning: a Strong, Simple Baseline](https://mlsanity.com/paper/2004.07211/) (2020)
- [PackNet: Adding Multiple Tasks to a Single Network by Iterative Pruning](https://mlsanity.com/paper/1711.05769/) (2017)
- [A Comprehensive Survey of Continual Learning: Theory, Method and Application](https://mlsanity.com/paper/2302.00487/) (2023)
- [Lifelong Learning with Dynamically Expandable Networks](https://mlsanity.com/paper/1708.01547/) (2017)
- [Variational Continual Learning](https://mlsanity.com/paper/1710.10628/) (2017)
- [A Theoretical Analysis of Catastrophic Forgetting through the NTK Overlap Matrix](https://mlsanity.com/paper/2010.04003/) (2020)
- [Continual Learning in the Teacher-Student Setup: Impact of Task Similarity](https://mlsanity.com/paper/2107.04384/) (2021)
- [How catastrophic can catastrophic forgetting be in linear regression?](https://mlsanity.com/paper/2205.09588/) (2022)
- [Theory on Forgetting and Generalization of Continual Learning](https://mlsanity.com/paper/2302.05836/) (2023)
- [Continual evaluation for lifelong learning: Identifying the stability gap](https://mlsanity.com/paper/2205.13452/) (2022)
- [Continual Learning in Linear Classification on Separable Data](https://mlsanity.com/paper/2306.03534/) (2023)
- [A Statistical Theory of Regularization-Based Continual Learning](https://mlsanity.com/paper/2406.06213/) (2024)
- [Understanding Forgetting in Continual Learning with Linear Regression](https://mlsanity.com/paper/2405.17583/) (2024)
- [Analysis of Catastrophic Forgetting for Random Orthogonal Transformation Tasks in the Overparameterized Regime](https://mlsanity.com/paper/2207.06475/) (2022)
- [The Joint Effect of Task Similarity and Overparameterization on Catastrophic Forgetting -- An Analytical Model](https://mlsanity.com/paper/2401.12617/) (2024)
- [Statistical Mechanical Analysis of Catastrophic Forgetting in Continual Learning with Teacher and Student Networks](https://mlsanity.com/paper/2105.07385/) (2021)
- [Disentangling and Mitigating the Impact of Task Similarity for Continual Learning](https://mlsanity.com/paper/2405.20236/) (2024)
- [Provable Continual Learning via Sketched Jacobian Approximations](https://mlsanity.com/paper/2112.05095/) (2021)
- [Optimal Protocols for Continual Learning via Statistical Physics and Control Theory](https://mlsanity.com/paper/2409.18061/) (2024)
- [Optimal Task Order for Continual Learning of Multiple Tasks](https://mlsanity.com/paper/2502.03350/) (2025)
- [From Continual Learning to SGD and Back: Better Rates for Continual Linear Models](https://mlsanity.com/paper/2504.04579/) (2025)
- [Theoretical Insights into Overparameterized Models in Multi-Task and Replay-Based Continual Learning](https://mlsanity.com/paper/2408.16939/) (2024)
- [Last Iterate Convergence of Incremental Methods and Applications in Continual Learning](https://mlsanity.com/paper/2403.06873/) (2024)
- [Randomized Kaczmarz converges along small singular vectors](https://mlsanity.com/paper/2006.16978/) (2020)
- [Convergence and Implicit Bias of Gradient Descent on Continual Linear Classification](https://mlsanity.com/paper/2504.12712/) (2025)
- [Optimal Rates in Continual Linear Regression via Increasing Regularization](https://mlsanity.com/paper/2506.06501/) (2025)

## Abstract (arXiv)

A central challenge in continual learning is forgetting: the loss of performance on previously learned tasks after learning new ones. Prior theory has analyzed forgetting under random orderings of fixed task collections in overparameterized linear regression. We shift the focus from task order to task distribution, asking how its structure determines forgetting. In the linear setting with a shared solution, i.i.d. task sampling, and sequential exact fitting, we derive an exact operator identity expressing historical forgetting directly in terms of the task distribution. Building on this identity, we establish an exponential decay guarantee for expected historical forgetting under every fixed task distribution in finite dimensions, characterize its asymptotic behavior, and relate decay to the distribution's coverage of observable directions. For an individual learned task, we show that subsequent tasks can collectively support recovery without exact revisits. We derive a lower bound on recovery time and construct a task distribution attaining its inverse-coverage scaling.