# The Sequential Price of Continual Learning

arXiv: https://arxiv.org/abs/2609.29674 · HTML: https://mlsanity.com/paper/2609.29674/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 50 of its 66 references that are in the map.
- First posted on arXiv 30 Aug 2026.

## Builds on (most cited first, 30 of 50)

- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Gradient Episodic Memory for Continual Learning](https://mlsanity.com/paper/1706.08840/) (2017)
- [Continual Lifelong Learning with Neural Networks: A Review](https://mlsanity.com/paper/1802.07569/) (2018)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [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)
- [A Comprehensive Survey of Continual Learning: Theory, Method and Application](https://mlsanity.com/paper/2302.00487/) (2023)
- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [Progress & Compress: A scalable framework for continual learning](https://mlsanity.com/paper/1805.06370/) (2018)
- [Conflict-Averse Gradient Descent for Multi-task Learning](https://mlsanity.com/paper/2110.14048/) (2021)
- [Online Continual Learning in Image Classification: An Empirical Survey](https://mlsanity.com/paper/2101.10423/) (2021)
- [Understanding the Role of Training Regimes in Continual Learning](https://mlsanity.com/paper/2006.06958/) (2020)
- [Linear Mode Connectivity in Multitask and Continual Learning](https://mlsanity.com/paper/2010.04495/) (2020)
- [Learning Fast, Learning Slow: A General Continual Learning Method based on Complementary Learning System](https://mlsanity.com/paper/2201.12604/) (2022)
- [A Theoretical Analysis of Catastrophic Forgetting through the NTK Overlap Matrix](https://mlsanity.com/paper/2010.04003/) (2020)
- [Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains](https://mlsanity.com/paper/1707.06386/) (2017)
- [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)
- [Wide Neural Networks Forget Less Catastrophically](https://mlsanity.com/paper/2110.11526/) (2021)
- [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)
- [Challenging Common Assumptions about Catastrophic Forgetting](https://mlsanity.com/paper/2207.04543/) (2022)
- [Disentangling and Mitigating the Impact of Task Similarity for Continual Learning](https://mlsanity.com/paper/2405.20236/) (2024)
- [Two Complementary Perspectives to Continual Learning: Ask Not Only What to Optimize, But Also How](https://mlsanity.com/paper/2311.04898/) (2023)
- [Lifelong Learning with Sketched Structural Regularization](https://mlsanity.com/paper/2104.08604/) (2021)

## Abstract (arXiv)

Sequential task updates are fundamental to continual learning, but their recency bias can impose a lasting performance cost. We study this cost in an overparameterized linear-regression model with i.i.d. task sampling. We prove that distribution-level forgetting and population loss converge to the same stationary limit. We quantify the additional loss incurred by sequential exact fitting, or the sequential price. In more homogeneous task geometries, it equals the intrinsic loss asymptotically attained by joint training, making the total loss twice as large. We further analyze fixed-strength elastic weight consolidation (EWC) under general task curvatures and characterize its stationary sequential price at every regularization strength. Under strong regularization, the price decays inversely with EWC strength while the mean-square coupling horizon grows proportionally. Experiments on Jester and Rotated MNIST support the predicted sequential price and its reduction by EWC, with quantitative agreement on real-world tasks satisfying the theory's assumptions and qualitative agreement under nonlinear finite-step training.