# Theoretical Foundations of Continual Learning via Drift-Plus-Penalty Source: [Theoretical Foundations of Continual Learning via Drift-Plus-Penalty]() Collection index: [llms.txt]() Data updated: 2026-06-10T05:01:35.776Z Authors: Nazreen Shah, Govinda Arya, Bharath B.N., Ranjitha Prasad Published: 2026-06-07T04:51:32.000Z ## Abstract In many real-world settings, data streams are nonstationary and arrive sequentially, requiring learning systems to adapt continuously without retraining from scratch. Continual learning (CL) addresses this challenge by incorporating new tasks while mitigating catastrophic forgetting, where learning new information degrades performance on previously acquired knowledge. We introduce a control-theoretic perspective on CL that explicitly regulates the evolution of forgetting, framing adaptation as a controlled process subject to long-term stability constraints. We focus on replay-based CL, where a finite memory buffer stores representative samples from prior tasks. We propose COntinual Learning with Drift-Plus-Penalty (COLD), a continual learning framework based on the Drift-Plus-Penalty (DPP) principle from stochastic optimization. To facilitate analysis, we also consider an oracle variant, COLD-ORACLE, as a reference benchmark. At each task, both methods minimize the current task loss while maintaining a virtual queue that tracks deviations from long-term stability on previously learned tasks, capturing the stability-plasticity trade-off as a regulated dynamical process. We establish stability and convergence guarantees that characterize this trade-off through a tunable control parameter. Experiments on standard benchmarks demonstrate that COLD consistently outperforms a broad range of state-of-the-art CL methods while providing competitive and controllable forgetting behavior through explicit regulation of stability and plasticity. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Task-Incremental Learning on Split-CIFAR10 M = 1k (test)](): Task-Incremental Learning; Split-CIFAR10 M = 1k (test); Accuracy: 87.94. - [Task-Incremental Learning on Split-CIFAR100 M = 5k (test)](): Task-Incremental Learning; Split-CIFAR100 M = 5k (test); Accuracy: 75.64. - [Task-Incremental Learning on Split-TinyImageNet M = 5k (test)](): Task-Incremental Learning; Split-TinyImageNet M = 5k (test); Accuracy: 62.21. - [Image Classification on PMNIST Domain-Incremental (test)](): Image Classification; PMNIST Domain-Incremental (test); Accuracy: 91.26. - [Task-incremental learning on Split Tiny-ImageNet (test)](): Task-Incremental Learning; Split Tiny-ImageNet (test); Average Accuracy (A\_T): 56.74. - [Task-incremental learning on Split CIFAR-100 (test)](): Task-Incremental Learning; Split CIFAR-100 (test); Average Accuracy (A\_T): 68.03. - [Task-incremental learning on CIFAR-10 Split (test)](): Task-Incremental Learning; CIFAR-10 Split (test); Average Accuracy: 85.68. ## Original sources and implementation [Original paper]() [PDF]()