Regularization-based methods in continual learning are designed to address the problem of catastrophic forgetting by constraining how much a model's parameters can change after learning previous tasks. The core idea is to identify which parameters are crucial for maintaining performance on earlier tasks and then penalize updates to those parameters during the learning of new tasks. By doing so, the model is less likely to overwrite knowledge that is important for previous tasks, thereby reducing forgetting.

Estimating the importance of each parameter is a key step in this process. One common approach is to use the Fisher information matrix, which provides a measure of how sensitive the loss function is to changes in each parameter. Parameters with high Fisher information are considered more important, as small changes to them can lead to large increases in loss. Another approach, used in methods like Synaptic Intelligence (SI), involves path integrals that accumulate information about how much each parameter contributes to reducing the loss over the course of training. Accurate estimation of parameter importance is essential, as it determines which parameters should be protected to preserve past knowledge.

Elastic Weight Consolidation (EWC) is a well-known regularization-based method that leverages the Fisher information matrix to estimate parameter importance. In EWC, after training on a task, the model computes the Fisher information for each parameter and uses it to construct a quadratic penalty that discourages changes to important parameters when learning new tasks. This penalty is added to the loss function for subsequent tasks, effectively anchoring the parameters that are critical for previous performance.

Synaptic Intelligence (SI), in contrast, tracks the contribution of each parameter to the reduction in loss throughout training by integrating over the parameter's trajectory. SI assigns higher importance to parameters that have played a significant role in minimizing loss, and it accumulates this information in an online manner. When learning new tasks, SI applies a regularization term that penalizes changes to these important parameters, but does so based on the parameter's entire training history rather than just a snapshot at the end of a task.

Despite their effectiveness, regularization-based methods have notable limitations. Because they constrain parameter updates, they can slow down learning on new tasks, especially when many parameters are deemed important for previous tasks. Furthermore, the success of these methods hinges on the accuracy of the parameter importance estimation; if the estimation is poor, the model may either over-constrain itself and fail to learn new tasks, or under-constrain and suffer from forgetting.

Key takeaways: regularization-based methods provide a principled framework for mitigating catastrophic forgetting by balancing the stability of past knowledge with the plasticity needed to learn new tasks. However, this balance involves trade-offs, and the effectiveness of these methods depends critically on how well parameter importance is estimated and incorporated into the learning process.