Continual learning can be understood through the lens of constrained optimization, where the central goal is to learn new tasks without degrading performance on previously learned tasks. From this perspective, each update to the model's parameters must not only minimize the loss for the current task, but also ensure that the loss on earlier tasks does not increase. This approach transforms the continual learning problem into a constrained minimization: the model seeks parameter updates that improve performance on the new data, subject to the constraint that they do not harm performance on any past data.

A key concept in this framework is that of safe subspaces. These are directions in the parameter space along which updates can be made without interfering with knowledge acquired from previous tasks. By restricting parameter changes to these safe subspaces, you can protect prior learning from being overwritten. However, this restriction comes at a cost.

There is a fundamental trade-off between rigidity and adaptability in continual learning. If you over-constrain updates—meaning you require the model to stay too close to previous solutions—you risk making the model too rigid. This rigidity can prevent the model from adapting sufficiently to new tasks, resulting in poor performance on tasks that require significant change. On the other hand, if you allow updates to be too flexible, the model may adapt quickly to new tasks but at the expense of increased forgetting of old tasks.

Structural constraints such as low-rank or sparsity assumptions in the parameter space can help manage this trade-off. By assuming that important knowledge is captured in a low-dimensional or sparse subspace, you can restrict updates to directions that are less likely to interfere with prior learning. However, these structural assumptions themselves introduce limitations: if the true solution does not fit the assumed structure, learning and retention may suffer.

Key takeaways: viewing continual learning as a constrained optimization problem provides both a geometric and mathematical framework for understanding and designing algorithms. Every constraint you impose—whether through safe subspaces, low-rank structures, or sparsity—introduces a balance between retaining old knowledge and acquiring new information. There is no perfect solution; all approaches involve trade-offs that must be managed based on the specific demands of the tasks and data at hand.