One of the most persistent challenges in continual learning research is the formal measurement of forgetting. While intuitive notions of forgetting are easy to grasp—such as a model losing the ability to perform a previously learned task—the precise, quantitative definition remains elusive. Metrics must capture both short-term forgetting, which can occur immediately after learning a new task, and long-term forgetting, which may only become apparent as the sequence of tasks grows. Traditional approaches often rely on tracking accuracy drops on earlier tasks, but these can miss subtle degradations or fail to account for knowledge that is only partially overwritten. The search for robust, theory-grounded metrics that can distinguish between temporary and irreversible forgetting, and that scale to complex, realistic scenarios, is an ongoing area of investigation.

Continual learning for large language models (LLMs) introduces a new set of difficulties. The sheer scale of LLMs—with billions of parameters and vast, diverse training corpora—means that naive approaches to sequential adaptation are often infeasible. Contextual dependencies in language data can span long sequences, making it hard to define discrete tasks or to isolate the effects of new data. Parameter sharing across many capabilities further complicates the picture: a small update intended to improve performance on one domain can have unpredictable effects elsewhere, due to the entangled representations learned by the model. This makes it particularly challenging to diagnose, measure, or mitigate forgetting in LLMs, and demands new experimental and theoretical tools.

The interplay between continual learning and modern fine-tuning strategies, such as parameter-efficient fine-tuning (PEFT), is also a frontier of research. Fine-tuning allows models to adapt to new data or domains, but often at the cost of some degree of forgetting. PEFT techniques, which modify only a small subset of parameters or introduce lightweight adapters, aim to minimize this disruption. However, it remains unclear how well continual learning principles—such as regularization, replay, or architectural isolation—translate to these adaptation paradigms. Questions about the optimal allocation of parameters, the limits of modularity, and the trade-offs between flexibility and stability are central to understanding how continual learning can be realized in practice with state-of-the-art models.

Theoretical work has also uncovered fundamental impossibility results that set boundaries on what continual learning can achieve. In certain settings, it is provably impossible to maintain perfect performance on all tasks indefinitely, especially when there are conflicting objectives or limited model capacity. These results often highlight the inherent trade-offs between stability (preserving old knowledge) and plasticity (acquiring new knowledge). They also underscore the importance of task similarity, data distribution shifts, and model size in determining the feasibility of continual learning. Recognizing these theoretical limits is crucial for setting realistic expectations and for guiding the search for practical algorithms that make the best use of available resources.

In summary, many foundational questions in continual learning remain open. Accurately measuring forgetting, scaling methods to massive models like LLMs, and understanding the theoretical limits of adaptation are all active areas of research. Progress in these domains will be essential for building systems that can truly learn continuously and robustly over time.