PMF-CL: Pareto-Minimal-Forgetting Continual Learner for Conflicting Tasks

In the literature, many *continual learning* (CL) algorithms have been proposed to address the issue of *catastrophic forgetting* in ML models (i.e., learning new tasks leads to the loss of performance on previously learned tasks). Although all CL approaches use some form of memory to retain information about past tasks, a grounded understanding of what information needs to be stored to minimize catastrophic forgetting remains elusive. Recently, it has been recognized that under the strong assumption of the existence of a common global minimizer over all tasks, catastrophic forgetting can be completely avoided. However, in practice, tasks rarely have a common global minimizer, and a certain amount of forgetting is inevitable. In this paper, we propose a foundational reframing of CL as balancing conflicting tasks in hindsight from a *multi-task learning* (MTL) perspective. The approach is based on finding *Pareto-optimal* solutions, i.e., the solutions which, by definition, minimally forget the previous tasks in the Pareto sense. We derive a Pareto-minimal-forgetting CL (PMF-CL) algorithm for linear and basis-function regression. Our algorithm naturally extends to loss functions with quadratic upper-bounds around their minimizers, despite consuming only a *static* memory footprint of $\mathcal{O}(d^2)$ for $d$ model parameters, *independent* of the number of sequentially occurring tasks $T$. In the quadratic-loss setting, we obtain exact hindsight Pareto-optimal guarantees on forgetting. In the quadratic-upper-bound setting, forgetting is provably bounded, with the tightness of the forgetting bound being directly determined by the tightness of the quadratic bound on the loss function. Our numerical results validate our theoretical claims of convergence to the Pareto-optimal solution, and demonstrate that we consistently outperform prior empirical methods while occupying lesser memory.