Control Theoretic Approach to Fine-Tuning and Transfer Learning

Given a training set in the form of a paired $(\mathcal{X},\mathcal{Y})$, we say that the control system $\dot{x} = f(x,u)$ has learned the paired set via the control $u^*$ if the system steers each point of $\mathcal{X}$ to its corresponding target in $\mathcal{Y}$. Most existing methods for finding a control function $u^*$ require learning of a new control function if the training set is updated. To overcome this limitation, we introduce the concept of $\textit{tuning without forgetting}$. We develop $\textit{an iterative algorithm}$ to tune the control function $u^*$ when the training set expands, whereby points already in the paired set are still matched, and new training samples are learned. More specifically, at each update of our method, the control $u^*$ is projected onto the kernel of the end-point mapping generated by the controlled dynamics at the learned samples. It ensures keeping the end points for the previously learned samples constant while iteratively learning additional samples. Our work contributes to the scalability of control methods, offering a novel approach to adaptively handle training set expansions.