Synchronizing pretrained kernel regressors with applications to American option pricing

American option pricing is a well-studied, computationally-heavy optimal stopping problem, which has motivated the need for light and accurate computational pipelines. Currently, the state-of-the-art light American option prices are built on randomly generated kernel regressors such as random neural networks; however, a single random (or pre-trained) kernel can yield unstable performance on new tasks. With this motivation, we study the problem of fine-tuning several pre-trained kernel regressors, trained on distinct datasets to a given novel dataset with a downstream view toward light American option pricing. This paper addresses the meta-optmization problem of the task of designing optimal transfer learning algorithms, which minimize a pathwise regret functional. Using techniques from optimal control, we construct the unique regret-optimal optimization algorithms for fine-tuning mixtures of pre-trained kernel regressors on $ \mathbb{R}^d $ sharing the same feature map. Our regret functional balances the objects of predictive power on the new dataset against transfer learning from other datasets under an algorithmic stability penalty. We show that an adversary which perturbs a proportion $ 0\leq q\leq 1 $ of training pairs by at most $ \varepsilon>0 $, across all training sets, cannot reduce the regret-optimal algorithm's regret by more than $ \mathcal{O}(\varepsilon (q \bar{N} )^{1/2} ) $, where $ \bar{N} $ is the aggregate number of training pairs. Further, we derive estimates of the computational complexity of our regret-optimal optimization algorithm. Our theoretical findings are used to improve state-of-the-art (SOTA) in computationally light American option pricing problems using random feature models, where we achieve SOTA performance compared to the benchmark light American option pricer of [32].