Regret-Optimal Federated Transfer Learning for Kernel Regression with
  Applications in American Option Pricing
Xuwei Yang Anastasis Kratsios Florian Krach Matheus Grasselli Aurelien Lucchi
We propose an optimal iterative scheme for federated transfer learning, where
a central planner has access to datasets ${\cal D}_1,\dots,{\cal D}_N$ for the
same learning model $f_{\theta}$. Our objective is to minimize the cumulative
deviation of the generated parameters $\{\theta_i(t)\}_{t=0}^T$ across all $T$
iterations from the specialized parameters
$\theta^\star_{1},\ldots,\theta^\star_N$ obtained for each dataset, while
respecting the loss function for the model $f_{\theta(T)}$ produced by the
algorithm upon halting. We only allow for continual communication between each
of the specialized models (nodes/agents) and the central planner (server), at
each iteration (round). For the case where the model $f_{\theta}$ is a
finite-rank kernel regression, we derive explicit updates for the
regret-optimal algorithm. By leveraging symmetries within the regret-optimal
algorithm, we further develop a nearly regret-optimal heuristic that runs with
$\mathcal{O}(Np^2)$ fewer elementary operations, where $p$ is the dimension of
the parameter space. Additionally, we investigate the adversarial robustness of
the regret-optimal algorithm showing that an adversary which perturbs $q$
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. To validate our theoretical findings, we conduct
numerical experiments in the context of American option pricing, utilizing a
randomly generated finite-rank kernel.