Regret-Optimal Federated Transfer Learning for Kernel Regression with Applications in American Option Pricing

ArXiv ID: 2309.04557 “View on arXiv”

Authors: Unknown

Abstract

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 $f_θ$. Our objective is to minimize the cumulative deviation of the generated parameters ${“θ_i(t)"}{“t=0”}^T$ across all $T$ iterations from the specialized parameters $θ^\star{“1”},\ldots,θ^\star_N$ obtained for each dataset, while respecting the loss function for the model $f_{“θ(T)”}$ 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 $f_θ$ 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.

Keywords: federated transfer learning, regret analysis, kernel regression, option pricing, machine learning

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper presents advanced mathematical derivations using optimal control theory to design a regret-optimal algorithm, with explicit complexity bounds and adversarial robustness analysis. The empirical section validates the theoretical claims with numerical experiments on American option pricing using a random feature kernel, demonstrating backtest-ready implementation.

  flowchart TD
    A["Research Goal: Derive regret-optimal federated transfer learning for kernel regression"] --> B["Methodology: Regret-optimal iterative scheme with continual client-server communication"]
    B --> C["Data: Multiple datasets $\mathcal{"D"}_1, \dots, \mathcal{"D"}_N$ for model $f_\theta$"]
    C --> D["Computations: Explicit updates for finite-rank kernel regression"]
    D --> E["Computations: Adversarial robustness analysis (perturbation $\varepsilon$)"]
    E --> F["Findings: Regret-optimal algorithm & nearly optimal heuristic with reduced complexity"]
    F --> G["Validation: Numerical experiments on American option pricing"]