Regret-Optimal Federated Transfer Learning for Kernel Regression: Theory and Application in American Option Pricing

核心概念

This paper introduces a novel regret-optimal algorithm for federated transfer learning with kernel regression, demonstrating its theoretical advantages and practical application in American option pricing.

摘要

• Bibliographic Information: Yang, X., Kratsios, A., Krach, F., Grasselli, M., & Lucchi, A. (2024). Regret-Optimal Federated Transfer Learning for Kernel Regression with Applications in American Option Pricing. arXiv preprint arXiv:2309.04557v2.
• Research Objective: The paper aims to develop a regret-optimal algorithm for federated transfer learning in the context of kernel regression, addressing the challenge of efficiently leveraging knowledge from multiple datasets to improve model performance on a focal task.
• Methodology: The authors employ techniques from optimal control theory to derive a closed-form expression for the regret-optimal algorithm. They analyze its optimality, computational complexity, and adversarial robustness. The algorithm is then applied to American option pricing problems using random feature models, comparing its performance against various baseline methods.
• Key Findings:
  • The proposed algorithm minimizes a regret functional that balances predictive power on the new dataset with transfer learning from other datasets under an algorithmic stability penalty.
  • The algorithm demonstrates adversarial robustness, with the regret being minimally affected by perturbations in the training data.
  • Empirical results on American option pricing tasks show that the algorithm outperforms traditional optimization methods, particularly when leveraging knowledge from similar datasets.
• Main Conclusions: The paper presents a theoretically grounded and practically effective approach for federated transfer learning with kernel regression. The regret-optimal algorithm exhibits superior performance compared to standard methods, highlighting its potential for real-world applications where data is scarce or distributed across multiple sources.
• Significance: This research contributes to the growing field of federated learning by providing a novel optimization framework specifically tailored for kernel regression. The application in American option pricing showcases its relevance to computationally challenging problems in quantitative finance.
• Limitations and Future Research: The paper primarily focuses on kernel regression models with shared feature maps. Future research could explore extensions to more general deep learning architectures and investigate the impact of heterogeneous data distributions on the algorithm's performance.

統計資料

The regret-optimal algorithm outperforms the local optimizer of the main dataset (LO-1) by 9% in American option pricing tasks.
Using 700 training samples from similar datasets, the regret-optimal algorithm achieves comparable performance to the local optimizer trained on the same number of samples from the main dataset, indicating efficient knowledge transfer.
The oracle local optimizer, trained on 50,000 samples from the main dataset, achieves approximately 10% better performance than the regret-optimal algorithm, highlighting the inherent limitations of transfer learning when the total available data is limited.

引述

"In transfer learning one is usually confronted with a focal task defined by a dataset D1 drawn from a distribution µ1 on Rd+1 and additional (more or less) related datasets D2, . . . , DN, where each Di is drawn from (possibly different) distributions µi on Rd+1."
"Our first main result (Theorem 1) is a statistical guarantee, which shows that the following choice of weights w⋆= (w⋆)N i=1 minimizes the worst-case generalization gap between the true risk and the weighted empirical risk of any L-Lipschitz learner (for a given L ≥0)."
"Our main motivation for this setting stems from the now typical setting, where one has access to a pre-trained foundation model, whose final linear layer is to be fine-tuned on several different datasets."

從以下內容提煉的關鍵洞見

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

by Xuwei Yang, ... 於 arxiv.org 10-04-2024

https://arxiv.org/pdf/2309.04557.pdf

深入探究

How could the proposed regret-optimal algorithm be adapted for online learning scenarios where data arrives sequentially?

Adapting the regret-optimal algorithm for online learning, where data arrives sequentially, presents an interesting challenge. Here's a breakdown of potential approaches and considerations:
Challenges:

Static Nature of Algorithm 2: The current formulation of Algorithm 2 relies on pre-computed matrices (P(t), S(t)) that depend on the entire dataset. This is not feasible in online settings.
Dynamically Changing Data Distribution: Online learning often involves concept drift, where the underlying data distribution changes over time. The current algorithm assumes a fixed distribution for each dataset.
Potential Adaptations:

Recursive Updates for P(t) and S(t): Instead of batch computation, explore recursive update rules for P(t) and S(t) that incorporate new data points as they arrive. This would involve leveraging techniques from online linear algebra and recursive least squares.
Sliding Window Approach: To handle concept drift, consider using a sliding window that only considers the most recent data points for computing the updates. This would allow the algorithm to adapt to changes in the data distribution.
Mini-Batch Processing: Process incoming data in small batches, striking a balance between computational efficiency and responsiveness to new information. This would involve updating P(t) and S(t) for each mini-batch.
Regret Analysis for Online Setting: The current regret analysis focuses on a fixed number of iterations (T). In an online setting, the regret should be analyzed in terms of the time horizon or the number of data points observed.

Additional Considerations:

Exploration-Exploitation Trade-off: In online learning, there's a trade-off between exploiting the current best model and exploring new data to potentially improve the model further. This aspect needs to be incorporated into the algorithm design.
Computational Complexity: Online algorithms need to be computationally efficient to handle real-time data streams. The complexity of the updates for P(t) and S(t) should be carefully considered.

Could the adversarial robustness of the algorithm be further enhanced by incorporating techniques from robust optimization or distributionally robust optimization?

Yes, the adversarial robustness of the algorithm can be significantly enhanced by incorporating techniques from robust optimization and distributionally robust optimization.
Robust Optimization (RO):

Incorporating Uncertainty Sets: Instead of assuming a fixed dataset, RO would involve defining uncertainty sets around the data points or the feature map ϕ. These sets would capture the potential adversarial perturbations.
Minimax Formulation: The optimization problem in Algorithm 2 could be reformulated as a minimax problem, where the goal is to minimize the worst-case regret over the uncertainty sets. This would lead to more conservative updates that are less sensitive to adversarial attacks.
Distributionally Robust Optimization (DRO):

Ambiguity Set of Distributions: DRO would involve defining an ambiguity set of probability distributions around the empirical distribution of the data. This set would capture the uncertainty about the true data-generating distribution.
Worst-Case Expected Regret: The optimization problem would then aim to minimize the worst-case expected regret over all distributions in the ambiguity set. This would lead to a more robust algorithm that is less sensitive to variations in the data distribution.
Specific Techniques:

Robust Kernel Methods: Explore robust variants of kernel ridge regression that are less sensitive to outliers or adversarial perturbations in the data.
Regularization Techniques: Introduce regularization terms in the objective function that penalize large deviations from the nominal data points or distributions. This can improve the robustness of the algorithm.
Benefits of RO/DRO:

Improved Robustness Guarantees: RO and DRO provide theoretical guarantees on the performance of the algorithm under adversarial attacks or distributional uncertainties.
Principled Handling of Uncertainty: These techniques offer a principled way to incorporate uncertainty and adversarial behavior into the algorithm design.

What are the potential ethical implications of using federated transfer learning in financial applications, particularly concerning data privacy and fairness?

Federated transfer learning in finance, while promising, raises significant ethical concerns regarding data privacy and fairness:
Data Privacy:

Indirect Data Leakage: Even without directly sharing raw data, federated learning can indirectly leak sensitive information about individual financial transactions or market positions through model updates.
Inference Attacks: Malicious actors could potentially infer sensitive information about the data held by other participants by analyzing the model updates or the final aggregated model.
Lack of Transparency: The decentralized nature of federated learning can make it challenging to track how data is being used and to ensure compliance with privacy regulations.
Fairness:

Bias Amplification: If the datasets held by different participants are biased, federated transfer learning could amplify these biases, leading to unfair or discriminatory outcomes. For example, loan applications might be unfairly rejected for certain demographic groups.
Unequal Benefits: Participants with larger or higher-quality datasets might benefit disproportionately from the federated learning process, potentially exacerbating existing inequalities in the financial system.
Lack of Accountability: The distributed nature of federated learning can make it difficult to assign responsibility for unfair or biased outcomes.
Mitigating Ethical Concerns:

Differential Privacy: Implement differential privacy mechanisms to add noise to model updates, making it harder to infer sensitive information about individual data points.
Federated Learning with Secure Aggregation: Employ cryptographic techniques like secure multi-party computation to aggregate model updates without revealing individual contributions.
Fairness-Aware Federated Learning: Develop fairness-aware algorithms that explicitly address potential biases in the data and strive for equitable outcomes for all stakeholders.
Regulatory Frameworks: Establish clear regulatory frameworks for federated learning in finance, addressing data privacy, fairness, and accountability.
Balancing Innovation and Ethics:
It's crucial to strike a balance between fostering innovation in financial services through federated transfer learning and upholding ethical considerations. This requires collaboration between researchers, practitioners, regulators, and ethicists to develop responsible and trustworthy AI systems.