Addressing Data Heterogeneity in Federated Learning of Cox Proportional Hazards Models (2407.14960v1)

Abstract: The diversity in disease profiles and therapeutic approaches between hospitals and health professionals underscores the need for patient-centric personalized strategies in healthcare. Alongside this, similarities in disease progression across patients can be utilized to improve prediction models in survival analysis. The need for patient privacy and the utility of prediction models can be simultaneously addressed in the framework of Federated Learning (FL). This paper outlines an approach in the domain of federated survival analysis, specifically the Cox Proportional Hazards (CoxPH) model, with a specific focus on mitigating data heterogeneity and elevating model performance. We present an FL approach that employs feature-based clustering to enhance model accuracy across synthetic datasets and real-world applications, including the Surveillance, Epidemiology, and End Results (SEER) database. Furthermore, we consider an event-based reporting strategy that provides a dynamic approach to model adaptation by responding to local data changes. Our experiments show the efficacy of our approach and discuss future directions for a practical application of FL in healthcare.

• The paper proves Algorithm 2 converges to the optimal CoxPH coefficients by showing exponential error reduction under smoothness and strong convexity assumptions.
• It employs local model updates and global aggregation to ensure precise parameter estimation across distributed datasets while preserving data privacy.
• The findings support robust, privacy-preserving survival analysis in healthcare, paving the way for practical federated learning implementations.

Convergence Analysis of Federated Learning for Cox Proportional Hazards Model

The paper, titled "Convergence Guarantee for Algorithm 2," primarily focuses on establishing the convergence properties of a federated learning algorithm when applied to the Cox Proportional Hazards (CoxPH) model. This paper is pivotal due to the growing relevance of federated learning in scenarios where data privacy is a concern, such as distributed healthcare datasets.

Cox Proportional Hazards Model: An Overview

The CoxPH model is a cornerstone of survival analysis. It utilizes a partial likelihood for parameter estimation, effectively bypassing the need to model the underlying survival times directly. Specifically, the partial likelihood function is:

$$L(\beta) = \prod_{i: \delta_i = 1} \frac{\exp(\beta^T x_i)}{\sum_{j \in R(t_i)} \exp(\beta^T x_j)},$$

where $x_i$ denotes the covariates for the $i$-th individual, $\beta$ represents the coefficient vector, $\delta_i$ is an event indicator, $t_i$ denotes the time of event or censoring, and $R(t_i)$ is the risk set at $t_i$. The loss function for parameter estimation is the negative log of this partial likelihood:

$$\mathcal{L}(\beta) = - \sum_{i: \delta_i = 1} \left[ \beta^T x_i - \log\left(\sum_{j \in R(t_i)} \exp(\beta^T x_j)\right) \right].$$

The Convergence Theorem

The paper proves that given the assumptions of smoothness, strong convexity, and proper initialization, Algorithm 2, which updates models in a federated learning context, converges to the optimal solution $\beta^*$. The convergence behavior is characterized by:

$$\mathbb{E}[\|\beta^{(t+1)} - \beta^*\|^2] \leq (1 - \eta \mu) \mathbb{E}[\|\beta^{(t)} - \beta^*\|^2] + \eta^2 L^2 \sigma^2,$$

where $\eta$ is the learning rate, $\mu$ the strong convexity parameter, $L$ the Lipschitz constant, and $\sigma^2$ the variance of the stochastic gradients.

Assumptions and Proof Outline

Assumptions:

1. Smoothness: The gradient of the loss function is Lipschitz continuous with constant $L$, ensuring:

$$\|\nabla \mathcal{L}(\beta_1) - \nabla \mathcal{L}(\beta_2)\| \leq L \|\beta_1 - \beta_2\|.$$

1. Strong Convexity: The loss function is strongly convex, guaranteeing that:

$$\mathcal{L}(\beta_1) \geq \mathcal{L}(\beta_2) + \nabla \mathcal{L}(\beta_2)^T(\beta_1 - \beta_2) + \frac{\mu}{2}\|\beta_1 - \beta_2\|^2.$$

1. Proper Initialization: Initial parameters are chosen close to the optimal values.

Proof Summary:

The proof involves updating the local models at each center $k$ and then aggregating them to update the global model. Key steps include:

1. Local Update:

$$\beta_k^{(t+1)} = \beta_k^{(t)} - \eta \nabla \mathcal{L}_k(\beta_k^{(t)}),$$

where $\nabla \mathcal{L}_k(\beta_k)$ is the gradient at center $k$.

1. Global Aggregation:

$$\beta^{(t+1)} = \frac{1}{K} \sum_{k=1}^K \beta_k^{(t+1)}.$$

1. Main Inequality: Through Lipschitz continuity and strong convexity properties, it establishes:

$$\|\beta^{(t+1)} - \beta^*\|^2 \leq (1 - \eta \mu) \|\beta^{(t)} - \beta^*\|^2 + \eta^2 L^2 \sigma^2.$$

1. Telescoping Technique: By summing the inequalities over iterations, it derives the relationship between the initial estimation error and the final error, confirming exponential convergence to the optimal solution $\beta^*$.

Implications and Future Directions

The convergence guarantee of Algorithm 2 under federated learning conditions indicates the robustness of the approach for distributed survival analysis, especially in sensitive fields like healthcare. The practical implications are substantial, enabling reliable and privacy-preserving survival predictions.

Future research could explore optimizing the learning rate $\eta$ and considering non-convex loss functions to expand the applicability of these results. Bridging the theoretical guarantees with empirical performance in real-world federated learning systems is another promising direction. Additionally, investigating the algorithm's resilience to communication delays and partial client participation could yield insights necessary for deploying federated learning in varying practical scenarios.

In conclusion, this paper contributes significantly to understanding the convergence dynamics of federated learning algorithms, especially in the context of the Cox Proportional Hazards model, paving the way for secure and efficient distributed learning in survival analysis.