[Paper Review] Rate-Distortion Theoretic Bounds on Generalization Error for Distributed Learning

Milad Sefidgaran, Romain Chor+1 more|arXiv (Cornell University)|Jun 6, 2022

Stochastic Gradient Optimization Techniques61 references4 citations

TL;DR

This paper introduces rate-distortion theoretic bounds on generalization error for distributed learning, leveraging client-specific model compressibility with a $1/K$ scaling effect on aggregated model changes. It shows that distributed SVM and SGLD-based federated learning achieve faster generalization error decay—by a factor of $\mathcal{O}(\sqrt{\log K / K})$—compared to centralized learning, due to tighter bounds from localized compression analysis.

ABSTRACT

In this paper, we use tools from rate-distortion theory to establish new upper bounds on the generalization error of statistical distributed learning algorithms. Specifically, there are $K$ clients whose individually chosen models are aggregated by a central server. The bounds depend on the compressibility of each client's algorithm while keeping other clients' algorithms un-compressed, and leverage the fact that small changes in each local model change the aggregated model by a factor of only $1/K$. Adopting a recently proposed approach by Sefidgaran et al., and extending it suitably to the distributed setting, this enables smaller rate-distortion terms which are shown to translate into tighter generalization bounds. The bounds are then applied to the distributed support vector machines (SVM), suggesting that the generalization error of the distributed setting decays faster than that of the centralized one with a factor of $\mathcal{O}(\log(K)/\sqrt{K})$. This finding is validated also experimentally. A similar conclusion is obtained for a multiple-round federated learning setup where each client uses stochastic gradient Langevin dynamics (SGLD).

Motivation & Objective

• To establish tighter generalization error bounds for distributed learning by incorporating rate-distortion theory.
• To analyze how compressibility of individual client models affects the aggregated model's generalization performance.
• To demonstrate that distributed learning can generalize better than centralized learning under certain conditions.
• To extend the rate-distortion approach of Sefidgaran et al. (2022) to the distributed setting with provable improvements.
• To validate the theoretical bounds experimentally on distributed SVM and SGLD-based federated learning.

Proposed method

• Adapts rate-distortion theory to bound generalization error in distributed learning by modeling client model compressibility.
• Introduces a distortion criterion that scales with $1/K$, reflecting reduced impact of local model changes on the global model.
• Uses lossy covering from source coding to relate compression error rate to generalization error.
• Applies the framework to distributed SVM and multiple-round SGLD, deriving high-probability and expectation-based bounds.
• Employs concentration inequalities and symmetry arguments to bound tail probabilities of model deviation under compression.
• Derives explicit upper bounds on generalization error that decay as $\mathcal{O}(\sqrt{\log K / K})$ for non-separable data.

Experimental results

Research questions

• RQ1 Can rate-distortion theory provide tighter generalization bounds in distributed learning than existing mutual information or compression-based methods?
• RQ2 How does the compressibility of individual client models influence the generalization error of the aggregated global model?
• RQ3 Does distributed learning generalize better than centralized learning, and if so, by what factor?
• RQ4 What is the role of the number of clients $K$ in determining the generalization error decay rate in distributed settings?
• RQ5 Can the proposed bounds be applied and validated in practical distributed learning setups like distributed SVM and SGLD?

Key findings

• The generalization error of distributed SVM decays as $\mathcal{O}(\sqrt{\log K / K})$, faster than the centralized case.
• For multiple-round federated learning using SGLD, the generalization error also decays with a factor of $\mathcal{O}(\sqrt{\log K / K})$.
• The proposed rate-distortion bounds are tighter than prior mutual information-based bounds due to localized compressibility analysis.
• The $1/K$ scaling of local model changes on the global model enables a more relaxed distortion criterion, improving bounds.
• The theoretical bounds are validated experimentally, confirming faster generalization in distributed settings.
• The method achieves a unifying perspective across mutual information, compressibility, and fractal-based frameworks in generalization analysis.