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Improved Information Theoretic Generalization Bounds for Distributed and Federated Learning

L. P. Barnes, Alex Dytso, H. V. Poor

TL;DR

This work develops tighter information-theoretic generalization bounds for distributed and federated learning by explicitly decomposing contributions from each node. The authors derive $O\left(\frac{1}{K}\right)$-scaling bounds that depend on per-node mutual information $I(W_k;S_k)$ for linear/Bregman losses and extend to Lipschitz nonlinear losses, as well as privacy and communication constraints. They further extend the analysis to iterative multi-round SGD, establishing per-round bounds that aggregate to an overall $1/K^2$-dependent term in certain regimes, and validate the theory with simulations showing close agreement with the true generalization error. The results offer practical guarantees for distributed systems under privacy/communication limits and illuminate how increased node count can tighten generalization bounds beyond previous end-to-end analyses.

Abstract

We consider information-theoretic bounds on expected generalization error for statistical learning problems in a networked setting. In this setting, there are $K$ nodes, each with its own independent dataset, and the models from each node have to be aggregated into a final centralized model. We consider both simple averaging of the models as well as more complicated multi-round algorithms. We give upper bounds on the expected generalization error for a variety of problems, such as those with Bregman divergence or Lipschitz continuous losses, that demonstrate an improved dependence of $1/K$ on the number of nodes. These "per node" bounds are in terms of the mutual information between the training dataset and the trained weights at each node, and are therefore useful in describing the generalization properties inherent to having communication or privacy constraints at each node.

Improved Information Theoretic Generalization Bounds for Distributed and Federated Learning

TL;DR

This work develops tighter information-theoretic generalization bounds for distributed and federated learning by explicitly decomposing contributions from each node. The authors derive -scaling bounds that depend on per-node mutual information for linear/Bregman losses and extend to Lipschitz nonlinear losses, as well as privacy and communication constraints. They further extend the analysis to iterative multi-round SGD, establishing per-round bounds that aggregate to an overall -dependent term in certain regimes, and validate the theory with simulations showing close agreement with the true generalization error. The results offer practical guarantees for distributed systems under privacy/communication limits and illuminate how increased node count can tighten generalization bounds beyond previous end-to-end analyses.

Abstract

We consider information-theoretic bounds on expected generalization error for statistical learning problems in a networked setting. In this setting, there are nodes, each with its own independent dataset, and the models from each node have to be aggregated into a final centralized model. We consider both simple averaging of the models as well as more complicated multi-round algorithms. We give upper bounds on the expected generalization error for a variety of problems, such as those with Bregman divergence or Lipschitz continuous losses, that demonstrate an improved dependence of on the number of nodes. These "per node" bounds are in terms of the mutual information between the training dataset and the trained weights at each node, and are therefore useful in describing the generalization properties inherent to having communication or privacy constraints at each node.
Paper Structure (8 sections, 8 theorems, 44 equations, 2 figures)

This paper contains 8 sections, 8 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Technical Preliminaries
  3. Distributed Learning and Model Aggregation
  4. Improved Bounds
  5. General Models and Losses
  6. Privacy and Communication Constraints
  7. Iterative Algorithms
  8. Simulations

Key Result

Theorem 1

Let $S^{(i)} = (Z_1,\ldots,Z_i',\ldots,Z_n)$ be a version of $S$ with $Z_i$ replaced by an i.i.d. copy $Z_i'$. Denote $S' = (Z'_1,\ldots,Z'_n)$. Then

Figures (2)

  • Figure 1: The distributed learning setting with model aggregation.
  • Figure 2: Information theoretic upper bounds and expected generalization error for a simulated linear regression example in linear (top) and log (bottom) scales.

Theorems & Definitions (12)

  • Theorem 1: Leave-one-out Expansion -- Lemma 11 in SSSS
  • proof
  • Theorem 2: Theorem 2 in veeravalli
  • Theorem 3
  • proof
  • Theorem 4: Linear or Location Models with Bregman Loss
  • proof
  • Theorem 5: Lipschitz Continuous Loss
  • proof
  • Corollary 1: Privacy Constraints
  • ...and 2 more