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Optimized Gradient Tracking for Decentralized Online Learning

Shivangi Dubey Sharma, Ketan Rajawat

TL;DR

The paper tackles decentralized online learning where $n$ nodes collaboratively track the time-varying minimizer of $f^k(x)=\frac{1}{n}\sum_{i=1}^n f_i^k(x)$. It introduces Generalized Gradient Tracking (GGT), a unified framework that blends consensus and gradient-tracking updates, and provides a novel SDP-based analysis yielding dynamic regret bounds without assuming gradient boundedness. A condensed, four-parameter version, oGGT, is developed and can be tuned offline to minimize regret, resulting in improved theoretical guarantees and empirical performance over state-of-the-art decentralized online algorithms. The work demonstrates the method’s effectiveness via synthetic target-tracking and real-room occupancy experiments, establishing oGGT as a practical and near-optimal template for dynamic decentralized learning with robust regret guarantees.

Abstract

This work considers the problem of decentralized online learning, where the goal is to track the optimum of the sum of time-varying functions, distributed across several nodes in a network. The local availability of the functions and their gradients necessitates coordination and consensus among the nodes. We put forth the Generalized Gradient Tracking (GGT) framework that unifies a number of existing approaches, including the state-of-the-art ones. The performance of the proposed GGT algorithm is theoretically analyzed using a novel semidefinite programming-based analysis that yields the desired regret bounds under very general conditions and without requiring the gradient boundedness assumption. The results are applicable to the special cases of GGT, which include various state-of-the-art algorithms as well as new dynamic versions of various classical decentralized algorithms. To further minimize the regret, we consider a condensed version of GGT with only four free parameters. A procedure for offline tuning of these parameters using only the problem parameters is also detailed. The resulting optimized GGT (oGGT) algorithm not only achieves improved dynamic regret bounds, but also outperforms all state-of-the-art algorithms on both synthetic and real-world datasets.

Optimized Gradient Tracking for Decentralized Online Learning

TL;DR

The paper tackles decentralized online learning where nodes collaboratively track the time-varying minimizer of . It introduces Generalized Gradient Tracking (GGT), a unified framework that blends consensus and gradient-tracking updates, and provides a novel SDP-based analysis yielding dynamic regret bounds without assuming gradient boundedness. A condensed, four-parameter version, oGGT, is developed and can be tuned offline to minimize regret, resulting in improved theoretical guarantees and empirical performance over state-of-the-art decentralized online algorithms. The work demonstrates the method’s effectiveness via synthetic target-tracking and real-room occupancy experiments, establishing oGGT as a practical and near-optimal template for dynamic decentralized learning with robust regret guarantees.

Abstract

This work considers the problem of decentralized online learning, where the goal is to track the optimum of the sum of time-varying functions, distributed across several nodes in a network. The local availability of the functions and their gradients necessitates coordination and consensus among the nodes. We put forth the Generalized Gradient Tracking (GGT) framework that unifies a number of existing approaches, including the state-of-the-art ones. The performance of the proposed GGT algorithm is theoretically analyzed using a novel semidefinite programming-based analysis that yields the desired regret bounds under very general conditions and without requiring the gradient boundedness assumption. The results are applicable to the special cases of GGT, which include various state-of-the-art algorithms as well as new dynamic versions of various classical decentralized algorithms. To further minimize the regret, we consider a condensed version of GGT with only four free parameters. A procedure for offline tuning of these parameters using only the problem parameters is also detailed. The resulting optimized GGT (oGGT) algorithm not only achieves improved dynamic regret bounds, but also outperforms all state-of-the-art algorithms on both synthetic and real-world datasets.
Paper Structure (22 sections, 11 theorems, 83 equations, 6 figures, 3 tables, 3 algorithms)

This paper contains 22 sections, 11 theorems, 83 equations, 6 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Review of decentralized optimization algorithms
  4. Review of decentralized online learning algorithms
  5. Unified Algorithm
  6. Regret Rate Analysis
  7. Assumptions
  8. Compact Form
  9. Preliminary Results
  10. Regret Analysis
  11. Optimized GGT
  12. Numerical Analysis
  13. Tracking Time Varying Target via Least-Squares
  14. Real Data : Online learning
  15. Conclusion
  16. ...and 7 more sections

Key Result

Lemma 1

If a function $f_i^k$ satisfies Assumption a1 such that $L_i{\mathbf{I}}_d - {\mathbf{M}}_i$ is invertible, then it holds that Where ${\mathbf{u}} = \nabla f_i^k({\mathbf{x}})$, ${\mathbf{v}} = \nabla f_i^k({\mathbf{y}})$, and ${\mathbf{L}}_{{\mathbf{M}}_i} = L_i {\mathbf{I}}_d - {\mathbf{M}}_i$, for all ${\mathbf{x}}$, ${\mathbf{y}} \in {\mathbb{R}}^d$.

Figures (6)

  • Figure 1: Block diagram for hyperparameter tuning of GGT
  • Figure 2: Convergence rate parameter ($\rho$) of various algorithms against the step-size parameter ($\eta$). The dashed line for oGGT corresponds to the optimal $\rho$ obtained after tuning all its hyperparameters.
  • Figure 3: Regret rates of the GGT variants for target tracking
  • Figure 4: Regret rates of state-of-the-art algorithms and oGGT for target tracking
  • Figure 5: Regret rates of the GGT variants for room occupancy prediction
  • ...and 1 more figures

Theorems & Definitions (12)

  • Lemma 1
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • proof
  • Corollary 3
  • Theorem 3
  • ...and 2 more