Distributed Optimization with Streaming Data: A Temporal Weighting Perspective

2026-08-10

AI Summary

This work addresses distributed optimization over decentralized streaming data in dynamic environments, with the goal of tracking time-varying learning targets. The authors propose a unified modeling framework based on time-weighted average loss and, for the first time, decompose the tracking error into a fixed-point tracking term and a bias term arising from decentralization and data heterogeneity. Explicit error bounds are derived for common weighting strategies—uniform, exponential discounting, and sliding window—revealing that uniform weighting achieves an $O(1/t)$ decay rate in tracking error, whereas both discounting and sliding window approaches incur a non-vanishing error floor dictated by their effective memory length. Experimental results corroborate the theoretical predictions.

Abstract

Optimization theory is a widely used tool for intelligent decision-making. While classical optimization deals with fixed, time-invariant objective functions, many modern applications operate in dynamic environments where data arrive sequentially, and the learning objective evolves over time, often under decentralized data and communication constraints. Motivated by these trends, we study decentralized optimization from streaming data through a structured time-varying formulation in which the global objective is a temporally weighted average of losses observed across the network. We analyze multi-iteration decentralized first-order methods, including decentralized gradient descent. For strongly convex and smooth losses, we develop guarantees for the Euclidean-norm \emph{tracking error} through a contraction-mapping viewpoint. The resulting bounds decompose the tracking error into a fixed-point tracking component and a bias term induced by decentralization and data heterogeneity. We specialize our analysis to uniform and exponentially discounted weights, as well as their finite-memory \emph{windowed} counterparts. The bounds explicitly characterize the roles of the temporal weighting rule, per-step iteration budget, step size, and network connectivity. Uniform weighting yields a vanishing fixed-point tracking contribution of order $\mathcal O(1/t)$, whereas discounted and windowed strategies generally induce non-vanishing tracking floors governed by the discount factor and effective memory, respectively. In all cases, decentralization induces an additional non-zero bias floor under a constant step size. Numerical experiments illustrate the predicted trends.

Problem

• distributed optimization
• streaming data
• time-varying objective
• decentralized learning
• temporal weighting

Innovation

• temporal weighting
• decentralized optimization
• tracking error
• streaming data
• contraction mapping