Decentralized Time-Varying Optimization for Streaming Data via Temporal Weighting

2026-05-07

AI Summary

This work addresses the challenge of tracking the minimizer of a time-varying global objective function in dynamic networks under streaming data, where communication and computation resources are limited. The authors analyze the tracking performance of decentralized gradient descent (DGD) when applied to objectives constructed via temporal weighting—either uniform or exponential discounting—with a fixed number of DGD iterations executed before each data update. Leveraging fixed-point theory, they characterize the tracking error as comprising a fixed-point tracking component and a bias term induced by data heterogeneity. Their analysis reveals that under uniform weighting, the error decays at an 𝒪(1/t) rate, whereas exponential discounting yields a non-vanishing error floor dictated by the discount factor. Moreover, with constant step sizes, decentralization inherently introduces additional bias. Numerical experiments corroborate the theoretical findings.

Abstract

Classical optimization theory largely focuses on fixed objective functions, whereas many modern learning systems operate in dynamic environments where data arrive sequentially and decisions must be updated continuously. In this work, we study optimization with streaming data over a distributed network of agents. We adopt a structured, weight-based formulation that explicitly captures the streaming-data origin of the time-varying objective: at each time step, every agent receives a new sample, and the network seeks to track the minimizer of a temporally weighted objective formed from all samples observed across the network so far. We focus on decentralized gradient descent (DGD) with a limited communication/computation budget, where at each time step, only a limited number of DGD iterations can be performed before the objective changes again. For strongly convex and smooth losses, we analyze the tracking error with respect to the time-varying minimizer through a fixed-point theory lens. Our analysis reveals that the tracking error decomposes into a fixed-point tracking term and a bias term induced by data heterogeneity across agents. We specialize the analysis to two natural weighting strategies: uniform weights, which treat all samples equally, and exponentially discounted weights, which geometrically decay the influence of older data. Under uniform weighting, DGD tracks the fixed-point at a rate $\mathcal{O}(1/t)$, whereas discounted weighting yields a non-vanishing fixed-point tracking floor controlled by the discount factor. In both cases, decentralization induces an additional non-zero bias floor under a constant step size. We validate our theoretical findings through numerical simulations.

Problem

• decentralized optimization
• time-varying optimization
• streaming data
• temporal weighting
• tracking error

Innovation

• decentralized optimization
• time-varying objectives
• streaming data
• temporal weighting
• tracking error analysis