Distributed Forgetting-factor Regret-based Online Optimization over Undirected Connected Networks

TL;DR

The paper tackles the challenge of assessing final-iteration performance in distributed online optimization by introducing distributed forgetting-factor regret (DFFR). It develops two algorithms—gradient-free for bandit feedback and projection-free for high-dimensional problems—on undirected connected networks and derives upper bounds showing that DFFR $R_T^F$ can converge to zero or remain tightly bounded as $T\to\infty$. The gradient-free method employs a $\delta$-smoothed gradient estimate, while the projection-free method adopts a Frank-Wolfe-like update; both are analyzed under mild convexity and network-connectivity assumptions. Numerical simulations demonstrate fast consensus and effective tracking, with the projection-free approach achieving faster convergence than the gradient-free one and DFFR providing a stronger signal of tracking performance than classical regret. Overall, the work advances scalable, information-efficient distributed online optimization with rigorous performance guarantees.

Abstract

The evaluation of final-iteration tracking performance is a formidable obstacle in distributed online optimization algorithms. To address this issue, this paper proposes a novel evaluation metric named distributed forgetting-factor regret (DFFR). It incorporates a weight into the loss function at each iteration, which progressively reduces the weights of historical loss functions while enabling dynamic weights allocation across optimization horizon. Furthermore, we develop two distributed online optimization algorithms based on DFFR over undirected connected networks: the Distributed Online Gradient-free Algorithm for bandit-feedback problems and the Distributed Online Projection-free Algorithm for high-dimensional problems. Through theoretical analysis, we derive the upper bounds of DFFR for both algorithms and further prove that under mild conditions, DFFR either converges to zero or maintains a tight upper bound as iterations approach infinity. Experimental simulation demonstrates the effectiveness of the algorithms and the superior performance of DFFR.

Figures (6)

• Figure 1: The topological structure of undirected graphs.
• Figure 2: The decisions of Algorithm 1.
• Figure 3: The decisions of Algorithm 2.
• Figure 4: DFFR of Algorithms.
• Figure 5: $\frac{1}{n}\sum\limits_{i=1}^{n}{\left\| x_{i}^{t}-x_{*}^{t} \right\|}$.

Theorems & Definitions (10)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Theorem 1
• proof
• proof
• Theorem 2
• proof