Optimal Decentralized Smoothed Online Convex Optimization

TL;DR

This paper addresses decentralized Smoothed Online Convex Optimization (SOCO) over dynamic networks, where each agent incurs a strong hitting cost, a switching penalty, and a dissimilarity penalty with neighbors. It introduces ACORD, a fully decentralized algorithm based on alternating coupled online regularized descent, which uses local computations and 1-hop communications to decouple the joint objective and achieve asymptotic optimality, with a competitive ratio approaching $CR_{*} = \tfrac{1}{2} + \tfrac{1}{2}\sqrt{1+\tfrac{4}{\min_i \mu_i}}$ as the horizon grows. The paper provides a comprehensive analysis of finite-time performance via a tunable $K_t = \mathcal{O}(\log T)$, graph-dependent iteration complexity (e.g., $K_t = \Theta(\mathcal{D}\log (N\mathcal{D}^2 T^4))$ for $\mathcal{D}$-regular graphs), and a lower-bound framework confirming asymptotic optimality. It also contrasts ACORD with LPC, showing that LPC can only match ACORD under full-network access and predictions, while ACORD delivers substantially lower computational and communication costs in practice, as corroborated by extensive numerical experiments across diverse network topologies. The results advance decentralized online optimization by enabling scalable, bias-free performance without exchanging hitting costs, and by providing a principled way to handle time-varying graphs and non-smooth costs.

Abstract

We study the multi-agent Smoothed Online Convex Optimization (SOCO) problem, where $N$ agents interact through a communication graph. In each round, each agent $i$ receives a strongly convex hitting cost function $f^i_t$ in an online fashion and selects an action $x^i_t \in \mathbb{R}^d$. The objective is to minimize the global cumulative cost, which includes the sum of individual hitting costs $f^i_t(x^i_t)$, a temporal "switching cost" for changing decisions, and a spatial "dissimilarity cost" that penalizes deviations in decisions among neighboring agents. We propose the first truly decentralized algorithm ACORD for multi-agent SOCO that provably exhibits asymptotic optimality. Our approach allows each agent to operate using only local information from its immediate neighbors in the graph. For finite-time performance, we establish that the optimality gap in the competitive ratio decreases with time horizon $T$ and can be conveniently tuned based on the per-round computation available to each agent. Our algorithm benefits from a provably scalable computational complexity that depends only logarithmically on the number of agents and almost linearly on their degree within the graph. Moreover, our results hold even when the communication graph changes arbitrarily and adaptively over time. Finally, ACORD, by virtue of its asymptotic-optimality, is shown to be provably superior to the state-of-the-art LPC algorithm, while exhibiting much lower computational complexity. Extensive numerical experiments across various network topologies further corroborate our theoretical claims.

Figures (6)

• Figure 1: All algorithms' performance for $N=20$, $\mathcal{D}=2$, $T=20$ and $\beta=50$
• Figure 2: $N=20$, $\mathcal{D}=2$, $T=20$ and different $\beta$ values.
• Figure 3: $N=40$, $\beta=50$, $T=20$ and different $\mathcal{D}$ values.
• Figure 4: acord v/s lpc for $T=20$, $\beta=50$, and different $N$ and $\mathcal{D}$. The trend varies with how $\frac{\mathcal{D}}{N}$ is set.
• Figure 5: acord v/s lpc for $N=50$, $T=20$, $\beta=50$, and different $r$ and $\mathcal{D}$.

Theorems & Definitions (38)

• Definition 2.1: Dissimilarity Costs
• Remark 2.2
• Definition 2.3: Adaptive Graph
• Definition 2.4: Asymptotic Competitive Ratio
• Theorem 3.1
• Remark 3.2
• Theorem 3.3
• Remark 3.4
• Corollary 3.5
• Definition 3.6