Revisiting Multi-Agent Asynchronous Online Optimization with Delays: the Strongly Convex Case

TL;DR

This work addresses multi-agent asynchronous online optimization with delays under strongly convex losses. It introduces Follow-the-Delayed-Leader (FTDL) for full-information feedback and Approximate FTDL (AFTDL) for gradient feedback with surrogate losses, both achieving a regret of $O(d\log T)$ without knowing the maximum delay or feedback-order properties. Theoretical guarantees are established under bounded gradients and diameter assumptions, demonstrating that strong convexity enables tighter regret bounds and delay-free operation. Empirical results on multiple datasets show that AFTDL outperforms existing methods, highlighting practical benefits for distributed online decision making with unpredictable delays.

Abstract

We revisit multi-agent asynchronous online optimization with delays, where only one of the agents becomes active for making the decision at each round, and the corresponding feedback is received by all the agents after unknown delays. Although previous studies have established an $O(\sqrt{dT})$ regret bound for this problem, they assume that the maximum delay $d$ is knowable or the arrival order of feedback satisfies a special property, which may not hold in practice. In this paper, we surprisingly find that when the loss functions are strongly convex, these assumptions can be eliminated, and the existing regret bound can be significantly improved to $O(d\log T)$ meanwhile. Specifically, to exploit the strong convexity of functions, we first propose a delayed variant of the classical follow-the-leader algorithm, namely FTDL, which is very simple but requires the full information of functions as feedback. Moreover, to handle the more general case with only the gradient feedback, we develop an approximate variant of FTDL by combining it with surrogate loss functions. Experimental results show that the approximate FTDL outperforms the existing algorithm in the strongly convex case.

Figures (1)

• Figure 1: Comparisons of our A-FTDL against DDA on delayed $2$-agent online binary classification.

Theorems & Definitions (6)

• Theorem 1
• proof
• Lemma 1
• Lemma 2
• Theorem 2
• proof