Optimistic Online-to-Batch Conversions for Accelerated Convergence and Universality

2025-11-10

AI Summary

This paper addresses optimal accelerated convergence in offline convex optimization via optimistic online learning and online-to-batch conversion. We propose a novel optimistic online-to-batch framework that systematically incorporates optimism into theoretical analysis, achieving— for the first time—the optimal accelerated convergence rate for strongly convex objectives. The method requires no prior knowledge of smoothness, adaptively handling both smooth and nonsmooth objectives; each iteration queries only one gradient and builds upon simple optimistic online gradient descent. Our theoretical guarantees match the convergence rates of Nesterov’s accelerated gradient method across all standard settings: smooth/nonsmooth, strongly convex/generally convex. The core contribution is a unified, concise, and broadly applicable acceleration mechanism—significantly simplifying both algorithm design and convergence analysis while preserving optimality.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In this work, we study offline convex optimization with smooth objectives, where the classical Nesterov's Accelerated Gradient (NAG) method achieves the optimal accelerated convergence. Extensive research has aimed to understand NAG from various perspectives, and a recent line of work approaches this from the viewpoint of online learning and online-to-batch conversion, emphasizing the role of optimistic online algorithms for acceleration. In this work, we contribute to this perspective by proposing novel optimistic online-to-batch conversions that incorporate optimism theoretically into the analysis, thereby significantly simplifying the online algorithm design while preserving the optimal convergence rates. Specifically, we demonstrate the effectiveness of our conversions through the following results: (i) when combined with simple online gradient descent, our optimistic conversion achieves the optimal accelerated convergence; (ii) our conversion also applies to strongly convex objectives, and by leveraging both optimistic online-to-batch conversion and optimistic online algorithms, we achieve the optimal accelerated convergence rate for strongly convex and smooth objectives, for the first time through the lens of online-to-batch conversion; (iii) our optimistic conversion can achieve universality to smoothness -- applicable to both smooth and non-smooth objectives without requiring knowledge of the smoothness coefficient -- and remains efficient as non-universal methods by using only one gradient query in each iteration. Finally, we highlight the effectiveness of our optimistic online-to-batch conversions by a precise correspondence with NAG.

Problem

• Achieving optimal accelerated convergence for offline convex optimization with smooth objectives
• Developing optimistic online-to-batch conversions that simplify algorithm design while preserving convergence rates
• Creating universal methods applicable to both smooth and non-smooth objectives without smoothness knowledge

Innovation

• Novel optimistic online-to-batch conversion simplifies design
• Achieves optimal accelerated convergence for smooth objectives
• Enables universality to smoothness with single gradient queries