Proximal Online Gradient Is Optimum for Dynamic Regret: A General Lower Bound

Area of Science:

• Machine Learning
• Online Learning Algorithms
• Optimization Theory

Background:

• Dynamic regret is crucial for online learning applications where reference solutions change over time, such as in evolving customer preferences for online recommendations.
• The optimal algorithm for dynamic regret remains an open question, unlike static regret where online gradient (OG) methods are known to be optimal.

Purpose of the Study:

• To determine the optimal algorithm for minimizing dynamic regret in online learning settings.
• To establish a new, general lower bound for dynamic regret to better understand the challenges of adapting to changing data.
• To demonstrate the optimality of proximal OG for dynamic regret.

Main Methods:

• Proving a new, general lower bound for dynamic regret.
• Establishing an upper bound for the proximal online gradient (OG) algorithm.
• Demonstrating that the upper bound achieved by proximal OG matches the proved lower bound.

Main Results:

• Proximal OG achieves an optimal dynamic regret, matching the newly established general lower bound.
• The study provides a theoretical foundation for the performance of proximal OG in dynamic online learning.
• A new general lower bound for dynamic regret is presented, offering deeper insights into online adaptation.

Conclusions:

• Proximal OG is identified as the optimal algorithm for dynamic regret in online learning.
• The findings enhance the understanding of the inherent difficulty in tracking dynamic changes in online environments.
• This research paves the way for more effective online learning strategies in adaptive systems.