No-Regret Algorithms

• No-regret algorithms are online decision-making techniques that guarantee the cumulative cost grows sublinearly compared to the best fixed policy in hindsight.
• They are fundamental to fields like online convex optimization, adaptive control, and game theory, providing robust performance in dynamic or adversarial environments.
• Their theoretical guarantees enable practical applications in AI and economics by ensuring learning efficacy and fostering equilibrium in multi-agent systems.

A no-regret algorithm is an online decision-making procedure that ensures the cumulative cost (or regret) incurred by the learner, as compared to the best possible fixed decision or policy in hindsight, grows sublinearly with time. This property enables the algorithm to asymptotically match the performance of the optimal offline strategy, even under adversarial or nonstationary environments. No-regret algorithms are foundational to online convex optimization, adaptive control, online combinatorial optimization, and repeated games, and are essential to theoretical and practical advances in both AI and economics. They underpin guarantees for learning efficacy, strategic robustness, and the emergence of equilibria in multi-agent systems.

1. Formal Regret Notions and Definitions

No-regret algorithms are defined relative to a regret metric that compares the cumulative reward (