No-regret algorithms guarantee that over time, your average performance matches the best fixed strategy in hindsight. This sounds like all no-regret algorithms are equivalent — they all converge to the same guarantee. Abdelraouf and Shamma (arXiv: 2603.03173) show they aren't.
Different no-regret algorithms facing identical environments can achieve different cumulative rewards. The guarantee is about asymptotic average performance; it says nothing about total reward along the way. The authors model generalized replicator dynamics as feedback systems and compare variants: standard, anticipatory, exponential, higher-order. The key result: anticipatory replicator dynamics globally dominate standard dynamics across all payoff settings. The minimum cumulative reward gap is exactly zero — anticipatory dynamics never do worse and sometimes do better.
The through-claim: an asymptotic guarantee hides transient inequality. “No regret” means no regret eventually. In finite time — the only time that exists in practice — the choice of algorithm matters. Two strategies that are equivalent in the limit can differ enormously on the path to the limit. The guarantee is about the destination; the experience is about the journey.
This has implications beyond game theory. Any system evaluated by asymptotic equivalence classes (convergence rates, Big-O complexity, consistency in statistics) faces the same issue: things that are “the same” in theory can be vastly different in practice, and the theory provides no guidance for choosing among them. The mathematics of limits is blind to the mathematics of paths.
Abdelraouf & Shamma, 2603.03173. Game theory / control theory / online learning.