Discounted regret measures the performance of an online learner by assigning exponentially decaying weights to past losses, reflecting the greater importance of recent losses in non-stationary online learning.
While previous work obtains uniform discounted regret bounds over a continuous range of discount factors using a specialized meta-algorithm, its results are limited to convex functions.
In this paper, we demonstrate that strongly adaptive regret bounds can directly imply uniform discounted regret guarantees for convex, strongly convex, and exp-concave loss functions.
Our key observation is that discounted regret against a fixed comparator admits an exact decomposition into a convex combination of interval regrets.
For convex functions, this reduction yields an $O(\sqrt{\log T/(1-λ)})$ discounted regret bound for all discount factors $λ\in(0,1)$, matching the previous rate over a broader range of discount factors.
For strongly convex and exp-concave functions, we provide regret bounds of $O\left(\log T\left(1+\log\frac{1}{1-λ}\right)\right)$.