Equipping Experts/Bandits with Long-term Memory
This work addresses a foundational challenge in online learning theory by providing the first reduction-based method for long-term memory, with implications for algorithm design in expert and bandit problems, though it is incremental in building on prior regret frameworks.
The paper tackles the problem of achieving long-term memory guarantees in online learning by proposing a reduction-based approach that converts it to switching regret, resulting in algorithms with regret bounds of order O(√T(S ln T + n ln K)) for expert problems and extensions to bandit settings. It resolves an open problem from 2014 and shows simultaneous adaptation to stochastic and adversarial environments.
We propose the first reduction-based approach to obtaining long-term memory guarantees for online learning in the sense of Bousquet and Warmuth, 2002, by reducing the problem to achieving typical switching regret. Specifically, for the classical expert problem with $K$ actions and $T$ rounds, using our framework we develop various algorithms with a regret bound of order $\mathcal{O}(\sqrt{T(S\ln T + n \ln K)})$ compared to any sequence of experts with $S-1$ switches among $n \leq \min\{S, K\}$ distinct experts. In addition, by plugging specific adaptive algorithms into our framework we also achieve the best of both stochastic and adversarial environments simultaneously. This resolves an open problem of Warmuth and Koolen, 2014. Furthermore, we extend our results to the sparse multi-armed bandit setting and show both negative and positive results for long-term memory guarantees. As a side result, our lower bound also implies that sparse losses do not help improve the worst-case regret for contextual bandits, a sharp contrast with the non-contextual case.