What Doubling Tricks Can and Can't Do for Multi-Armed Bandits
This work addresses the problem of designing anytime algorithms for researchers and practitioners in online reinforcement learning, providing theoretical insights into when doubling tricks are effective, but it is incremental as it builds on and generalizes prior known results.
The paper investigates the capabilities and limitations of doubling tricks for converting non-anytime algorithms into anytime algorithms in multi-armed bandits, showing that geometric doubling tricks conserve minimax regret bounds of O(√T) but not distribution-dependent bounds of O(log T), while exponential doubling tricks can conserve O(log T) bounds and nearly conserve O(√T) bounds.
An online reinforcement learning algorithm is anytime if it does not need to know in advance the horizon T of the experiment. A well-known technique to obtain an anytime algorithm from any non-anytime algorithm is the "Doubling Trick". In the context of adversarial or stochastic multi-armed bandits, the performance of an algorithm is measured by its regret, and we study two families of sequences of growing horizons (geometric and exponential) to generalize previously known results that certain doubling tricks can be used to conserve certain regret bounds. In a broad setting, we prove that a geometric doubling trick can be used to conserve (minimax) bounds in $R\_T = O(\sqrt{T})$ but cannot conserve (distribution-dependent) bounds in $R\_T = O(\log T)$. We give insights as to why exponential doubling tricks may be better, as they conserve bounds in $R\_T = O(\log T)$, and are close to conserving bounds in $R\_T = O(\sqrt{T})$.