Impossible Tuning Made Possible: A New Expert Algorithm and Its Applications
This solves a long-standing theoretical problem in online learning, enabling improved regret bounds and new results for expert and online linear optimization problems.
The paper resolves the 'impossible tuning' issue in the expert problem by achieving regret O(√((ln d)∑_t ℓ_{t,i}^2)) for all experts, and generalizes this to O(√((ln d)∑_t (ℓ_{t,i} - m_{t,i})^2)) with a master algorithm that combines base algorithms with minimal overhead.
We resolve the long-standing "impossible tuning" issue for the classic expert problem and show that, it is in fact possible to achieve regret $O\left(\sqrt{(\ln d)\sum_t \ell_{t,i}^2}\right)$ simultaneously for all expert $i$ in a $T$-round $d$-expert problem where $\ell_{t,i}$ is the loss for expert $i$ in round $t$. Our algorithm is based on the Mirror Descent framework with a correction term and a weighted entropy regularizer. While natural, the algorithm has not been studied before and requires a careful analysis. We also generalize the bound to $O\left(\sqrt{(\ln d)\sum_t (\ell_{t,i}-m_{t,i})^2}\right)$ for any prediction vector $m_t$ that the learner receives, and recover or improve many existing results by choosing different $m_t$. Furthermore, we use the same framework to create a master algorithm that combines a set of base algorithms and learns the best one with little overhead. The new guarantee of our master allows us to derive many new results for both the expert problem and more generally Online Linear Optimization.