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We develop the first general semi-bandit algorithm that simultaneouslyachieves $\mathcal{O}(\log T)$ regret for stochastic environments and$\mathcal{O}(\sqrt{T})$ regret for adversarial environments without knowledgeof the regime or the number of rounds $T$. The leading problem-dependentconstants of our bounds are not only optimal in some worst-case sense studiedpreviously, but also optimal for two concrete instances of semi-banditproblems. Our algorithm and analysis extend the recent work of (Zimmert &Seldin, 2019) for the special case of multi-armed bandit, but importantlyrequires a novel hybrid regularizer designed specifically for semi-bandit.Experimental results on synthetic data show that our algorithm indeed performswell uniformly over different environments. We finally provide a preliminaryextension of our results to the full bandit feedback.

Beating Stochastic and Adversarial Semi-bandits Optimally and Simultaneously