In this paper, we analyze the problem of online convex optimization in different settings. We show that any algorithm for online linear optimization with fully adaptive adversaries is an algorithm for online convex optimization. We also show that any such algorithm that requires full-information feedback may be transformed to an algorithm with semi-bandit feedback with comparable regret bound. We further show that algorithms that are designed for fully adaptive adversaries using deterministic semi-bandit feedback can obtain similar bounds using only stochastic semi-bandit feedback when facing oblivious adversaries. We use this to describe general meta-algorithms to convert first order algorithms to zeroth order algorithms with comparable regret bounds. Our framework allows us to analyze online optimization in various settings, such full-information feedback, bandit feedback, stochastic regret, adversarial regret and various forms of non-stationary regret. Using our analysis, we provide the first efficient projection-free online convex optimization algorithm using linear optimization oracles.

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