New Projection-free Algorithms for Online Convex Optimization with Adaptive Regret Guarantees

• URL: http://arxiv.org/abs/2202.04721v3
• Date: Sun, 19 Mar 2023 10:11:15 GMT
• Title: New Projection-free Algorithms for Online Convex Optimization with Adaptive Regret Guarantees
• Authors: Dan Garber, Ben Kretzu
• Abstract summary: We present new efficient textitprojection-free algorithms for online convex optimization (OCO) Our algorithms are based on the textitonline gradient descent algorithm with a novel and efficient approach to computing so-called textitinfeasible projections We present algorithms which, using overall $O(T)$ calls to the separation oracle, guarantee $O(sqrtT)$ adaptive regret and $O(T3/4)$ adaptive expected regret.
• Score: 21.30065439295409
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present new efficient \textit{projection-free} algorithms for online convex optimization (OCO), where by projection-free we refer to algorithms that avoid computing orthogonal projections onto the feasible set, and instead relay on different and potentially much more efficient oracles. While most state-of-the-art projection-free algorithms are based on the \textit{follow-the-leader} framework, our algorithms are fundamentally different and are based on the \textit{online gradient descent} algorithm with a novel and efficient approach to computing so-called \textit{infeasible projections}. As a consequence, we obtain the first projection-free algorithms which naturally yield \textit{adaptive regret} guarantees, i.e., regret bounds that hold w.r.t. any sub-interval of the sequence. Concretely, when assuming the availability of a linear optimization oracle (LOO) for the feasible set, on a sequence of length $T$, our algorithms guarantee $O(T^{3/4})$ adaptive regret and $O(T^{3/4})$ adaptive expected regret, for the full-information and bandit settings, respectively, using only $O(T)$ calls to the LOO. These bounds match the current state-of-the-art regret bounds for LOO-based projection-free OCO, which are \textit{not adaptive}. We also consider a new natural setting in which the feasible set is accessible through a separation oracle. We present algorithms which, using overall $O(T)$ calls to the separation oracle, guarantee $O(\sqrt{T})$ adaptive regret and $O(T^{3/4})$ adaptive expected regret for the full-information and bandit settings, respectively.

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