Gradient-Variation Bound for Online Convex Optimization with Constraints

• URL: http://arxiv.org/abs/2006.12455v2
• Date: Tue, 23 Aug 2022 13:12:53 GMT
• Title: Gradient-Variation Bound for Online Convex Optimization with Constraints
• Authors: Shuang Qiu, Xiaohan Wei, Mladen Kolar
• Abstract summary: We study online convex optimization with constraints consisting of multiple functional constraints and a relatively simple constraint set, such as a Euclidean ball. First-order methods achieve an $mathcalO(sqrtT)$ regret and an $mathcalO(1)$ constraint violation, but do not take into account the structural information of the problem. In this paper, we provide an emphinstance-dependent bound for online convex optimization with complex constraints obtained by a novel online primal-dual mirror-prox algorithm.
• Score: 25.002868073267464
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study online convex optimization with constraints consisting of multiple functional constraints and a relatively simple constraint set, such as a Euclidean ball. As enforcing the constraints at each time step through projections is computationally challenging in general, we allow decisions to violate the functional constraints but aim to achieve a low regret and cumulative violation of the constraints over a horizon of $T$ time steps. First-order methods achieve an $\mathcal{O}(\sqrt{T})$ regret and an $\mathcal{O}(1)$ constraint violation, which is the best-known bound, but do not take into account the structural information of the problem. Furthermore, the existing algorithms and analysis are limited to Euclidean space. In this paper, we provide an \emph{instance-dependent} bound for online convex optimization with complex constraints obtained by a novel online primal-dual mirror-prox algorithm. Our instance-dependent regret is quantified by the total gradient variation $V_*(T)$ in the sequence of loss functions. The proposed algorithm works in \emph{general} non-Euclidean spaces and simultaneously achieves an $\mathcal{O}(\sqrt{V_*(T)})$ regret and an $\mathcal{O}(1)$ constraint violation, which is never worse than the best-known $( \mathcal{O}(\sqrt{T}), \mathcal{O}(1) )$ result and improves over previous works that applied mirror-prox-type algorithms for this problem achieving $\mathcal{O}(T^{2/3})$ regret and constraint violation. Finally, our algorithm is computationally efficient, as it only performs mirror descent steps in each iteration instead of solving a general Lagrangian minimization problem.

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