Online Convex Optimization with Classical and Quantum Evaluation Oracles

As a fundamental tool in AI, convex optimization has been a significant research field for many years, and the same goes for its online version. Recently, general convex optimization problem has been accelerated with the help of quantum computing, and the technique of online convex optimization has been used for accelerating the online quantum state learning problem, thus we want to study whether online convex optimization (OCO) model can also be benefited from quantum computing. In this paper, we consider the OCO model, which can be described as a $T$ round iterative game between the player and the adversary. A key factor for measuring the performance of an OCO algorithm ${\cal A}$ is the regret denoted by $\text{regret}_{T}(\mathcal{A})$, and it is said to perform well if its regret is sublinear as a function of $T$. Another factor is the computational cost (e.g., query complexity) of the algorithm. We give a quantum algorithm for the online convex optimization model with only zeroth-order oracle available, which can achieve $O(\sqrt{T})$ and $O(\log{T})$ regret for general convex loss functions and $\alpha$-strong loss functions respectively, where only $O(1)$ queries are needed in each round. Our results show that the zeroth-order quantum oracle is as powerful as the classical first-order oracle, and show potential advantages of quantum computing over classical computing in the OCO model where only zeroth-order oracle available.