Optimistic Algorithms for Safe Linear Bandits Under General Constraints

The stochastic linear bandit problem has emerged as a fundamental building-block in machine learning and control, and a realistic model for many applications. By equipping this classical problem with safety constraints, the safe linear bandit problem further broadens its relevance to safety-critical applications. However, most existing algorithms for safe linear bandits only consider linear constraints, making them inadequate for many real-world applications, which often have non-linear constraints. To alleviate this limitation, we study the problem of safe linear bandits under general (non-linear) constraints. Under a novel constraint regularity condition that is weaker than convexity, we give two algorithms with $\tilde{\mathcal {O}}(d \sqrt{T})$ regret. We then give efficient implementations of these algorithms for several specific settings. Lastly, we give simulation results demonstrating the effectiveness of our algorithms in choosing dynamic pricing signals for a demand response problem under distribution power flow constraints.