Rectified Pessimistic-Optimistic Learning for Stochastic Continuum-armed Bandit with Constraints
This work addresses constrained optimization in continuous spaces for applications like robotics or resource allocation, but it is incremental as it builds on existing GP bandit methods with a penalty-based approach.
The paper tackles the problem of optimizing a black-box reward function subject to a black-box constraint in stochastic continuum-armed bandits, proposing the Rectified Pessimistic-Optimistic Learning (RPOL) framework, which achieves sublinear regret and minimal cumulative constraint violation, with experiments showing it outperforms baseline algorithms.
This paper studies the problem of stochastic continuum-armed bandit with constraints (SCBwC), where we optimize a black-box reward function $f(x)$ subject to a black-box constraint function $g(x)\leq 0$ over a continuous space $\mathcal X$. We model reward and constraint functions via Gaussian processes (GPs) and propose a Rectified Pessimistic-Optimistic Learning framework (RPOL), a penalty-based method incorporating optimistic and pessimistic GP bandit learning for reward and constraint functions, respectively. We consider the metric of cumulative constraint violation $\sum_{t=1}^T(g(x_t))^{+},$ which is strictly stronger than the traditional long-term constraint violation $\sum_{t=1}^Tg(x_t).$ The rectified design for the penalty update and the pessimistic learning for the constraint function in RPOL guarantee the cumulative constraint violation is minimal. RPOL can achieve sublinear regret and cumulative constraint violation for SCBwC and its variants (e.g., under delayed feedback and non-stationary environment). These theoretical results match their unconstrained counterparts. Our experiments justify RPOL outperforms several existing baseline algorithms.