Beyond Monotonicity: Stochastic Online Optimization under Semi-Bandit Feedback

Tsubasa Harada, Yasushi Kawase, Hanna Sumita

We study stochastic online optimization (SOO) under semi-bandit feedback: in each of $T$ rounds, a policy adaptively observes $n$ independent random variables with unknown distributions, learning only from observed values. First-order stochastic dominance (FSD) monotonicity enables general reductions for learning SOO, but can fail when observations change future feasibility, resource use, or reference points. Our algorithm runs the dynamic program for the optimal policy, replacing each expectation by its most optimistic value over a confidence set of distributions. Its regret depends on the level-set complexity $s_i$ of the computed values as functions of the $i$th variable: the number of intervals needed to describe their superlevel sets. For rewards in $[0,1]$, the regret is $\tilde{O}(\sqrt{ST})$ without this monotonicity, where $S=\sum_{i=1}^n\min\{s_i,k_i\}$ and $k_i$ is the support size of the $i$th distribution, which may be infinite when $s_i$ is finite. For constant level-set complexities, regret is $\tilde{O}(\sqrt{nT})$, even for continuous distributions. The algorithm requires no knowledge of $s_i$, $k_i$, or the supports, and it also achieves regret…