revenue management teams

Abstract: 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 $\tilde{O}(\sqrt{nkT})$ for supports of size at most $k$ without any level-set condition, which improves the previous bound $\tilde{O}(n(\sqrt{kT}+k))$ for FSD-monotone problems. We also prove a lower bound of $Ω(\min\{T,\sqrt{ST}\})$ for all positive integers $s_1,\ldots,s_n$ with $k_i=2s_i+1$, which holds even under full feedback and matches the upper bound up to logarithmic factors. We obtain $\tilde{O}(\sqrt{nT})$ regret for these problems, all nonmonotone under FSD: stochastic Boolean function evaluation, the full-information secretary problem, optimal stopping with loss aversion, scheduling with deadlines and correlated stochastic knapsack with unimodal rewards, and overbooking with group reservations.

Abstract: We show upper and lower bounds on the regret of $m$-set adversarial bandits for different utilities (winner reward or sum of rewards) and feedback models (winner index, winner reward, sum of rewards, and their combinations). By comparing to standard bounds for combinatorial and MNL bandits, our results reveal how subtle changes in the setting can have a dramatic impact on the learning rates. Our main technical contributions are the information-theoretic lower bounds on the regret. Experiments on synthetic data confirm our theoretical analyses.

Abstract: This paper proposes a new functional approximation approach to reduce Bellman error in high-dimensional dynamic programming and Reinforcement Learning problems. Using a classic dynamic programming problem (network capacity control in revenue management) as the motivational example, the paper illustrates that deep neural networks and linear programming approximation algorithms can be combined to derive approximate solutions to dynamic programming problems. Simulation results show the proposed approximation algorithms achieves competitive performance when compared with benchmark.