Online Resource Allocation under Horizon Uncertainty

Online Allocation with Replenishable Budgets: Worst Case and Beyond

“…which, if satisfied at round 𝑡, guarantees the existence of at least one feasible decision that can still satisfy the constraint. In other words, if the decisions 𝑥 𝑡 are chosen out of the constrained set (8) for round 𝑡 ∈ [𝑇 ], worst-case utility constraint (7b) can be satisfied at the end of any sequence 𝑦 ∈ Y. To our knowledge, the design of Δ(𝑥 𝑡 ) for constructing a constrained decision set ( 8) is novel for online allocation with replenishable budgets and also differs from many prior learningaugmented algorithms (e.g., [5] uses a pre-determined threshold for dynamically switching between ML prediction x𝑡 and the worst-case robust action 𝑥 † 𝑡 ).…”

“…In LA-OACP, the competitive algorithm (i.e., 𝜋 † ) runs independently for the purpose of bounding the worst-case utility constraint (7b), and the ML predictor π takes the actual online information 𝑦 1:𝑡 (including the actual remaining budget 𝐵 𝑡 and replenishment 𝐸 𝑡 ) as its input and generates its prediction x𝑡 as advice to LA-OACP. Then, x𝑡 is projected into a constrained decision set (8) to find the actual decision 𝑥 𝑡 that guarantees the worst-case utility constraint. The purpose of the projection in LA-OACP is to ensure that 𝑥 𝑡 is both close to the ML prediction x𝑡 to exploit its potential for improving the average utility, while still staying inside the constrained decision set (8) for worst-case utility constraint (7b).…”

“…Then, x𝑡 is projected into a constrained decision set (8) to find the actual decision 𝑥 𝑡 that guarantees the worst-case utility constraint. The purpose of the projection in LA-OACP is to ensure that 𝑥 𝑡 is both close to the ML prediction x𝑡 to exploit its potential for improving the average utility, while still staying inside the constrained decision set (8) for worst-case utility constraint (7b).…”

“…As formally stated below, our results highlight that LA-OACP guarantees the worst-case utility constraint for any sequence 𝑦 ∈ Y and meanwhile is able to exploit the benefits of ML predictions to improve the average utility. The proof of Theorem 4.1 is available in Appendix C and shows that, based on the design of Δ(𝑥 𝑡 ), if (8) is satisfied for round 𝑡, then there must always exist a feasible solution satisfying (8) for round 𝑡 + 1.…”

“…As such, given an optimally trained ML model, we measure the average-case consistency of LA-OACP with respect to the optimal unconstrained ML model π * in terms of the average utility difference between LA-OACP and π * . Our consideration of an optimal ML model essentially parallels the assumption of "perfect ML prediction" for worst-case consistency analysis of learning-augmented algorithms [52,57], while noting that our optimality is in the average sense subject to our designed constrained decision set (8).…”