Leveraging Recourse and Size Bounds for OnlineKnapsack with Removable Items

We study new algorithms for Contextual Bandits with Knapsack. In these problems, there are finitely many types of customers, products, and resources. Each product is made from a fixed combination of resources, and resources have finite capacity. A decision maker must assign each arriving customer one out of a set of multiple possible products. Every assignment of a customer to a product will generate a random reward, which equals an unknown linear function of customer and product features, plus a noise term. The objective is to jointly learn the mean reward function, and to make online assignments to minimize the expected revenue loss relative to an optimal policy that knows the reward function. We propose a natural and simple extension of the Upper-Confidence-Bound (UCB) family of algorithms and apply re-optimization techniques. We show that by taking advantage of re-optimization, our algorithm achieves an average regret of $O(\frac{(\ln T)^3}{T})$ where $T$ is the horizon length. Our bound significantly reduces the $O(\frac{1}{\sqrt{T}})$ bound in the literature for closely related dynamic-pricing problems that are based on re-optimization.

We study satisficing regret minimization in bandits with knapsacks (BwK), where a learner sequentially selects randomized policies, i.e., distributions over arms, to obtain mean rewards above a satisficing threshold as frequently as possible while respecting finite resource budgets. The performance is measured by satisficing regret, the cumulative deficits of the selected policies' mean rewards below this threshold. Resource constraints introduce two difficulties: an algorithm can incur linear satisficing regret even with small standard regret as deficits to the threshold cannot be compensated, and random consumption can cause early resource depletion even when policies are feasible in expectation. To address these challenges, we propose SELECTwK, a general algorithmic template that uses a BwK learning oracle algorithm with sub-linear expected (standard) regret upper bound to search for and test candidate satisficing policies (i.e., policies whose mean rewards exceed satisficing threshold and whose expected resource consumption is feasible over the time horizon). Our regret analysis, supported by an LP-duality argument, establishes that SELECTwK attains constant expected satisficing regret when a satisficing policy exists. We further introduce a stronger regret condition that bounds the expected sum of positive reward deficits to the optimal one at each time step, under which SELECTwK achieves a sharper constant bound together with a sub-linear bound in the time horizon. As a complement, SELECTwK still preserves the oracle's sub-linear expected standard regret bound if the satisficing threshold is unattainable. Finally, we establish the oracle conditions and corresponding satisficing regret bound for existing finite-armed BwK and dynamic pricing algorithms, and conduct numerical experiments on dynamic pricing with limited inventory to demonstrate the empirical performance of SELECTwK.

Network routers that enforce Quality-of-Service (QoS) guarantees must decide, at every clock cycle, which expiring packet of information to transmit, even when the value of the packet is unknown until it is processed. We frame this problem as the Online Packet Scheduling with Deadlines (OPSD) problem under Partial Feedback: packets arrive at every clock cycle, with different deadlines, but the weights are only observed after execution. Under a stochastic assumption on the unknown weights, we explore different variants of the OPSD problem with bandit feedback. We establish a connection between our setting and the sleeping bandits problem, and set our learning goal to $α$-regret minimization. We provide algorithms with provable $α$-regret guarantees under different spans of slackness, distinguishing systems allowing for randomization and systems that do not. In every scenario, our algorithms achieve an $α$-regret upper bound of $\widetilde{\mathcal{O}}\left(\sqrt{KT}\right)$, matching the lower bound for the standard bandit setting. In the practically relevant case of $2$-bounded deadline instances, where the deadline is set at most one clock cycle away from the arrival, our deterministic algorithm achieves the provably tightest possible competitive ratio. Remarkably, when the number of distinct packet types $K\ge 2$ is finite, it is possible to break the well-established $Φ= \frac{1+\sqrt{5}}{2}$ competitive ratio barrier and attain a tighter competitive ratio $θ_K$ ranging in $[\sqrt{2}, Φ)$.