Lag: A Mean-Field Optimal Policy for Online Assignment with Reusable Resources

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 finite-horizon online resource allocation with i.i.d. requests and an endogenous Markov state on a finite state space: each action affects the transition of the state that governs future rewards and resource consumption. In this problem, a transient fluid LP benchmark upper bounds the expected reward of every nonanticipating policy, while a stationary LP supplies randomized state-dependent controls. We assume that the stationary LP has a unique optimum and identify primal nondegeneracy and irreducibility of the optimal induced kernel as important regularity conditions in this framework. With a known request prior, we show that, under nondegeneracy and irreducibility, both frequent and infrequent re-solving attain $O(1)$ regret. However, under a degenerate optimum, irreducibility yields the sharp worst-case $Θ(\sqrt{T})$ rate for infrequent re-solving, while frequent re-solving can incur $Ω(T)$ regret. Thus, more frequent optimization can perform asymptotically worse. With an unknown request prior, we develop a three-phase U-shaped infrequent re-solving policy that coordinates learning and inventory correction with $O(\log\log T)$ LP solves. When the optimal induced kernel is irreducible and the algorithm is given the optimal target state class and a constant-cost entrance policy, it attains $O(1)$ regret under nondegeneracy and $O(\sqrt{T})$ regret under degeneracy. Without the target-class information, linear minimax regret is unavoidable. Numerical experiments further illustrate the instability of round-by-round re-solving relative to epoch-wise infrequent re-solving, show that thresholding greatly mitigates its loss, and find that infrequent schemes remain dominant under both known and estimated priors.

We study contextual queueing bandits, where a learner schedules jobs while learning unknown service rates modeled by logistic functions of job-server features. Performance is measured by queue length regret, the expected excess queue length at round $t$ relative to an oracle that knows the service rates. Existing decaying-regret guarantees either have a suboptimal decay rate or require a known fixed horizon. They also assume context-wise slack and a strictly positive minimum eigenvalue of the feature covariance. In this paper, we propose WISE (Widest Interval Selection with Elimination), achieving rate-optimal $\widetilde{\mathcal O}(t^{-1/2})$ queue length regret at every sufficiently large time without knowing the horizon. We assume capacity slack, meaning that expected incoming workload under best-server service is below service capacity, and impose no covariance lower bound. Our analysis uses a workload potential measuring the expected service attempts needed by waiting jobs on their best servers. Its drift on nonempty rounds combines a negative term ensured by capacity slack with errors from suboptimal service choices. Then an elliptical potential count bounds how often WISE selects wide confidence intervals, thereby limiting the number of rounds with large service errors. We also sharpen the arrival-rate dependence of an existing lower bound and make its dependence on feature dimension and server count explicit. We prove another lower bound that quantifies the increase in regret as the normalized capacity slack decreases; to our knowledge, this is the first such lower bound for CQB. Simulations show small regret even when context-wise slack fails.