We study online decision making problems under resource constraints, where both reward and cost functions are drawn from distributions that may change adversarially over time. We focus on two canonical settings: (i) online resource allocation where rewards and costs are observed before action selection, and (ii)online learning with resource constraints where they are observed after action selection, under full feedback or bandit feedback. It is well known that achieving sublinear regret in these settings is impossible when reward and cost distributions may change arbitrarily over time. To address this challenge, we analyze a framework in which the learner is guided by a spending plan--a sequence prescribing expected resource usage across rounds. We design general (primal-)dual methods that achieve sublinear regret with respect to baselines that follow the spending plan. Crucially, the performance of our algorithms improves when the spending plan ensures a well-balanced distribution of the budget across rounds. We additionally provide a robust variant of our methods to handle worst-case scenarios where the spending plan is highly imbalanced. To conclude, we study the regret of our algorithms when competing against benchmarks that deviate from the prescribed spending plan.

Heterogeneity poses a fundamental challenge for many real-world large-scale decision-making problems but remains largely understudied. In this paper, we study the fully heterogeneous setting of a prominent class of such problems, known as weakly-coupled Markov decision processes (WCMDPs). Each WCMDP consists of N arms (or subproblems), which have distinct model parameters in the fully heterogeneous setting, leading to the curse of dimensionality when N is large. We show that, under mild assumptions, an efficiently computable policy achieves an O(1/ N) optimality gap in the long-run average reward per arm for fully heterogeneous WCMDPs as N becomes large. This is the first asymptotic optimality result for fully heterogeneous average-reward WCMDPs. Our main technical innovation is the construction of projection-based Lyapunov functions that certify the convergence of rewards and costs to an optimal region, even under full heterogeneity.1

We begin by characterizing the space of elements that test an agent's ability to optimally allocate1031 their limited resources to goods and services they desire. In economics and decision theory, the1032 most primitive approach to describing the preferences of decision-makers is to use a function that1033 maps a set of possible choices to the agent's optimal choice within that set. Under a set of intuitive1034 assumptions, such as transitivity (i.e., if bundle X is preferred to bundle Y, and Y is preferred to1035 bundle Z, then X must be preferred to Z), it becomes possible to "rationalize" preferences by instead1036 describing a utility function. This function assigns a real number to each bundle, and the agent selects1037 the bundle with the highest utility.1038 In this paper, we focus on these "rationalizable" preferences, where agent choice can be implemented1039 as utility maximization constrained by prices and income. The solution to these consumer choice1040 problems provides ...

Designing effective agentic systems requires the seamless composition and integration of agents, tools, and models within dynamic and uncertain environments. Most existing methods rely on static, semantic retrieval approaches for tool or agent discovery. However, effective reuse and composition of existing components remain challenging due to incomplete capability descriptions and the limitations of retrieval methods. Component selection suffers because the decisions are not based on capability, cost, and real-time utility. To address these challenges, we introduce a structured, automated framework for agentic system composition that is inspired by the knapsack problem. Our framework enables a composer agent to systematically identify, select, and assemble an optimal set of agentic components by jointly considering performance, budget constraints, and compatibility.

Evaluating large language models increasingly relies on LLM-as-a-judge protocols, but such evaluations remain costly: different judges have different prices and reliabilities, and the difficulty of each prompt-response pair can vary substantially. This raises a basic allocation question: under a fixed budget, how should one distribute evaluation queries across heterogeneous judges and instances to obtain the most accurate score estimates? We formalize this question as *budgeted heteroskedastic multi-judge estimation*. Given $K$ prompt-response pairs, $J$ judges with known costs, and unknown query-judge variances, the goal is to estimate a bounded score vector while minimizing an $\ell_p$-error. Our first contribution is to analyze the inverse-variance weighted estimator (IVWE) and to derive the oracle allocation that minimizes its error rate. Since this allocation depends on the unknown variances, we then address the practical unknown-variance setting by proposing EST-IVWE, an adaptive algorithm that constructs and leverages *optimistically biased* variance estimates to stabilize the empirical allocation. We prove that EST-IVWE matches the oracle IVWE rate up to lower-order terms in the budget. Our second and central theoretical contribution is a matching *local* minimax lower bound, which establishes the instance-optimality of the proposed algorithms. A key technical insight is that Fano-type high-probability arguments are too coarse for this problem: their packing construction loses the local variance structure that governs the optimal allocation. We instead use an Assouad-type in-expectation argument, based on local perturbations, which preserves this structure and yields the sharp allocation-dependent lower bound. Finally, we numerically validate the superiority of our approach over naïve uniform allocation on synthetic and HelpSteer2 datasets.

We study optimal policy learning under combined budget and minimum coverage constraints. We show that the problem admits a knapsack-type structure and that the optimal policy can be characterized by an affine threshold rule involving both budget and coverage shadow prices. We establish that the linear programming relaxation of the combinatorial solution has an O(1) integrality gap, implying asymptotic equivalence with the optimal discrete allocation. Building on this result, we analyze two implementable approaches: a Greedy-Lagrangian (GLC) and a rank-and-cut (RC) algorithm. We show that the GLC closely approximates the optimal solution and achieves near-optimal performance in finite samples. By contrast, RC is approximately optimal whenever the coverage constraint is slack or costs are homogeneous, while misallocation arises only when cost heterogeneity interacts with a binding coverage constraint. Monte Carlo evidence supports these findings.

Post-deployment monitoring of healthcare AI requires statistically valid, label-efficient methods, but gold-standard labels from clinician chart review are expensive. Prediction-powered inference (PPI) and active statistical inference (ASI) reduce label cost by combining a small labeled sample with abundant model predictions, but both are restricted to a single predictor, a poor fit for modern clinical pipelines that have multiple predictors of differing cost and accuracy available at inference time. We propose Active Multiple-Prediction-Powered Inference (AM-PPI), which routes each instance to a cost-appropriate predictor subset, samples gold-standard labels in proportion to the chosen subset's residual uncertainty, and reweights predictions to minimize estimator variance, all under a single deployment-time budget. AM-PPI generalizes ASI to leverage multiple predictors and extends Multiple-PPI from global per-predictor allocation to per-instance adaptive routing. We derive closed-form Karush-Kuhn-Tucker (KKT) conditions for all three decisions and prove, via biconvexity and strong duality, that the resulting fixed point is a global optimum despite the joint problem being non-jointly-convex. We establish asymptotic normality with valid coverage, minimum-variance unbiasedness within the linear-prediction augmented inverse propensity weighted (AIPW) class, and a closed-form criterion identifying when multiple predictors help. On synthetic data and three healthcare monitoring tasks, AM-PPI produces 10 to 40 percent narrower confidence intervals (CIs) than single-predictor ASI in the budget regime where routing matters, and matches the better baseline elsewhere.

We study the worst-case adaptive optimization problem with budget constraint that is useful for modeling various practical applications in artificial intelligence and machine learning. We investigate the near-optimality of greedy algorithms for this problem with both modular and non-modular cost functions. In both cases, we prove that two simple greedy algorithms are not near-optimal but the best between them is near-optimal if the utility function satisfies pointwise submodularity and pointwise cost-sensitive submodularity respectively. This implies a combined algorithm that is near-optimal with respect to the optimal algorithm that uses half of the budget. We discuss applications of our theoretical results and also report experiments comparing the greedy algorithms on the active learning problem.

Treatment allocation under budget constraints is a central challenge in digital advertising: advertisers must decide which users to show ads to while spending a limited budget wisely. The standard approach follows a two-stage offline pipeline - first collect historical data to estimate heterogeneous treatment effects (HTE), then solve a constrained optimization to allocate the budget. This works well with abundant data, but fails in cold-start settings such as new campaigns, new markets, or new customer segments where little historical data exists. We propose Budget-Constrained Causal Bandits (BCCB), an online framework that learns which users respond to ads while simultaneously spending the budget, making treatment decisions one user at a time. BCCB unifies three components into a single sequential process: learning individual-level ad effectiveness, exploring users whose response is uncertain, and pacing the budget over time. We evaluated on the Criteo Uplift dataset, a large-scale advertising dataset from a real randomized controlled trial. Our key finding is a data-efficiency crossover: offline methods require approximately 10,000 historical observations to produce reliable results, while BCCB operates effectively from the very first user. Furthermore, BCCB exhibits 3-5x lower performance variance between runs, making it more practical for real campaign planning. Among purely online methods, BCCB consistently outperforms standard Thompson Sampling, budgeted Thompson Sampling, and greedy HTE estimation across all budget levels tested.