Constrained Online Decision-Making: A Unified Framework

Contextual online decision-making problems with constraints appear in various real-world applications, such as personalized recommendation with resource limits and dynamic pricing with fairness constraints. In this paper, we investigate a general formulation of sequential decision-making with stage-wise feasibility constraints, where at each round, the learner must select an action based on observed context while ensuring a problem-specific feasibility criterion. We propose a unified algorithmic framework that captures many existing constrained learning problems, including constrained bandits, stream active learning, online hypothesis testing, and model calibration. Central to our approach is the concept of upper counterfactual confidence bound, which enables the design of practically efficient online algorithms using any offline conditional density estimation oracle. Technically, to handle feasibility constraints, we introduce a generalized notion of the eluder dimension, extending it from the classical setting based on square loss to a broader class of metric-like probability divergences, which could capture the complexity of various density function classes and characterize the loss incurred due to feasibility constraint uncertainty. Our result offers a principled foundation for constrained sequential decision-making in both theory and practice.

Constrained Online Decision-Making with Density Estimation Oracles

Contextual online decision-making problems with constraints appear in a wide range of real-world applications, such as personalized recommendation with resource limits, adaptive experimental design, and decision-making under safety or fairness requirements. In this paper, we investigate a general formulation of sequential decision-making with stage-wise feasibility constraints, where at each round, the learner must select an action based on observed context while ensuring that a problem-specific feasibility criterion is satisfied. We propose a unified algorithmic framework that captures many existing constrained learning problems, including constrained bandits, active learning with label budgets, online hypothesis testing with Type I error control, and model calibration. Central to our approach is the concept of upper counterfactual confidence bounds, which enables the design of practically efficient online algorithms with strong theoretical guarantee using any offline conditional density estimation oracle. Technically, to handle feasibility constraints in complex environments, we introduce a generalized notion of the eluder dimension - extending it from the classical setting based on square loss to a broader class of metric-like probability divergences. This allows us to capture the complexity of various density function classes and characterize the utility regret incurred due to feasibility constraint uncertainty. Our result offers a principled foundation for constrained sequential decision-making in both theory and practice.

Optimizing LLM Inference: Fluid-Guided Online Scheduling with Memory Constraints

Large Language Models (LLMs) are indispensable in today's applications, but their inference procedure -- generating responses by processing text in segments and using a memory-heavy Key-Value (KV) cache -- demands significant computational resources, particularly under memory constraints. This paper formulates LLM inference optimization as a multi-stage online scheduling problem where sequential prompt arrivals and KV cache growth render conventional scheduling ineffective. We develop a fluid dynamics approximation to provide a tractable benchmark that guides algorithm design. Building on this, we propose the Waiting for Accumulated Inference Threshold (WAIT) algorithm, which uses multiple thresholds to schedule incoming prompts optimally when output lengths are known, and extend it to Nested WAIT for cases with unknown output lengths. Theoretical analysis shows that both algorithms achieve near-optimal performance against the fluid benchmark in heavy traffic conditions, balancing throughput, latency, and Time to First Token (TTFT). Experiments with the Llama-7B model on an A100 GPU using both synthetic and real-world datasets demonstrate improved throughput and reduced latency relative to established baselines like vLLM and Sarathi. This work bridges operations research and machine learning, offering a rigorous framework for the efficient deployment of LLMs under memory constraints.

Near-Optimal Private Learning in Linear Contextual Bandits

We analyze the problem of private learning in generalized linear contextual bandits. Our approach is based on a novel method of re-weighted regression, yielding an efficient algorithm with regret of order $\sqrt{T}+\frac{1}{\alpha}$ and $\sqrt{T}/\alpha$ in the joint and local model of $\alpha$-privacy, respectively. Further, we provide near-optimal private procedures that achieve dimension-independent rates in private linear models and linear contextual bandits. In particular, our results imply that joint privacy is almost "for free" in all the settings we consider, partially addressing the open problem posed by Azize and Basu (2024).

Contextual Online Decision Making with Infinite-Dimensional Functional Regression

Contextual sequential decision-making problems play a crucial role in machine learning, encompassing a wide range of downstream applications such as bandits, sequential hypothesis testing and online risk control. These applications often require different statistical measures, including expectation, variance and quantiles. In this paper, we provide a universal admissible algorithm framework for dealing with all kinds of contextual online decision-making problems that directly learns the whole underlying unknown distribution instead of focusing on individual statistics. This is much more difficult because the dimension of the regression is uncountably infinite, and any existing linear contextual bandits algorithm will result in infinite regret. To overcome this issue, we propose an efficient infinite-dimensional functional regression oracle for contextual cumulative distribution functions (CDFs), where each data point is modeled as a combination of context-dependent CDF basis functions. Our analysis reveals that the decay rate of the eigenvalue sequence of the design integral operator governs the regression error rate and, consequently, the utility regret rate. Specifically, when the eigenvalue sequence exhibits a polynomial decay of order $\frac{1}{\gamma}\ge 1$, the utility regret is bounded by $\tilde{\mathcal{O}}\Big(T^{\frac{3\gamma+2}{2(\gamma+2)}}\Big)$. By setting $\gamma=0$, this recovers the existing optimal regret rate for contextual bandits with finite-dimensional regression and is optimal under a stronger exponential decay assumption. Additionally, we provide a numerical method to compute the eigenvalue sequence of the integral operator, enabling the practical implementation of our framework.

Learning to Price with Resource Constraints: From Full Information to Machine-Learned Prices

We study the dynamic pricing problem with knapsack, addressing the challenge of balancing exploration and exploitation under resource constraints. We introduce three algorithms tailored to different informational settings: a Boundary Attracted Re-solve Method for full information, an online learning algorithm for scenarios with no prior information, and an estimate-then-select re-solve algorithm that leverages machine-learned informed prices with known upper bound of estimation errors. The Boundary Attracted Re-solve Method achieves logarithmic regret without requiring the non-degeneracy condition, while the online learning algorithm attains an optimal $O(\sqrt{T})$ regret. Our estimate-then-select approach bridges the gap between these settings, providing improved regret bounds when reliable offline data is available. Numerical experiments validate the effectiveness and robustness of our algorithms across various scenarios. This work advances the understanding of online resource allocation and dynamic pricing, offering practical solutions adaptable to different informational structures.

Prediction-Guided Active Experiments

In this work, we introduce a new framework for active experimentation, the Prediction-Guided Active Experiment (PGAE), which leverages predictions from an existing machine learning model to guide sampling and experimentation. Specifically, at each time step, an experimental unit is sampled according to a designated sampling distribution, and the actual outcome is observed based on an experimental probability. Otherwise, only a prediction for the outcome is available. We begin by analyzing the non-adaptive case, where full information on the joint distribution of the predictor and the actual outcome is assumed. For this scenario, we derive an optimal experimentation strategy by minimizing the semi-parametric efficiency bound for the class of regular estimators. We then introduce an estimator that meets this efficiency bound, achieving asymptotic optimality. Next, we move to the adaptive case, where the predictor is continuously updated with newly sampled data. We show that the adaptive version of the estimator remains efficient and attains the same semi-parametric bound under certain regularity assumptions. Finally, we validate PGAE's performance through simulations and a semi-synthetic experiment using data from the US Census Bureau. The results underscore the PGAE framework's effectiveness and superiority compared to other existing methods.

The Power of Adaptivity in Experimental Design

Given n experiment subjects with potentially heterogeneous covariates and two possible treatments, namely active treatment and control, this paper addresses the fundamental question of determining the optimal accuracy in estimating the treatment effect. Furthermore, we propose an experimental design that approaches this optimal accuracy, giving a (non-asymptotic) answer to this fundamental yet still open question. The methodological contribution is listed as following. First, we establish an idealized optimal estimator with minimal variance as benchmark, and then demonstrate that adaptive experiment is necessary to achieve near-optimal estimation accuracy. Secondly, by incorporating the concept of doubly robust method into sequential experimental design, we frame the optimal estimation problem as an online bandit learning problem, bridging the two fields of statistical estimation and bandit learning. Using tools and ideas from both bandit algorithm design and adaptive statistical estimation, we propose a general low switching adaptive experiment framework, which could be a generic research paradigm for a wide range of adaptive experimental design. Through information-theoretic lower bound combined with Bayes risk analysis, we demonstrate the optimality of our proposed experiment. Numerical result indicates that the estimation accuracy approaches optimal with as few as two or three policy updates.

Experimenting on Markov Decision Processes with Local Treatments

As service systems grow increasingly complex and dynamic, many interventions become localized, available and taking effect only in specific states. This paper investigates experiments with local treatments on a widely-used class of dynamic models, Markov Decision Processes (MDPs). Particularly, we focus on utilizing the local structure to improve the inference efficiency of the average treatment effect. We begin by demonstrating the efficiency of classical inference methods, including model-based estimation and temporal difference learning under a fixed policy, as well as classical A/B testing with general treatments. We then introduce a variance reduction technique that exploits the local treatment structure by sharing information for states unaffected by the treatment policy. Our new estimator effectively overcomes the variance lower bound for general treatments while matching the more stringent lower bound incorporating the local treatment structure. Furthermore, our estimator can optimally achieve a linear reduction with the number of test arms for a major part of the variance. Finally, we explore scenarios with perfect knowledge of the control arm and design estimators that further improve inference efficiency.

Offline Oracle-Efficient Learning for Contextual MDPs via Layerwise Exploration-Exploitation Tradeoff

Motivated by the recent discovery of a statistical and computational reduction from contextual bandits to offline regression (Simchi-Levi and Xu, 2021), we address the general (stochastic) Contextual Markov Decision Process (CMDP) problem with horizon H (as known as CMDP with H layers). In this paper, we introduce a reduction from CMDPs to offline density estimation under the realizability assumption, i.e., a model class M containing the true underlying CMDP is provided in advance. We develop an efficient, statistically near-optimal algorithm requiring only O(HlogT) calls to an offline density estimation algorithm (or oracle) across all T rounds of interaction. This number can be further reduced to O(HloglogT) if T is known in advance. Our results mark the first efficient and near-optimal reduction from CMDPs to offline density estimation without imposing any structural assumptions on the model class. A notable feature of our algorithm is the design of a layerwise exploration-exploitation tradeoff tailored to address the layerwise structure of CMDPs. Additionally, our algorithm is versatile and applicable to pure exploration tasks in reward-free reinforcement learning.