Complementarily, we present a minimax lower bound for this problem.

We study fixed-confidence best-arm identification (BAI) where a cheap but potentially biased proxy (e.g., LLM judge) is available for every sample, while an expensive ground-truth label can only be acquired selectively when using a human for auditing. Unlike classical multi-fidelity BAI, the proxy is biased (arm- and context-dependent) and ground truth is selectively observed. Consequently, standard multi-fidelity methods can mis-select the best arm, and uniform auditing, though accurate, wastes scarce resources and is inefficient. We prove that without bias correction and propensity adjustment, mis-selection probability may not vanish (even with unlimited proxy data). We then develop an estimator for the mean of each arm that combines proxy scores with inverse-propensity-weighted residuals and form anytime-valid confidence sequences for that estimator. Based on the estimator and confidence sequence, we propose an algorithm that adaptively selects and audits arms. The algorithm concentrates audits on unreliable contexts and close arms and we prove that a plug-in Neyman rule achieves near-oracle audit efficiency. Numerical experiments confirm the theoretical guarantees and demonstrate the superior empirical performance of the proposed algorithm.

In bandit best-arm identification, an algorithm is tasked with finding the arm with highest mean reward with a specified accuracy as fast as possible. We study multi-fidelity best-arm identification, in which the algorithm can choose to sample an arm at a lower fidelity (less accurate mean estimate) for a lower cost. Several methods have been proposed for tackling this problem, but their optimality remain elusive, notably due to loose lower bounds on the total cost needed to identify the best arm. Our first contribution is a tight, instance-dependent lower bound on the cost complexity. The study of the optimization problem featured in the lower bound provides new insights to devise computationally efficient algorithms, and leads us to propose a gradient-based approach with asymptotically optimal cost complexity. We demonstrate the benefits of the new algorithm compared to existing methods in experiments. Our theoretical and empirical findings also shed light on an intriguing concept of optimal fidelity for each arm.

We consider a stochastic bandit problem with a possibly infinite number of arms. We write $p^*$ for the proportion of optimal arms and $\Delta$ for the minimal mean-gap between optimal and sub-optimal arms. We characterize the optimal learning rates both in the cumulative regret setting, and in the best-arm identification setting in terms of the problem parameters $T$ (the budget), $p^*$ and $\Delta$. For the objective of minimizing the cumulative regret, we provide a lower bound of order $\Omega(\log(T)/(p^*\Delta))$ and a UCB-style algorithm with matching upper bound up to a factor of $\log(1/\Delta)$. Our algorithm needs $p^*$ to calibrate its parameters, and we prove that this knowledge is necessary, since adapting to $p^*$ in this setting is impossible. For best-arm identification we also provide a lower bound of order $\Omega(\exp(-cT\Delta^2p^*))$ on the probability of outputting a sub-optimal arm where $c> 0$ is an absolute constant. We also provide an elimination algorithm with an upper bound matching the lower bound up to a factor of order $\log(T)$ in the exponential, and that does not need $p^*$ or $\Delta$ as parameter. Our results apply directly to the three related problems of competing against the $j$-th best arm, identifying an $\epsilon$ good arm, and finding an arm with mean larger than a quantile of a known order.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. In this problem there is a hidden unknown parameter \theta*\in R^d and a finite set of arms X\subseteq R^d. When an arm is pulled you observe a reward x^T\theta*+epsilon where epsilon is a zero mean i.i.d noise with bounded range. The goal is to identify argmax_{x\in X} x^T\theta* with the least number of samples. Results: The paper characterizes the sample complexity of static and dynamic allocation strategies to identify the best arm.

We consider a stochastic bandit problem with a possibly infinite number of arms.

We study the Stochastic Multi-armed Bandit problem under bounded arm-memory.

This study investigates adaptive experimental design for treatment choice, also known as fixed-budget best-arm identification. We consider an adaptive procedure consisting of a treatment-allocation phase followed by a treatment-choice phase, and we design an adaptive experiment for this setup to efficiently identify the best treatment arm, defined as the one with the highest expected outcome. In our designed experiment, the treatment-allocation phase consists of two stages. The first stage is a pilot phase, where we allocate each treatment arm uniformly with equal proportions to eliminate clearly suboptimal arms and estimate outcome variances. In the second stage, we allocate treatment arms in proportion to the variances estimated in the first stage. After the treatment-allocation phase, the procedure enters the treatment-choice phase, where we choose the treatment arm with the highest sample mean as our estimate of the best treatment arm. We prove that this single design is simultaneously asymptotically minimax and Bayes optimal for the simple regret, with upper bounds that match our lower bounds up to exact constants. Therefore, our designed experiment achieves the sharp efficiency limits without requiring separate tuning for minimax and Bayesian objectives.