Selective Sampling for Online Best-arm Identification

TL;DR

This paper studies an adaptive sampling method for efficiently identifying the best option in a set, balancing the number of samples taken and the confidence in the decision, with applications to classification.

Contribution

It introduces a nearly optimal algorithm for selective sampling that characterizes the trade-off between label complexity and stopping time, with a simple geometric decision rule.

Findings

Provides a theoretical characterization of the label-sampling trade-off.

Proposes an algorithm that nearly minimizes label complexity for a given stopping time.

Extends the framework to binary classification problems.

Abstract

This work considers the problem of selective-sampling for best-arm identification. Given a set of potential options $\mathcal{Z}\subset\mathbb{R}^d$, a learner aims to compute with probability greater than $1-\delta$, $\arg\max_{z\in \mathcal{Z}} z^{\top}\theta_{\ast}$ where $\theta_{\ast}$ is unknown. At each time step, a potential measurement $x_t\in \mathcal{X}\subset\mathbb{R}^d$ is drawn IID and the learner can either choose to take the measurement, in which case they observe a noisy measurement of $x^{\top}\theta_{\ast}$, or to abstain from taking the measurement and wait for a potentially more informative point to arrive in the stream. Hence the learner faces a fundamental trade-off between the number of labeled samples they take and when they have collected enough evidence to declare the best arm and stop sampling. The main results of this work precisely characterize this trade-off between labeled samples and stopping time and provide an algorithm that nearly-optimally achieves the minimal label complexity given a desired stopping time. In addition, we show that the optimal decision rule has a simple geometric form based on deciding whether a point is in an ellipse or not. Finally, our framework is general enough to capture binary classification improving upon previous works.