Optimal Decision Tree and Adaptive Submodular Ranking with Noisy Outcomes
Su Jia, Fatemeh Navidi, Viswanath Nagarajan, R. Ravi
TL;DR
This work studies Optimal Decision Tree with Noise (ODTN) and its generalization Adaptive Submodular Ranking with Noise (ASRN), addressing the challenge of persistent, noisy test outcomes. It develops a unified framework that connects ODTN to ASRN and to Submodular Ranking with Noise (SFRN), enabling noise-tolerant approximation guarantees across non-adaptive and adaptive settings. The results include a polynomial-time $O(\log \frac{1}{\varepsilon})$-approximation for non-adaptive instances, adaptive guarantees $O(\min\{c,r\} + \log \frac{m}{\varepsilon})$ under low noise, and a sparsity-based $O(m^{\alpha} + \log m \cdot OPT)$-type bound for high-noise, along with a robust Membership Oracle and a Stochastic Set Cover perspective. Experiments on toxic chemicals and linear classifiers show practical performance close to information-theoretic lower bounds, highlighting the approach’s potential for real-world noisy diagnostic and learning tasks.
Abstract
In pool-based active learning, the learner is given an unlabeled data set and aims to efficiently learn the unknown hypothesis by querying the labels of the data points. This can be formulated as the classical Optimal Decision Tree (ODT) problem: Given a set of tests, a set of hypotheses, and an outcome for each pair of test and hypothesis, our objective is to find a low-cost testing procedure (i.e., decision tree) that identifies the true hypothesis. This optimization problem has been extensively studied under the assumption that each test generates a deterministic outcome. However, in numerous applications, for example, clinical trials, the outcomes may be uncertain, which renders the ideas from the deterministic setting invalid. In this work, we study a fundamental variant of the ODT problem in which some test outcomes are noisy, even in the more general case where the noise is persistent, i.e., repeating a test gives the same noisy output. Our approximation algorithms provide guarantees that are nearly best possible and hold for the general case of a large number of noisy outcomes per test or per hypothesis where the performance degrades continuously with this number. We numerically evaluated our algorithms for identifying toxic chemicals and learning linear classifiers, and observed that our algorithms have costs very close to the information-theoretic minimum.