Optimal Sequential Multi-class Diagnosis

TL;DR

This paper develops a low-dimensional manifold approach for optimal sequential multi-class diagnosis, enabling efficient computation of policies in high-dimensional belief spaces with broad applications.

Contribution

It introduces a matrix factorization and belief reconstruction method exploiting structural properties to reduce dimensionality in multi-hypothesis testing.

Findings

Beliefs can be restricted to low-dimensional manifolds with closed-form expressions.

Exact belief reconstruction for common univariate distributions.

Proposed low-rank approximation improves diagnosis speed and accuracy.

Abstract

Sequential multi-class diagnosis, also known as multi-hypothesis testing, is a classical sequential decision problem with broad applications. However, the optimal solution remains, in general, unknown as the dynamic program suffers from the curse of dimensionality in the posterior belief space. We consider a class of practical problems in which the observation distributions associated with different classes are related through exponential tilting, and show that the reachable beliefs could be restricted on, or near, a set of low-dimensional, time-dependent manifolds with closed-form expressions. This sparsity is driven by the low dimensionality of the observation distributions (which is intuitive) as well as by specific structural interrelations among them (which is less intuitive). We use a matrix factorization approach to uncover the potential low dimensionality hidden in high-dimensional beliefs and reconstruct the beliefs using a diagnostic statistic in lower dimension. For common univariate distributions, e.g., normal, binomial, and Poisson, the belief reconstruction is exact, and the optimal policies can be efficiently computed for a large number of classes. We also characterize the structure of the optimal policy in the reduced dimension. For multivariate distributions, we propose a low-rank matrix approximation scheme that works well when the beliefs are near the low-dimensional manifolds. The optimal policy significantly outperforms the state-of-the-art heuristic policy in quick diagnosis with noisy data. (forthcoming in Operations Research)