Approximate Supermodularity Bounds for Experimental Design

TL;DR

This paper establishes performance guarantees for greedy algorithms in experimental design by leveraging approximate supermodularity, especially in low SNR scenarios, explaining their empirical success.

Contribution

It introduces the concept of approximate supermodularity to derive non-asymptotic bounds for greedy A- and E-optimal designs, linking their effectiveness to SNR levels.

Findings

Greedy solutions approach (1-1/e)-optimality at low SNR.

Performance bounds are derived for non-supermodular criteria.

Empirical success of greedy designs is explained through approximate supermodularity.

Abstract

This work provides performance guarantees for the greedy solution of experimental design problems. In particular, it focuses on A- and E-optimal designs, for which typical guarantees do not apply since the mean-square error and the maximum eigenvalue of the estimation error covariance matrix are not supermodular. To do so, it leverages the concept of approximate supermodularity to derive non-asymptotic worst-case suboptimality bounds for these greedy solutions. These bounds reveal that as the SNR of the experiments decreases, these cost functions behave increasingly as supermodular functions. As such, greedy A- and E-optimal designs approach (1-1/e)-optimality. These results reconcile the empirical success of greedy experimental design with the non-supermodularity of the A- and E-optimality criteria.