Sufficient conditions for submodularity-ratio lower bounds beyond naive Bayes

Identify sufficient structural conditions weaker than naive Bayes conditional independence that guarantee a nontrivial lower bound on the submodularity ratio of the mutual-information set function $A \mapsto I(Y;X_A)$ for active feature acquisition.

Background

The paper establishes that A↦I(Y;XA)A \mapsto I(Y;X_A) is submodular under naive Bayes conditional independence, while showing that submodularity fails in general, including for jointly Gaussian variables.

Because the clinical experiments exhibit feature synergies and do not establish a global submodularity-ratio lower bound, the authors state that they do not know sufficient conditions under weaker dependence assumptions that would preserve a quantitative approximation guarantee.

References

We do not, however, derive sufficient conditions for $\gamma \geq \gamma_0$ under weaker structure than na\"ive Bayes (Proposition~\ref{prop:nb_submodular}), which we leave open.

— The Lagrangian Structure of Cost-Penalized Active Feature Acquisition  (2610.21919 - Bingham, 15 Oct 2026) in Empirical submodularity ratio, Section 6.3

Several empirical quantities that would further sharpen the picture remain unmeasured on the flagship instance: bootstrap confidence intervals for the submodularity ratio $\gamma_{f,k}$; the greedy-gap decomposition of Theorem~\ref{thm:lag_bound} on the full MIMIC-IV instance (we measure $\Delta_\lambda$ exactly on heterogeneous-cost synthetic instances in Section~\ref{sec:exp_hetcost}, but $L(\lambda)$ is intractable at full clinical scale); and a forced-budget frontier that quantifies the cost of voluntary stopping on the clinical data.

— The Lagrangian Structure of Cost-Penalized Active Feature Acquisition  (2610.21919 - Bingham, 15 Oct 2026) in Limitations, Section 7 (Discussion)