Budgeted Active Learning for Agriculture Soil Fertility Mapping Under Costly Laboratory Assays: An Artificial Intelligence Information-Gain Regret Bound

Soil fertility mapping in smallholder agriculture is constrained by the cost of laboratory assays, which limits the number of plots that can be sampled and therefore the accuracy of the resulting maps. The framework is analysed through an information-gain regret bound that decomposes the excess risk of the learned map into an approximation term, a noise term and an information term governed by the mutual information between selected plots and the fertility field. The bound is derived under a Gaussian process prior over the fertility field and a submodular information-gain objective, and it is shown that greedy selection attains a constant-factor approximation to the optimal information gain, which yields a regret bound that scales with the square root of the budget rather than with the number of candidate plots. The analysis is complemented by a design-based evaluation of the before-after-control-impact estimator used to assess mapping accuracy, with a between-cluster standard deviation of 6.0, a residual standard deviation of 4.5 and an intraclass correlation of 0.64; power was estimated across 2000 design cells and 52 clusters were required for a 5 percentage point effect at 80% power. The results clarify when active selection is worth its computational and logistical cost and when uniform sampling is preferable.