Sparse-Reward Reinforcement Learning for Climate-Adaptive Irrigation Scheduling Under Groundwater Uncertainty: Regret Bounds and Smallholder Water-Allocation Equity

Smallholder irrigation scheduling under groundwater uncertainty is a sequential decision problem in which the reward signal is sparse: crop yield responds to cumulative water deficit only at the end of a season, while the aquifer state that constrains pumping is only partially observed. The irrigation decision process is modelled as a finite-horizon Markov decision process with a delayed, thresholded reward and a partially observable groundwater state; a sparse-reward upper confidence bound algorithm is proposed and analysed. The main result is a regret bound of order the square root of the product of the number of episodes and the number of state-action pairs, inflated by a term that measures the mismatch between the exploration distribution and the water-allocation rule applied to smallholders. A second result shows that any scheduling policy that is regret-optimal in the aggregate can concentrate pumping on a subset of plots, and a fairness-constrained variant is derived whose regret penalty is bounded by the equity tolerance. The framework is illustrated with reference to reanalysed temperature records for Accra, Ghana (2000-2023; 312 monthly observations; mean 26.521 degrees Celsius; warming trend 0.1622 degrees Celsius per decade), which motivate the climate-adaptive component without serving as experimental evidence for the learning algorithm. The contribution is a bound that makes the trade-off between regret and allocation equity explicit and auditable, rather than a claim of field performance.