Non-Linear Model-Based Sequential Decision-Making in Agriculture

TL;DR

Nonlinear model-based bandit algorithms are developed as a framework for adaptive fertilizer management under uncertainty to illustrate how interpretable, uncertainty-aware sequential decision rules can support economically sustainable fertilizer recommendations and contribute to more efficient agricultural input use.

Abstract

Sequential decision-making is central to sustainable agricultural management and precision agriculture, where resource inputs must be optimized under uncertainty and over time. However, such decisions must often be made with limited observations, whereas classical bandit and reinforcement learning approaches typically rely on either linear or black-box reward models that may misrepresent domain knowledge or require large amounts of data. We propose a family of \emph{nonlinear, model-based bandit algorithms} that embed domain-specific response curves directly into the exploration-exploitation loop. By coupling (i) principled uncertainty quantification with (ii) closed-form or rapidly computable profit optima, these algorithms achieve sublinear regret and near-optimal sample complexity while preserving interpretability. Theoretical analysis establishes regret and sample complexity bounds, and extensive simulations emulating real-world fertilizer-rate decisions show consistent improvements over both linear and nonparametric baselines (such as linear UCB and $k$-NN UCB) in the low-sample regime, under both well-specified and shape-compatible misspecified models. Because our approach leverages mechanistic insight rather than large data volumes, it is especially suited to resource-constrained settings, supporting sustainable, inclusive, and transparent sequential decision-making across agriculture, environmental management, and allied applications.

Figures (12)

• Figure 1: Illustration of model-dependent upper bounds on sequential Rademacher complexity. We plot $B_{\mathcal{F}}\sqrt{\log(t)/t}$ (up to a universal constant common to all models) for the four yield-response families used in the simulations, where $B_{\mathcal{F}}=\sup_{x\in[0,250]}|f(x; \theta_{\text{true}})|$ is computed from the parameter settings in Section 5. The bound decreases with the number of rounds $t$; differences across model families appear through the constant $B_{\mathcal{F}}$. The dashed vertical line marks the horizon $T=30$ used in the well-specified experiments.
• Figure 4: How the algorithms' fertilizer-rate choices evolve over time (well-specified quadratic plateau). Actions are chosen from $\mathcal{X}=\{0,50,\ldots,250\}$ lb N/ac. (a) For $p_x=\$0.5$/lb N, the running proportion of times each nitrogen rate has been selected up to round $t$ is shown for $\epsilon$-greedy, nonlinear-UCB, and ViOlin (averaged over 10 replicates). (b) The most frequently selected nitrogen rate at each round is shown for fertilizer prices $p_x\in\{0.3,0.5,0.7\}$ $/lb N, illustrating how higher fertilizer cost shifts the learned decision toward lower nitrogen rates.
• Figure 5: Model-parameter learning over time (well-specified quadratic-plateau). For $p_x=\$0.5$/lb N and $T=30$, we plot estimated $(a,b,c,x_0)$ for $\epsilon$-greedy, nonlinear-UCB, and ViOlin; The horizontal red line denotes the true value. Parameter estimates stabilize over time, indicating that the response curve can be learned from sequential data within the horizon considered. (One representative replicate is shown; similar behavior occurs across runs. )
• Figure 6: Profit regret under model misspecification. Data are generated from a Mitscherlich (truth) yield curve, but the learner fits a different parametric family. We use $p_y=\$5$/bu, $p_x=\$0.7$/lb N, $\mathcal{X}=\{0,50,\ldots,250\}$ lb N /ac, $\sigma=0.5$, and $T=100$. Left: quadratic-plateau fit; Right: Michaelis--Menten fit. Curves show mean cumulative profit regret (in $/ac) over 10 replicates for $\epsilon$-greedy, nonlinear-UCB, ViOlin, LinUCB, and kNN-UCB. Under misspecification, regret increases for all methods, but nonlinear model-based policies remain competitive.
• Figure 7: Offline replay on real multi-site corn nitrogen trials (2014--2016): profit regret under a quadratic-plateau model. Reward is per-acre profit $\Pi(x)=p_y Y(x)-p_x x$, where $p_y$ is the corn price (USDA--NASS Crop Values) and $p_x$ is the nitrogen cost as in Table \ref{['tab:realdata_prices_urea']}. Panels (a)--(b) show the data-limited Urbana, IL case study with rounds defined by (Year, Block) ($T=12$). Panels (c)--(d) show the pooled low-productivity evaluation with rounds defined by (State, Site, Year, Block) (longer horizon). Curves are means over replay replications; shaded bands are pointwise 95% confidence intervals.

Theorems & Definitions (8)

• Remark 1: Scope
• Remark 2: Bandit (partial) feedback
• Definition 1: Sequential Rademacher Complexity
• Theorem 1: Sample Complexity for Model-based Nonlinear Bandits dong2021provable
• Theorem 2: Sequential Rademacher Complexity for Bounded Functions
• Corollary 1: Sample Complexity for online learning of Bounded Non-linear Reward classes
• Remark 3
• proof : Proof of Theorem \ref{['thm: SeqRademacherForBoundedFunctions']}