Learning in models with discrete latent variables is challenging due to high variance gradient estimators. Generally, approaches have relied on control variates to reduce the variance of the REINFORCE estimator. Recent work (Jang et al., 2016; Maddi-son et al., 2016) has taken a different approach, introducing a continuous relaxation of discrete variables to produce low-variance, but biased, gradient estimates.

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 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. This methodology directly contributes to SDG 2 (Zero Hunger) and SDG 12 (Responsible Consumption and Production) by enabling data-driven, less wasteful agricultural practices.

Thank you for your encouraging comments. Thank you for your thorough and helpful review. We appreciate all of your feedback. This is explained in the "sensor selection" paragraph at the end of the paper and We are glad that you understand and appreciate the significance of Theorem 2. Empirical results/better demonstrations It also suggests that with the "right" constraints put in place, a nonlinear method should do very well. For example, we can try multiple process models on the flu data.

When inverse kinematics (IK) is adopted to control robotic arms in manipulation tasks, there is often a discrepancy between the end effector (EE) position of the robot model in the simulator and the physical EE in reality. In most robotic scenarios with sim-to-real transfer, we have information about joint positions in both simulation and reality, but the EE position is only available in simulation. We developed a novel method to overcome this difficulty based on haptic feedback calibration, using a touchscreen in front of the robot that provides information on the EE position in the real environment. During the calibration procedure, the robot touches specific points on the screen, and the information is stored. In the next stage, we build a transformation function from the data based on linear transformation and neural networks that is capable of outputting all missing variables from any partial input (simulated/real joint/EE position). Our results demonstrate that a fully nonlinear neural network model performs best, significantly reducing positioning errors.

A fundamental challenge in the cognitive sciences is discovering the dynamics that govern behaviour. Take the example of spoken language, which is characterised by a highly variable and complex set of physical movements that map onto the small set of cognitive units that comprise language. What are the fundamental dynamical principles behind the movements that structure speech production? In this study, we discover models in the form of symbolic equations that govern articulatory gestures during speech. A sparse symbolic regression algorithm is used to discover models from kinematic data on the tongue and lips. We explore these candidate models using analytical techniques and numerical simulations, and find that a second-order linear model achieves high levels of accuracy, but a nonlinear force is required to properly model articulatory dynamics in approximately one third of cases. This supports the proposal that an autonomous, nonlinear, second-order differential equation is a viable dynamical law for articulatory gestures in speech. We conclude by identifying future opportunities and obstacles in data-driven model discovery and outline prospects for discovering the dynamical principles that govern language, brain and behaviour.

Continual learning is an emerging subject in machine learning that aims to solve multiple tasks presented sequentially to the learner without forgetting previously learned tasks. Recently, many deep learning based approaches have been proposed for continual learning, however the mathematical foundations behind existing continual learning methods remain underdeveloped. On the other hand, adaptive filtering is a classic subject in signal processing with a rich history of mathematically principled methods. However, its role in understanding the foundations of continual learning has been underappreciated. In this tutorial, we review the basic principles behind both continual learning and adaptive filtering, and present a comparative analysis that highlights multiple connections between them. These connections allow us to enhance the mathematical foundations of continual learning based on existing results for adaptive filtering, extend adaptive filtering insights using existing continual learning methods, and discuss a few research directions for continual learning suggested by the historical developments in adaptive filtering.

Modeling the associations between real world entities from their multivariate cross-sectional profiles can provide cues into the concerted working of these entities as a system. Several techniques have been proposed for deciphering these associations including constraint-based Bayesian structure learning (BSL) algorithms that model them as directed acyclic graphs. Benchmarking these algorithms have typically focused on assessing the variation in performance measures such as sensitivity as a function of the dimensionality represented by the number of nodes in the DAG, and sample size. The present study elucidates the importance of network topology in benchmarking exercises. More specifically, it investigates variations in sensitivity across distinct network topologies while constraining the nodes, edges, and sample-size to be identical, eliminating these as potential confounders. Sensitivity of three popular constraint-based BSL algorithms (Peter-Clarke, Grow-Shrink, Incremental Association Markov Blanket) in learning the network structure from multivariate cross-sectional profiles sampled from network models with sub-linear, linear, and super-linear DAG topologies generated using preferential attachment is investigated. Results across linear and nonlinear models revealed statistically significant $(\alpha=0.05)$ decrease in sensitivity estimates from sub-linear to super-linear topology constitutively across the three algorithms. These results are demonstrated on networks with nodes $(N_{nods}=48,64)$, noise strengths $(\sigma =3,6)$ and sample size $(N = 2^{10})$. The findings elucidate the importance of accommodating the network topology in constraint-based BSL benchmarking exercises.