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Accurate environmental monitoring relies on Informative Path Planning (IPP) to deploy autonomous systems that maximize information gain. While Gaussian Processes (GPs) provide a robust mathematical foundation for modeling the environment, contemporary IPP approaches remain hindered by heuristic operational constraints and the lack of simultaneous theoretical guarantees for worst-case estimation error and approximation quality. Rather than introducing yet another heuristic, this paper presents a rigorous theoretical analysis of existing error bounds and algorithms to further develop the mathematical foundation for IPP. By evaluating probabilistic and deterministic bounds on reconstruction error, we establish that predictive variance is the theoretically ideal optimization objective for GP-based IPP. Furthermore, we analyze GreedyCover, a recent approach for optimizing predictive variance, and demonstrate that its near-optimal approximation guarantee matches the fundamental hardness-of-approximation limit for the problem. Comprehensive benchmarking confirms that this approach delivers superior estimation accuracy and computational efficiency compared to prominent information-theoretic objectives and optimization methods used in IPP. Finally, this theoretical analysis establishes a rigorous, near-optimal foundation for discrete offline IPP under bounded uncertainty constraints, offering clear and principled guidance for future work in continuous-space and online formulations.

Reinforcement learning (RL) has become a standard post-training method for diffusion models, but most RL objectives roll out the current policy at every iteration, and with a many-step sampler these rollouts typically dominate the cost of training. Few-step models cut this cost, yet they model an average velocity over an interval, whereas existing RL objectives are built for an instantaneous velocity with a many-step sampler. We adapt RL objectives to few-step models systematically in three phases, namely a transformation, a decomposition and an adaptation. The transformation to $x$-space expresses each objective we consider through a common policy prediction, at the clean end by default and at an explicitly marked level for transition or flow-map objectives, so that objectives stated over densities, velocities or rewards become comparable. The decomposition writes the gradient of each of these objectives as the sum of a self-regression toward a reference and a reward augmentation, so that a method reduces, at the gradient level, to a choice of anchors and weights. The adaptation computes each anchor from a few-step model at the end of its transport or along its tangent, with the instantaneous velocity recovered from the network itself or from an auxiliary network. This yields few-step versions of FlowGRPO, AWM, DiffusionNFT, DiffusionOPSD and ReFL in one modular codebase. In our experiments, reward-gradient-based methods are the most effective when the reward is differentiable and its gradient is affordable; otherwise DiffusionNFT-style methods generally reach high rewards in relatively few iterations despite a sometimes higher per-iteration cost. Extra inference steps show no systematic aggregate benefit in our experiments, although selected outputs visibly improve with two to four times as many sampling steps as in training.