Nonstationarity is prevalent in sequential decision making, which poses a critical challenge to the vast majority of existing approaches developed offline.

Large language models (LLMs) have recently demonstrated great success in generating and understanding natural language. While they have also shown potential beyond the domain of natural language, it remains an open question as to what extent and in which way these LLMs can plan.

Statistical learning under distributional drift remains insufficiently characterized: when each observation alters the data-generating law, classical generalization bounds can collapse. We introduce a new statistical primitive, the reproducibility budget $C_T$, which quantifies a system's finite capacity for statistical reproducibility - the extent to which its sampling process can remain governed by a consistent underlying distribution in the presence of both exogenous change and endogenous feedback. Formally, $C_T$ is defined as the cumulative Fisher-Rao path length of the coupled learner-environment evolution, measuring the total distributional motion accumulated during learning. From this construct we derive a drift-feedback generalization bound of order $O(T^{-1/2} + C_T/T)$, and we prove a matching minimax lower bound showing that this rate is minimax-optimal. Consequently, the results establish a reproducibility speed limit: no algorithm can achieve smaller worst-case generalization error than that imposed by the average Fisher-Rao drift rate $C_T/T$ of the data-generating process. The framework situates exogenous drift, adaptive data analysis, and performative prediction within a common geometric structure, with $C_T$ emerging as the intrinsic quantity measuring distributional motion across these settings.

Abstract-- Hard-constraint trajectory planners often rely on commercial solvers and demand substantial computational resources. Existing soft-constraint methods achieve faster computation, but either (1) decouple spatial and temporal optimization or (2) restrict the search space. T o overcome these limitations, we introduce MIGHTY, a Hermite spline-based planner that performs spatiotemporal optimization while fully leveraging the continuous search space of a spline. In simulation, MIGHTY achieves a 9.3% reduction in computation time and a 13.1% reduction in travel time over state-of-the-art baselines, with a 100% success rate. In hardware, MIGHTY completes multiple high-speed flights up to 6.7 m/s in a cluttered static environment and long-duration flights with dynamically added obstacles. Trajectory planning for autonomous navigation has been extensively studied, with a wide variety of parameterizations and formulations [3], [6], [7], [9], [12], [14]-[17], [19]-[23].

Path planning in grid maps, arising from various applications, has garnered significant attention. Existing methods, such as A*, Dijkstra, and their variants, work well for small-scale maps but fail to address large-scale ones due to high search time and memory consumption. Recently, Large Language Models (LLMs) have shown remarkable performance in path planning but still suffer from spatial illusion and poor planning performance. Among all the works, LLM-A* \cite{meng2024llm} leverages LLM to generate a series of waypoints and then uses A* to plan the paths between the neighboring waypoints. In this way, the complete path is constructed. However, LLM-A* still suffers from high computational time for large-scale maps. To fill this gap, we conducted a deep investigation into LLM-A* and found its bottleneck, resulting in limited performance. Accordingly, we design an innovative LLM-enhanced algorithm, abbr. as iLLM-A*. iLLM-A* includes 3 carefully designed mechanisms, including the optimization of A*, an incremental learning method for LLM to generate high-quality waypoints, and the selection of the appropriate waypoints for A* for path planning. Finally, a comprehensive evaluation on various grid maps shows that, compared with LLM-A*, iLLM-A* \textbf{1) achieves more than $1000\times$ speedup on average, and up to $2349.5\times$ speedup in the extreme case, 2) saves up to $58.6\%$ of the memory cost, 3) achieves both obviously shorter path length and lower path length standard deviation.}

Obstacle avoidance in complex and dynamic environments is a critical challenge for real-time robot navigation. Model-based and learning-based methods often fail in highly dynamic scenarios because traditional methods assume a static environment and cannot adapt to real-time changes, while learning-based methods rely on single-frame observations for motion constraint estimation, limiting their adaptability. To overcome these limitations, this paper proposes a novel framework that leverages multi-frame point constraints, including current and future frames predicted by a dedicated module, to enable proactive end-to-end navigation. By incorporating a prediction module that forecasts the future path of moving obstacles based on multi-frame observations, our method allows the robot to proactively anticipate and avoid potential dangers. This proactive planning capability significantly enhances navigation robustness and efficiency in unknown dynamic environments. Simulations and real-world experiments validate the effectiveness of our approach.

For example, most of the gradient in a residual network with 110 layers comes from paths that are only 10-34 layers deep.

The goal is to find the model, i.e. hyper-parameters, with the highest validation accuracy.