Diffusion models have demonstrated remarkable performance in modeling complex data priors, catalyzing their widespread adoption in solving various inverse problems. However, the inherently iterative nature of diffusion-based inverse algorithms often requires hundreds to thousands of steps, with performance degradation occurring under fewer steps which limits their practical applicability. While high-order diffusion ODE solvers have been extensively explored for efficient diffusion sampling without observations, their application to inverse problems remains underexplored due to the diverse forms of inverse algorithms and their need for repeated trajectory correction based on observations. To address this gap, we first introduce a canonical form that decomposes existing diffusion-based inverse algorithms into three modules to unify their analysis. Inspired by the linear subspace search strategy in the design of high-order diffusion ODE solvers, we propose the Learnable Linear Extrapolation (LLE) method, a lightweight approach that universally enhances the performance of any diffusion-based inverse algorithm that fits the proposed canonical form. Extensive experiments demonstrate consistent improvements of the proposed LLE method across multiple algorithms and tasks, indicating its potential for more efficient solutions and boosted performance of diffusion-based inverse algorithms with limited steps. Codes for reproducing our experiments are available at https://github.com/weigerzan/LLE_inverse_problem.

A fundamental problem in statistics and machine learning is to estimate a function $f$ from possibly noisy observations of its point samples. The goal is to design a numerical algorithm to construct an approximation $\hat f$ to $f$ in a prescribed norm that asymptotically achieves the best possible error (as a function of the number $m$ of observations and the variance $\sigma^2$ of the noise). This problem has received considerable attention in both nonparametric statistics (noisy observations) and optimal recovery (noiseless observations). Quantitative bounds require assumptions on $f$, known as model class assumptions. Classical results assume that $f$ is in the unit ball of a Besov space. In nonparametric statistics, the best possible performance of an algorithm for finding $\hat f$ is known as the minimax rate and has been studied in this setting under the assumption that the noise is Gaussian. In optimal recovery, the best possible performance of an algorithm is known as the optimal recovery rate and has also been determined in this setting. While one would expect that the minimax rate recovers the optimal recovery rate when the noise level $\sigma$ tends to zero, it turns out that the current results on minimax rates do not carefully determine the dependence on $\sigma$ and the limit cannot be taken. This paper handles this issue and determines the noise-level-aware (NLA) minimax rates for Besov classes when error is measured in an $L_q$-norm with matching upper and lower bounds. The end result is a reconciliation between minimax rates and optimal recovery rates. The NLA minimax rate continuously depends on the noise level and recovers the optimal recovery rate when $\sigma$ tends to zero.

Solving Partially Observable Markov Decision Processes (POMDPs) in continuous state, action and observation spaces is key for autonomous planning in many real-world mobility and robotics applications. Current approaches are mostly sample based, and cannot hope to reach near-optimal solutions in reasonable time. We propose two complementary theoretical contributions. First, we formulate a novel Multiple Importance Sampling (MIS) tree for value estimation, that allows to share value information between sibling action branches. The novel MIS tree supports action updates during search time, such as gradient-based updates. Second, we propose a novel methodology to compute value gradients with online sampling based on transition likelihoods. It is applicable to MDPs, and we extend it to POMDPs via particle beliefs with the application of the propagated belief trick. The gradient estimator is computed in practice using the MIS tree with efficient Monte Carlo sampling. These two parts are combined into a new planning algorithm Action Gradient Monte Carlo Tree Search (AGMCTS). We demonstrate in a simulated environment its applicability, advantages over continuous online POMDP solvers that rely solely on sampling, and we discuss further implications.

Human reasoning relies on constructing and manipulating mental models-simplified internal representations of situations that we use to understand and solve problems. Conceptual diagrams (for example, sketches drawn by humans to aid reasoning) externalize these mental models, abstracting irrelevant details to efficiently capture relational and spatial information. In contrast, Large Language Models (LLMs) and Large Multimodal Models (LMMs) predominantly reason through textual representations, limiting their effectiveness in complex multi-step combinatorial and planning tasks. In this paper, we propose a zero-shot fully automatic framework that enables LMMs to reason through multiple chains of self-generated intermediate conceptual diagrams, significantly enhancing their combinatorial planning capabilities. Our approach does not require any human initialization beyond a natural language description of the task. It integrates both textual and diagrammatic reasoning within an optimized graph-of-thought inference framework, enhanced by beam search and depth-wise backtracking. Evaluated on multiple challenging PDDL planning domains, our method substantially improves GPT-4o's performance (for example, from 35.5% to 90.2% in Blocksworld). On more difficult planning domains with solution depths up to 40, our approach outperforms even the o1-preview reasoning model (for example, over 13% improvement in Parking). These results highlight the value of conceptual diagrams as a complementary reasoning medium in LMMs.

Target search problems are central to a wide range of fields, from biological foraging to the optimization algorithms. Recently, the ability to reset the search has been shown to significantly improve the searcher's efficiency. However, the optimal resetting strategy depends on the specific properties of the search problem and can often be challenging to determine. In this work, we propose a reinforcement learning (RL)-based framework to train agents capable of optimizing their search efficiency in environments by learning how to reset. First, we validate the approach in a well-established benchmark: the Brownian search with resetting. There, RL agents consistently recover strategies closely resembling the sharp resetting distribution, known to be optimal in this scenario. We then extend the framework by allowing agents to control not only when to reset, but also their spatial dynamics through turning actions. In this more complex setting, the agents discover strategies that adapt both resetting and turning to the properties of the environment, outperforming the proposed benchmarks. These results demonstrate how reinforcement learning can serve both as an optimization tool and a mechanism for uncovering new, interpretable strategies in stochastic search processes with resetting.

Constrained pathfinding is a well-studied, yet challenging network optimisation problem that can be seen in a broad range of real-world applications. Pathfinding with multiple resource limits, which is known as the Resource Constrained Shortest Path Problem (RCSP), aims to plan a cost-optimum path subject to limited usage of resources. Given the recent advances in constrained and multi-criteria search with A*, this paper introduces a new resource constrained search framework on the basis of A* to tackle RCSP in large networks, even in the presence of negative cost and negative resources. We empirically evaluate our new algorithm on a set of large instances and show up to two orders of magnitude faster performance compared to state-of-the-art RCSP algorithms in the literature.

This paper addresses the problem of optimizing communicated information among heterogeneous, resource-aware robot teams to facilitate their navigation. In such operations, a mobile robot compresses its local map to assist another robot in reaching a target within an uncharted environment. The primary challenge lies in ensuring that the map compression step balances network load while transmitting only the most essential information for effective navigation. We propose a communication framework that sequentially selects the optimal map compression in a task-driven, communication-aware manner. It introduces a decoder capable of iterative map estimation, handling noise through Kalman filter techniques. The computational speed of our decoder allows for a larger compression template set compared to previous methods, and enables applications in more challenging environments. Specifically, our simulations demonstrate a remarkable 98% reduction in communicated information, compared to a framework that transmits the raw data, on a large Mars inclination map and an Earth map, all while maintaining similar planning costs. Furthermore, our method significantly reduces computational time compared to the state-of-the-art approach.

This work concerns the path-star task, a minimal example of searching over a graph. The graph, $G$, is star-shaped with $D$ arms radiating from a start node, $s$. A language model (LM) is given $G$, $s$, and a target node $t$, which ends one of the arms and is tasked with generating the arm containing $t$. The minimal nature of this task means only a single choice needs to be made: which of the $D$ arms contains $t$? Decoder-only LMs fail to solve this elementary task above $1/D$ chance due to a learned shortcut that absorbs training supervision. We show how this pathology is caused by excess supervision and we present a series of solutions demonstrating that the task is solvable via decoder-only LMs. We find that the task's minimal nature causes its difficulty, as it prevents task decomposition. Our solutions provide insight into the pathology and its implications for LMs trained via next-token prediction.

In many engineering applications, processes must be followed precisely, making conformance checking between event logs and declarative process models crucial for ensuring adherence to desired behaviors. This is a critical area where Artificial Intelligence (AI) plays a pivotal role in driving effective process improvement. However, computing optimal alignments poses significant computational challenges due to the vast search space inherent in these models. Consequently, existing approaches often struggle with scalability and efficiency, limiting their applicability in real-world settings. This paper introduces DeclareAligner, a novel algorithm that uses the A* search algorithm, an established AI pathfinding technique, to tackle the problem from a fresh perspective leveraging the flexibility of declarative models. Key features of DeclareAligner include only performing actions that actively contribute to fixing constraint violations, utilizing a tailored heuristic to navigate towards optimal solutions, and employing early pruning to eliminate unproductive branches, while also streamlining the process through preprocessing and consolidating multiple fixes into unified actions. The proposed method is evaluated using 8,054 synthetic and real-life alignment problems, demonstrating its ability to efficiently compute optimal alignments by significantly outperforming the current state of the art. By enabling process analysts to more effectively identify and understand conformance issues, DeclareAligner has the potential to drive meaningful process improvement and management.

The application of learning based methods to vehicle routing problems has emerged as a pivotal area of research in combinatorial optimization. These problems are characterized by vast solution spaces and intricate constraints, making traditional approaches such as exact mathematical models or heuristic methods prone to high computational overhead or reliant on the design of complex heuristic operators to achieve optimal or near optimal solutions. Meanwhile, although some recent learning-based methods can produce good performance for VRP with straightforward constraint scenarios, they often fail to effectively handle hard constraints that are common in practice. This study introduces a novel end-to-end framework that combines constraint-oriented hypergraphs with reinforcement learning to address vehicle routing problems. A central innovation of this work is the development of a constraint-oriented dynamic hyperedge reconstruction strategy within an encoder, which significantly enhances hypergraph representation learning. Additionally, the decoder leverages a double-pointer attention mechanism to iteratively generate solutions. The proposed model is trained by incorporating asynchronous parameter updates informed by hypergraph constraints and optimizing a dual loss function comprising constraint loss and policy gradient loss. The experiment results on benchmark datasets demonstrate that the proposed approach not only eliminates the need for sophisticated heuristic operators but also achieves substantial improvements in solution quality.