Higher-dimensional sliding puzzles are constructed on the vertices of a $d$-dimensional hypercube, where $2^d-l$ vertices are distinctly coloured. Rings with the same colours are initially set randomly on the vertices of the hypercube. The goal of the puzzle is to move each of the $2^d-l$ rings to pre-defined target vertices on the cube. In this setting, the $k$-rule constraint represents a generalisation of edge collision for the movement of colours between vertices, allowing movement only when a hypercube face of dimension $k$ containing a ring is completely free of other rings. Starting from an initial configuration, what is the minimum number of moves needed to make ring colours match the vertex colours? An algorithm that provides us with such a number is called God's algorithm. When such an algorithm exists, it does not have a polynomial time complexity, at least in the case of the 15-puzzle corresponding to $k=1$ in the cubical puzzle. This paper presents a comprehensive computational study of different scenarios of the higher-dimensional puzzle. A benchmark of three computational techniques, an exact algorithm (the A* search) and two approximately optimal search techniques (an evolutionary algorithm (EA) and reinforcement learning (RL)) is presented in this work. The experiments show that all three methods can successfully solve the puzzle of dimension three for different face dimensions and across various difficulty levels. When the dimension increases, the A* search fails, and RL and EA methods can still provide a generally acceptable solution, i.e. a distribution of a number of moves with a median value of less than $30$. Overall, the EA method consistently requires less computational time, while failing in most cases to minimise the number of moves for the puzzle dimensions $d=4$ and $d=5$.

Graph learning is naturally well suited for use in planning due to its ability to exploit relational structures exhibited in planning domains and to take as input planning instances with arbitrary number of objects. In this paper, we study the usage of graph learning for planning thus far by studying the theoretical and empirical effects on learning and planning performance of (1) graph representations of planning tasks, (2) graph learning architectures, and (3) optimisation formulations for learning. Our studies accumulate in the GOOSE framework which learns domain knowledge from small planning tasks in order to scale up to much larger planning tasks. In this paper, we also highlight and propose the 5 open challenges in the general Learning for Planning field that we believe need to be addressed for advancing the state-of-the-art.

Causal discovery is a crucial endeavor in many scientific fields. Specifically, it focuses on revealing the causal structures within the data. Generally, causal discovery can be carried out through one of two data collection approaches - observational data-based discovery and interventional or experimental data-based discovery. Most of past research employs observational methods, such as those using conditional independence tests, to provide valuable insights into causal structure. However, these methods have significant limitations, as they often face challenges from confounding variables and their inability to determine causality conclusively [1, 2].

Euclidean Steiner trees are relevant to model minimal networks in real-world applications ubiquitously. In this paper, we study the feasibility of a hierarchical approach embedded with bundling operations to compute multiple and mutually disjoint Euclidean Steiner trees that avoid clutter and overlapping with obstacles in the plane, which is significant to model the decentralized and the multipoint coordination of agents in constrained 2D domains. Our computational experiments using arbitrary obstacle configuration with convex and non-convex geometries show the feasibility and the attractive performance when computing multiple obstacle-avoiding Steiner trees in the plane. Our results offer the mechanisms to elucidate new operators for obstacle-avoiding Steiner trees.

Ionizable lipids are essential in developing lipid nanoparticles (LNPs) for effective messenger RNA (mRNA) delivery. While traditional methods for designing new ionizable lipids are typically time-consuming, deep generative models have emerged as a powerful solution, significantly accelerating the molecular discovery process. However, a practical challenge arises as the molecular structures generated can often be difficult or infeasible to synthesize. This project explores Monte Carlo tree search (MCTS)-based generative models for synthesizable ionizable lipids. Leveraging a synthetically accessible lipid building block dataset and two specialized predictors to guide the search through chemical space, we introduce a policy network guided MCTS generative model capable of producing new ionizable lipids with available synthesis pathways.

Scenario-based virtual testing is one of the most significant methods to test and evaluate the safety of automated driving systems (ADSs). However, it is impractical to enumerate all concrete scenarios in a logical scenario space and test them exhaustively. Recently, Black-Box Optimization (BBO) was introduced to accelerate the scenario-based test of ADSs by utilizing the historical test information to generate new test cases. However, a single optimum found by the BBO algorithm is insufficient for the purpose of a comprehensive safety evaluation of ADSs in a logical scenario. In fact, all the subspaces representing danger in the logical scenario space, rather than only the most critical concrete scenario, play a more significant role for the safety evaluation. Covering as many of the critical concrete scenarios in a logical scenario space through a limited number of tests is defined as the Black-Box Coverage (BBC) problem in this paper. We formalized this problem in a sample-based search paradigm and constructed a coverage criterion with Confusion Matrix Analysis. Furthermore, we propose LAMBDA (Latent-Action Monte-Carlo Beam Search with Density Adaption) to solve BBC problems. LAMBDA can quickly focus on critical subspaces by recursively partitioning the logical scenario space into accepted and rejected parts. Compared with its predecessor LaMCTS, LAMBDA introduces sampling density to overcome the sampling bias from optimization and Beam Search to obtain more parallelizability. Experimental results show that LAMBDA achieves state-of-the-art performance among all baselines and can reach at most 33 and 6000 times faster than Random Search to get 95% coverage of the critical areas in 2- and 5-dimensional synthetic functions, respectively. Experiments also demonstrate that LAMBDA has a promising future in the safety evaluation of ADSs in virtual tests.

To efficiently factorize high-dimensional distributed representations to the constituent atomic vectors, one can exploit the compute-in-superposition capabilities of vector-symbolic architectures (VSA). Such factorizers however suffer from the phenomenon of limit cycles. Applying noise during the iterative decoding is one mechanism to address this issue. In this paper, we explore ways to further relax the noise requirement by applying noise only at the time of VSA's reconstruction codebook initialization. While the need for noise during iterations proves analog in-memory computing systems to be a natural choice as an implementation media, the adequacy of initialization noise allows digital hardware to remain equally indispensable. This broadens the implementation possibilities of factorizers. Our study finds that while the best performance shifts from initialization noise to iterative noise as the number of factors increases from 2 to 4, both extend the operational capacity by at least 50 times compared to the baseline factorizer resonator networks. Our code is available at: https://github.com/IBM/in-memory-factorizer

Bootstrap percolation, introduced by Chalupa, Leath, and Reich in their 1979 work [4], serves as a simplified model for ferromagnetic dynamics. Since then, it has been applied in numerous fields of physics and has become a significant topic of interest in mathematics. The process starts with an initial set of "infected" vertices in a graph G, where at each step, any vertex with at least r infected neighbors also becomes infected. A key problem in this framework is determining the minimum size of an initial set that results in the entire graph becoming infected, known as the percolating set. This minimum is denoted by m(G, r).

The Rubik's Cube, with its vast state space and sparse reward structure, presents a significant challenge for reinforcement learning (RL) due to the difficulty of reaching rewarded states. Previous research addressed this by propagating cost-to-go estimates from the solved state and incorporating search techniques. These approaches differ from human strategies that start from fully scrambled cubes, which can be tricky for solving a general sparse-reward problem. In this paper, we introduce a novel RL algorithm using policy gradient methods to solve the Rubik's Cube without relying on near solved-state sampling. Our approach employs a neural network to predict cost patterns between states, allowing the agent to learn directly from scrambled states. Our method was tested on the 2x2x2 Rubik's Cube, where the cube was scrambled 50,000 times, and the model successfully solved it in over 99.4% of cases. Notably, this result was achieved using only the policy network without relying on tree search as in previous methods, demonstrating its effectiveness and potential for broader applications in sparse-reward problems.

We study the kernel instrumental variable algorithm of \citet{singh2019kernel}, a nonparametric two-stage least squares (2SLS) procedure which has demonstrated strong empirical performance. We provide a convergence analysis that covers both the identified and unidentified settings: when the structural function cannot be identified, we show that the kernel NPIV estimator converges to the IV solution with minimum norm. Crucially, our convergence is with respect to the strong $L_2$-norm, rather than a pseudo-norm. Additionally, we characterize the smoothness of the target function without relying on the instrument, instead leveraging a new description of the projected subspace size (this being closely related to the link condition in inverse learning literature). With the subspace size description and under standard kernel learning assumptions, we derive, for the first time, the minimax optimal learning rate for kernel NPIV in the strong $L_2$-norm. Our result demonstrates that the strength of the instrument is essential to achieve efficient learning. We also improve the original kernel NPIV algorithm by adopting a general spectral regularization in stage 1 regression. The modified regularization can overcome the saturation effect of Tikhonov regularization.