We propose an unsupervised approach for learning vertex orderings for the maximum clique problem by framing it within a permutation-based framework. We transform the combinatorial constraints into geometric relationships such that the ordering of vertices aligns with the clique structures. By integrating this clique-oriented ordering into branch-and-bound search, we improve search efficiency and reduce the number of computational steps. Our results demonstrate how unsupervised learning of vertex ordering can enhance search efficiency across diverse graph instances. We further study the generalization across different sizes.

The study of propositional logic -- fundamental to the theory of computing -- is a cornerstone of the undergraduate computer science curriculum. Learning to solve logical proofs requires repeated guided practice, but undergraduate students often lack access to on-demand tutoring in a judgment-free environment. In this work, we highlight the need for guided practice tools in undergraduate mathematics education and outline the desiderata of an effective practice tool. We accordingly develop LogicLearner, a web application for guided logic proof practice. LogicLearner consists of an interface to attempt logic proofs step-by-step and an automated proof solver to generate solutions on the fly, allowing users to request guidance as needed. We pilot LogicLearner as a practice tool in two semesters of an undergraduate discrete mathematics course and receive strongly positive feedback for usability and pedagogical value in student surveys. To the best of our knowledge, LogicLearner is the only learning tool that provides an end-to-end practice environment for logic proofs with immediate, judgment-free feedback.

Proving Rubik's Cube theorems at the high level represents a notable milestone in human-level spatial imagination and logic thinking and reasoning. Traditional Rubik's Cube robots, relying on complex vision systems and fixed algorithms, often struggle to adapt to complex and dynamic scenarios. To overcome this limitation, we introduce CubeRobot, a novel vision-language model (VLM) tailored for solving 3x3 Rubik's Cubes, empowering embodied agents with multimodal understanding and execution capabilities. We used the CubeCoT image dataset, which contains multiple-level tasks (43 subtasks in total) that humans are unable to handle, encompassing various cube states. We incorporate a dual-loop VisionCoT architecture and Memory Stream, a paradigm for extracting task-related features from VLM-generated planning queries, thus enabling CubeRobot to independent planning, decision-making, reflection and separate management of high- and low-level Rubik's Cube tasks. Furthermore, in low-level Rubik's Cube restoration tasks, CubeRobot achieved a high accuracy rate of 100%, similar to 100% in medium-level tasks, and achieved an accuracy rate of 80% in high-level tasks.

Large Vision-Language Models (VLMs), such as GPT-4, have achieved remarkable success across various fields. However, there are few studies on 3D indoor scene generation with VLMs. This paper considers this task as a planning problem subject to spatial and layout common sense constraints. To solve the problem with a VLM, we propose a new global-local tree search algorithm. Globally, the method places each object sequentially and explores multiple placements during each placement process, where the problem space is represented as a tree. To reduce the depth of the tree, we decompose the scene structure hierarchically, i.e. room level, region level, floor object level, and supported object level. The algorithm independently generates the floor objects in different regions and supported objects placed on different floor objects. Locally, we also decompose the sub-task, the placement of each object, into multiple steps. The algorithm searches the tree of problem space. To leverage the VLM model to produce positions of objects, we discretize the top-down view space as a dense grid and fill each cell with diverse emojis to make to cells distinct. We prompt the VLM with the emoji grid and the VLM produces a reasonable location for the object by describing the position with the name of emojis. The quantitative and qualitative experimental results illustrate our approach generates more plausible 3D scenes than state-of-the-art approaches. Our source code is available at https://github.com/dw-dengwei/TreeSearchGen .

This paper presents an efficient preference elicitation framework for uncertain matroid optimization, where precise weight information is unavailable, but insights into possible weight values are accessible. The core innovation of our approach lies in its ability to systematically elicit user preferences, aligning the optimization process more closely with decision-makers' objectives. By incrementally querying preferences between pairs of elements, we iteratively refine the parametric uncertainty regions, leveraging the structural properties of matroids. Our method aims to achieve the exact optimum by reducing regret with a few elicitation rounds. Additionally, our approach avoids the computation of Minimax Regret and the use of Linear programming solvers at every iteration, unlike previous methods. Experimental results on four standard matroids demonstrate that our method reaches optimality more quickly and with fewer preference queries than existing techniques.

In this paper, we study the problem of (finite sum) minimax optimization in the Differential Privacy (DP) model. Unlike most of the previous studies on the (strongly) convex-concave settings or loss functions satisfying the Polyak-Lojasiewicz condition, here we mainly focus on the nonconvex-strongly-concave one, which encapsulates many models in deep learning such as deep AUC maximization. Specifically, we first analyze a DP version of Stochastic Gradient Descent Ascent (SGDA) and show that it is possible to get a DP estimator whose $l_2$-norm of the gradient for the empirical risk function is upper bounded by $\tilde{O}(\frac{d^{1/4}}{({n\epsilon})^{1/2}})$, where $d$ is the model dimension and $n$ is the sample size. We then propose a new method with less gradient noise variance and improve the upper bound to $\tilde{O}(\frac{d^{1/3}}{(n\epsilon)^{2/3}})$, which matches the best-known result for DP Empirical Risk Minimization with non-convex loss. We also discussed several lower bounds of private minimax optimization. Finally, experiments on AUC maximization, generative adversarial networks, and temporal difference learning with real-world data support our theoretical analysis.

We study the problem of sequential probability assignment under logarithmic loss, both with and without side information. Our objective is to analyze the minimax regret -- a notion extensively studied in the literature -- in terms of geometric quantities, such as covering numbers and scale-sensitive dimensions. We show that the minimax regret for the case of no side information (equivalently, the Shtarkov sum) can be upper bounded in terms of sequential square-root entropy, a notion closely related to Hellinger distance. For the problem of sequential probability assignment with side information, we develop both upper and lower bounds based on the aforementioned entropy. The lower bound matches the upper bound, up to log factors, for classes in the Donsker regime (according to our definition of entropy).

Model customization requires high-quality and diverse datasets, but acquiring such data remains challenging and costly. Although large language models (LLMs) can synthesize training data, current approaches are constrained by limited seed data, model bias and insufficient control over the generation process, resulting in limited diversity and biased distribution with the increase of data scales. To tackle this challenge, we present TreeSynth, a tree-guided subspace-based data synthesis framework that recursively partitions the entire data space into hierar-chical subspaces, enabling comprehensive and diverse scaling of data synthesis. Briefly, given a task-specific description, we construct a data space partitioning tree by iteratively executing criteria determination and subspace coverage steps. This hierarchically divides the whole space (i.e., root node) into mutually exclusive and complementary atomic subspaces (i.e., leaf nodes). By collecting synthesized data according to the attributes of each leaf node, we obtain a diverse dataset that fully covers the data space. Empirically, our extensive experiments demonstrate that TreeSynth surpasses both human-designed datasets and the state-of-the-art data synthesis baselines, achieving maximum improvements of 45.2% in data diversity and 17.6% in downstream task performance across various models and tasks. Hopefully, TreeSynth provides a scalable solution to synthesize diverse and comprehensive datasets from scratch without human intervention.

Deep learning approaches are becoming increasingly attractive for equation discovery. We show the advantages and disadvantages of using neural-guided equation discovery by giving an overview of recent papers and the results of experiments using our modular equation discovery system MGMT ($\textbf{M}$ulti-Task $\textbf{G}$rammar-Guided $\textbf{M}$onte-Carlo $\textbf{T}$ree Search for Equation Discovery). The system uses neural-guided Monte-Carlo Tree Search (MCTS) and supports both supervised and reinforcement learning, with a search space defined by a context-free grammar. We summarize seven desirable properties of equation discovery systems, emphasizing the importance of embedding tabular data sets for such learning approaches. Using the modular structure of MGMT, we compare seven architectures (among them, RNNs, CNNs, and Transformers) for embedding tabular datasets on the auxiliary task of contrastive learning for tabular data sets on an equation discovery task. For almost all combinations of modules, supervised learning outperforms reinforcement learning. Moreover, our experiments indicate an advantage of using grammar rules as action space instead of tokens. Two adaptations of MCTS -- risk-seeking MCTS and AmEx-MCTS -- can improve equation discovery with that kind of search.

Strength estimation and adjustment are crucial in designing human-AI interactions, particularly in games where AI surpasses human players. This paper introduces a novel strength system, including a strength estimator (SE) and an SE-based Monte Carlo tree search, denoted as SE-MCTS, which predicts strengths from games and offers different playing strengths with human styles. The strength estimator calculates strength scores and predicts ranks from games without direct human interaction. SE-MCTS utilizes the strength scores in a Monte Carlo tree search to adjust playing strength and style. We first conduct experiments in Go, a challenging board game with a wide range of ranks. Our strength estimator significantly achieves over 80% accuracy in predicting ranks by observing 15 games only, whereas the previous method reached 49% accuracy for 100 games. For strength adjustment, SE-MCTS successfully adjusts to designated ranks while achieving a 51.33% accuracy in aligning to human actions, outperforming a previous stateof-the-art, with only 42.56% accuracy. To demonstrate the generality of our strength system, we further apply SE and SE-MCTS to chess and obtain consistent results. These results show a promising approach to strength estimation and adjustment, enhancing human-AI interactions in games. Artificial intelligence has achieved superhuman performance in various domains in recent years, especially in games (Silver et al., 2018; Schrittwieser et al., 2020; Vinyals et al., 2019; OpenAI et al., 2019). These achievements have raised interests within the community in exploring AI programs for human interactions, particularly in estimating human players' strengths and offering corresponding levels to increase entertainment or improve skills (Demediuk et al., 2017; Fan et al., 2019; Moon & Seo, 2020; Gusmão et al., 2015; Silva et al., 2015; Hunicke & Chapman, 2004).