Planning with pretrained diffusion models has emerged as a promising approach for solving test-time guided control problems. Standard gradient guidance typically performs optimally under convex, differentiable reward landscapes. However, it shows substantially reduced effectiveness in real-world scenarios with non-convex objectives, non-differentiable constraints, and multi-reward structures. Furthermore, recent supervised planning approaches require task-specific training or value estimators, which limits test-time flexibility and zero-shot generalization. We propose a Tree-guided Diffusion Planner (TDP), a zero-shot test-time planning framework that balances exploration and exploitation through structured trajectory generation. We frame test-time planning as a tree search problem using a bi-level sampling process: (1) diverse parent trajectories are produced via training-free particle guidance to encourage broad exploration, and (2) sub-trajectories are refined through fast conditional denoising guided by task objectives. TDP addresses the limitations of gradient guidance by exploring diverse trajectory regions and harnessing gradient information across this expanded solution space using only pretrained models and test-time reward signals. We evaluate TDP on three diverse tasks: maze gold-picking, robot arm block manipulation, and AntMaze multi-goal exploration. TDP consistently outperforms state-of-the-art approaches on all tasks.

Clustering is a fundamental problem that has been extensively studied over past few decades, with most research focusing on point-based clustering such as $k$-means, $k$-median, and $k$-center. However, numerous real-world applications, such as motion analysis, computer vision, and missing data analysis, require clustering over structured data, including lines, time series and affine subspaces (flats), where traditional point-based clustering algorithms often fall short. In this paper, we study the $k$-means of lines problem, where the input is a set $L$ of lines in $\mathbb{R}^d$, and the goal is to find $k$ centers $C$ in $\mathbb{R}^d$ such that the sum of squared distances from each line in $L$ to its nearest center in $C$ is minimized. The local search algorithm is a well-established strategy for point-based $k$-means clustering, known for its efficiency and provable approximation guarantees. However, extending local search algorithm to the $k$-means of lines problem is nontrivial, as the capture relation used in point-based clustering does not generalize to the line setting.

Contrastive learning is a powerful technique for discovering meaningful data representations by optimizing objectives based on $\textit{contrastive information}$, often given as a set of weighted triplets $\{(x_i, y_i^+, z_{i}^-)\}_{i = 1}^m$ indicating that an anchor $x_i$ is more similar to a positive example $y_i$ than to a negative example $z_i$. The goal is to find representations (e.g., embeddings in $\mathbb{R}^d$ or a tree metric) where anchors are placed closer to positive than to negative examples. While finding $\textit{global}$ optima of contrastive objectives is $\mathsf{NP}$-hard, the complexity of finding $\text{\textit{local}}$ optima---representations that do not improve by local search algorithms such as gradient-based methods---remains open. Our work settles the complexity of finding local optima in various contrastive learning problems by proving $\mathsf{PLS}$-hardness in discrete settings (e.g., maximize satisfied triplets) and $\mathsf{CLS}$-hardness in continuous settings (e.g., minimize Triplet Loss), where $\mathsf{PLS}$ (Polynomial Local Search) and $\mathsf{CLS}$ (Continuous Local Search) are well-studied complexity classes capturing local search dynamics in discrete and continuous optimization, respectively.

Local search is a powerful clustering technique that provides high-quality solutions with theoretical guarantees. With distance-based sampling strategies, local search methods can achieve constant approximations for clustering with linear running time in data size. Despite their effectiveness, existing algorithms still face scalability issues as they require scanning the entire dataset for iterative center swaps. This typically leads to an O(ndk) running time, where n is the data size, d is the dimension, k is the number of clusters. To further improve the efficiency of local search algorithms, we propose new methods based on adaptive sampling and bandit strategies.

Symbolic regression aims to discover concise, interpretable mathematical expressions that satisfy desired objectives, such as fitting data, posing a highly combinatorial optimization problem. While genetic programming has been the dominant approach, recent efforts have explored reinforcement learning methods for improving search efficiency. Monte Carlo Tree Search (MCTS), with its ability to balance exploration and exploitation through guided search, has emerged as a promising technique for symbolic expression discovery.

In modern sequential decision-making systems, the construction of an optimal candidate action space is critical to efficient inference. However, existing approaches either rely on manually defined action spaces that lack scalability or utilize unstructured spaces that render exhaustive search computationally prohibitive. In this paper, we propose a novel framework named \textsc{DynaAct} for automatically constructing a compact action space to enhance sequential reasoning in complex problem-solving scenarios. Our method first estimates a proxy for the complete action space by extracting general sketches observed in a corpus covering diverse complex reasoning problems using large language models. We then formulate a submodular function that jointly evaluates candidate actions based on their utility to the current state and their diversity, and employ a greedy algorithm to select an optimal candidate set. Extensive experiments on six diverse standard benchmarks demonstrate that our approach significantly improves overall performance, while maintaining efficient inference without introducing substantial latency.

Recent work on neural scaling laws demonstrates that model performance scales predictably with compute budget, model size, and dataset size. In this work, we develop scaling laws based on problem complexity. We analyze two fundamental complexity measures: solution space size and representation space size. Using the Traveling Salesman Problem (TSP) as a case study, we show that combinatorial optimization promotes smooth cost trends, and therefore meaningful scaling laws can be obtained even in the absence of an interpretable loss. We then show that suboptimality grows predictably for fixed-size models when scaling the number of TSP nodes or spatial dimensions, independent of whether the model was trained with reinforcement learning or supervised fine-tuning on a static dataset. We conclude with an analogy to problem complexity scaling in local search, showing that a much simpler gradient descent of the cost landscape produces similar trends.

Signed networks, where edges are labeled as positive or negative to represent friendly or antagonistic interactions, offer a natural framework for analyzing polarization, trust, and conflict in social systems. Detecting meaningful group structures in such networks is crucial for understanding online discourse, political divisions, and trust dynamics. A key challenge is to identify communities that are internally cohesive and externally antagonistic, while allowing for neutral or unaligned vertices. In this paper, we propose a method for identifying $k$ polarized communities that addresses a major limitation of prior methods: their tendency to produce highly size-imbalanced solutions. We introduce a novel optimization objective that avoids such imbalance. In addition, it is well known that approximation algorithms based on *local search* are highly effective for clustering signed networks when neutral vertices are not allowed. We build on this idea and design the first local search algorithm that extends to the setting with neutral vertices while scaling to large networks. By connecting our approach to block-coordinate Frank-Wolfe optimization, we prove a linear convergence rate, enabled by the structure of our objective. Experiments on real-world and synthetic datasets demonstrate that our method consistently outperforms state-of-the-art baselines in solution quality, while remaining competitive in computational efficiency.

Symbolic regression (SR) seeks interpretable analytical expressions that uncover the governing relationships within data, providing mechanistic insight beyond'black-box' models. However, existing SR methods often suffer from two key limitations: (1) *redundant representations* that fail to capture mathematical equivalences and higher-order operand relations, breaking permutation invariance and hindering efficient learning; and (2) *sparse rewards* caused by incomplete incorporation of constraints that can only be evaluated on full expressions, such as constant fitting or physical-law verification. To address these challenges, we propose a unified framework, **Graph-based Symbolic Regression (GSR)**, which compresses the search space through the permutation-invariant representations, expression graphs (EGs), that intrinsically encode expression equivalences via a term-rewriting system (TRS) and a directed acyclic graph (DAG) structure. GSR mitigates reward sparsity by employing a hybrid neural-guided Monte Carlo tree search (hnMCTS) on EGs, where constraint-informed neural guidance enables the direct incorporation of expression-level constraint priors, and an adaptive $\epsilon$-UCB policy balances exploration and exploitation. Theoretical analyses establish the uniqueness of our proposed EG representation and the convergence of the hnMCTS algorithm. Experiments on synthetic and real-world scientific datasets demonstrate the efficiency and accuracy of GSR in discovering underlying expressions and adhering to physical laws, offering practical solutions for scientific discovery.

Bayesian optimization (BO) is a popular and effective approach for tuning expensive, noisy experiments, but requires the formulation of an explicit objective function. Preferential BO (PBO) removes this requirement by learning from pairwise human feedback, yet existing methods struggle to efficiently optimize beyond low- and medium-dimensional problems due to their global search approaches. We address this limitation by developing a family of local PBO methods that transfer key ideas from high-dimensional BO to the preferential setting. In particular, we introduce local PBO methods which adapt trust-region and derivative-informed local search to pairwise preference feedback, where the latter exploits first- and second-order derivatives of the Laplace-approximated GP posterior. Our benchmark on GP sample paths, standard optimization benchmark functions, and policy-search tasks shows that local PBO methods are especially effective in high-dimensional and complex landscapes with steep optima. Compared with global preference-based baselines, they can substantially reduce cumulative regret, making them particularly useful for real-world preference-based optimization tasks such as policy search.