Computational methods for learning to sample from the Boltzmann distribution-- where the target distribution is known only up to an unnormalized energy function-- have advanced significantly recently. Due to the lack of explicit target samples, however, prior diffusion-based methods, known as diffusion samplers, often require importance-weighted estimation or complicated learning processes.

Graph condensation, which reduces the size of a large-scale graph by synthesizing a small-scale condensed graph as its substitution, has immediate benefits for various graph learning tasks. However, existing graph condensation methods rely on the joint optimization of nodes and structures in the condensed graph, and overlook critical issues in effectiveness and generalization ability. In this paper, we advocate a new Structure-Free Graph Condensation paradigm, named SFGC, to distill a largescale graph into a small-scale graph node set without explicit graph structures, i.e., graph-free data. Our idea is to implicitly encode topology structure information into the node attributes in the synthesized graph-free data, whose topology is reduced to an identity matrix.

In recent years, there has been much interest in understanding the generalization behavior of interpolating predictors, which overfit on noisy training data. Whereas standard analyses are concerned with whether a method is consistent or not, recent observations have shown that even inconsistent predictors can generalize well. In this work, we revisit the classic interpolating Nadaraya-Watson (NW) estimator (also known as Shepard's method), and study its generalization capabilities through this modern viewpoint. In particular, by varying a single bandwidth-like hyperparameter, we prove the existence of multiple overfitting behaviors, ranging non-monotonically from catastrophic, through benign, to tempered. Our results highlight how even classical interpolating methods can exhibit intricate generalization behaviors. In addition, for the purpose of tuning the hyperparameter, the results suggest that over-estimating the intrinsic dimension of the data is less harmful than under-estimating it. Numerical experiments complement our theory, demonstrating the same phenomena.

Sampling from high dimensional distributions is a computational bottleneck in many scientific applications. Hamiltonian Monte Carlo (HMC), and in particular the No-U-Turn Sampler (NUTS), are widely used, yet they struggle on problems with a very large number of parameters or a complicated geometry. Microcanonical Langevin Monte Carlo (MCLMC) has been recently proposed as an alternative which shows striking gains in efficiency over NUTS, especially for high-dimensional problems. However, it produces biased samples, with a bias that is hard to control in general. We introduce the Metropolis-Adjusted Microcanonical sampler (MAMS), which relies on the same dynamics as MCLMC, but introduces a Metropolis-Hastings step and thus produces asymptotically unbiased samples. We develop an automated tuning scheme for the hyperparameters of the algorithm, making it applicable out of the box. We demonstrate that MAMS outperforms NUTS across the board on benchmark problems of varying complexity and dimensionality, achieving up to a factor of seven speedup.

Large language models are increasingly used as judges (LLM-as-a-judge) to evaluate model outputs at scale, but their assessments often diverge systematically from human judgments.

Even in low dimensions, sampling from multi-modal distributions is challenging. We provide the first sampling algorithm for a broad class of distributions -- including all Gaussian mixtures -- with a query complexity that is polynomial in the parameters governing multi-modality, assuming fixed dimension. Our sampling algorithm simulates a time-reversed diffusion process, using a self-normalized Monte Carlo estimator of the intermediate score functions. Unlike previous works, it avoids metastability, requires no prior knowledge of the mode locations, and relaxes the well-known log-smoothness assumption which excluded general Gaussian mixtures so far.

Recent advances in deep learning optimization have unveiled two intriguing phenomena under large learning rates: Edge of Stability (EoS) and Progressive Sharpening (PS), challenging classical Gradient Descent (GD) analyses.

Although recent methods have made progress toward unified solutions, their reliance on post hoc processing introduces considerable application inconvenience and compromises forensic reliability.

Bilevel optimization has emerged as a technique for addressing a wide range of machine learning problems that involve an outer objective implicitly determined by the minimizer of an inner problem. While prior works have primarily focused on the parametric setting, a learning-theoretic foundation for bilevel optimization in the nonparametric case remains relatively unexplored. In this paper, we take a first step toward bridging this gap by studying Kernel Bilevel Optimization (KBO), where the inner objective is optimized over a reproducing kernel Hilbert space. This setting enables rich function approximation while providing a foundation for rigorous theoretical analysis. In this context, we derive novel finite-sample generalization bounds for KBO, leveraging tools from empirical process theory. These bounds further allow us to assess the statistical accuracy of gradient-based methods applied to the empirical discretization of KBO. We numerically illustrate our theoretical findings on a synthetic instrumental variable regression task.

Irregular data in the real world are usually organized as heterogeneous graphs consisting of multiple types of nodes and edges. However, current heterogeneous graph research confronts three fundamental challenges: i) Benchmark Deficiency, ii) Semantic Disalignment, and iii) Propagation Degradation. In this paper, we construct a large-scale, universal, and joint multi-domain heterogeneous graph dataset named UniHG to facilitate heterogeneous graph representation learning and cross-domain knowledge mining. Overall, UniHG contains 77.31 million nodes and 564 million directed edges with thousands of labels and attributes, which is currently the largest universal heterogeneous graph dataset available to the best of our knowledge. To perform effective learning and provide comprehensively benchmarks on UniHG, two key measures are taken, including i) the semantic alignment strategy for multi-attribute entities, which projects the feature description of multi-attribute nodes and edges into a common embedding space to facilitate information aggregation; ii) proposing the novel Heterogeneous Graph Decoupling (HGD) framework with a specifically designed Anisotropy Feature Propagation (AFP) module for learning effective multi-hop anisotropic propagation kernels. These two strategies enable efficient information propagation among a tremendous number of multi-attribute entities and meanwhile mine multi-attribute association adaptively through the multi-hop aggregation in large-scale heterogeneous graphs. Comprehensive benchmark results demonstrate that our model significantly outperforms existing methods with an accuracy improvement of 28.93%. And the UniHG can facilitate downstream tasks, achieving an NDCG@20 improvement rate of 11.48% and 11.71%.