Markov networks are probabilistic graphical models that employ undirected graphs to depict conditional independence relationships among variables. Our focus lies in constraint-based structure learning, which entails learning the undirected graph from data through the execution of conditional independence tests. We establish theoretical limits concerning two critical aspects of constraint-based learning of Markov networks: the number of tests and the sizes of the conditioning sets. These bounds uncover an exciting interplay between the structural properties of the graph and the amount of tests required to learn a Markov network. The starting point of our work is that the graph parameter maximum pairwise connectivity, $κ$, that is, the maximum number of vertex-disjoint paths connecting a pair of vertices in the graph, is responsible for the sizes of independence tests required to learn the graph. On one hand, we show that at least one test with the size of the conditioning set at least $κ$ is always necessary. On the other hand, we prove that any graph can be learned by performing tests of size at most $κ$. This completely resolves the question of the minimum size of conditioning sets required to learn the graph. When it comes to the number of tests, our upper bound on the sizes of conditioning sets implies that every $n$-vertex graph can be learned by at most $n^κ$ tests with conditioning sets of sizes at most $κ$. We show that for any upper bound $q$ on the sizes of the conditioning sets, there exist graphs with $O(n q)$ vertices that require at least $n^{Ω(κ)}$ tests to learn. This lower bound holds even when the treewidth and the maximum degree of the graph are at most $κ+2$. On the positive side, we prove that every graph of bounded treewidth can be learned by a polynomial number of tests with conditioning sets of sizes at most $2κ$.
Controlling Exploration-Exploitation in GFlowNets via Markov Chain Perspectives
LUMIA Lab, School of Artificial Intelligence, SJTU（LUMIA实验室，人工智能学院，上海交通大学）；School of Mathematical Sciences, SJTU（数学科学学院，上海交通大学）；Shanghai Advanced Institute of Finance, SJTU（上海先进金融研究院，上海交通大学）；School of Computer Science, SJTU（计算机科学学院，上海交通大学）；Shanghai AI Laboratory（上海人工智能实验室）；Shanghai Innovation Institute（上海创新研究院）；Shanghai Center for Data Science, NYU Shanghai（上海数据科学中心，纽约大学上海学院）；NYU-ECNU Institute of Mathematical Sciences, NYU Shanghai（纽约大学-华东师范大学数学科学研究院，纽约大学上海学院）
Safety Alignment Depth in Large Language Models: A Markov Chain Perspective
Institute of Information Science (IIS), Academia Sinica（中央研究院资讯科学研究所）；National Taiwan University（国立台湾大学）；National Yang Ming Chiao Tung University（国立阳明交通大学）
On the physics of nested Markov models: a generalized probabilistic theory perspective
QICI Quantum Information and Computation Initiative, School of Computing and Data Science, The University of Hong Kong, Hong Kong SAR, China（量子信息与计算倡议，计算与数据科学学院，香港大学，香港特别行政区，中国）；Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China（交叉信息科学研究所，清华大学，北京，中国）；Shanghai Qi Zhi Institute, Shanghai, China（上海启智研究所，上海，中国）；Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada（理论物理研究所，滑铁卢，安大略，加拿大）
A Graph Perspective to Probe Structural Patterns of Knowledge in Large Language Models
University of Oregon（俄勒冈大学）；Adobe Research（Adobe研究）；Cisco AI Research（思科人工智能研究）；Pacific Northwest National Laboratory（太平洋西北国家实验室）；Rensselaer Polytechnic Institute（拉特格斯理工学院）
Rethinking Generalization in Graph Neural Networks: A Structural Complexity Perspective
Institute of Intelligent Information Processing（智能信息处理研究院）；Shanxi University（山西大学）；Alliance Manchester Business School University of Manchester（曼彻斯特大学曼彻斯特商业学院）；University of Manchester（曼彻斯特大学）；Department of Mathematics（数学系）；Hong Kong Baptist University（香港 Baptist 大学）
Manipulating hidden-Markov-model inferences by corrupting batch data
United States Air Force Academy（美国空军学院）；Institute of Mathematical Sciences (ICMAT)（数学科学研究所）；McCoy College of Business, Texas State University（德克萨斯州立大学麦科伊商学院）；CUNEF Universidad（库内夫大学）
