We consider the problem of graph searching with prediction recently introduced by Banerjee et al. (2023). In this problem, an agent starting at some vertex r has to traverse a (potentially unknown) graph G to find a hidden goal node g while minimizing the total distance traveled. We study a setting in which at any node v, the agent receives a noisy estimate of the distance from v to g. We design algorithms for this search task on unknown graphs. We establish the first formal guarantees on unknown weighted graphs and provide lower bounds showing that the algorithms we propose have optimal or nearly-optimal dependence on the prediction error. Further, we perform numerical experiments demonstrating that in addition to being robust to adversarial error, our algorithms perform well in typical instances in which the error is stochastic. Finally, we provide simpler performance bounds on the algorithms of Banerjee et al. (2023) for the case of searching on a known graph and establish new lower bounds for this setting.

A fundamental approach to semi-supervised learning is to leverage the structure of the sample space to diffuse label information from annotated examples to unlabeled points. Traditional methods model the input data points as a graph and rely on fast algorithms for solving Laplacian systems of equations, such as those defining PageRank. However, previous work has demonstrated that graph-based models fail to capture higher-order relations, such as group membership, which are better modeled by hypergraphs. Unfortunately, the scalable application of hypergraph models has been hampered by the non-linearity of the hypergraph Laplacian. In this paper, we present highly scalable algorithms for hypergraph primitives, such as hypergraph PageRank vectors and hypergraph Laplacian systems, over general families of hypergraphs. In addition to giving strong theoretical guarantees, we empirically showcase the speed of our algorithms on benchmark instances of semi-supervised learning on categorical data. We exploit their generality to improve semi-supervised manifold clustering via hypergraph models. By providing significant speed-ups on fundamental hypergraph tasks, our algorithms enable the deployment of hypergraph models on a massive scale.

We consider a clustering problem where a learner seeks to partition a finite set by querying a faulty oracle. This models applications where learners crowdsource information from non-expert human workers or conduct noisy experiments to determine group structure. The learner aims to exactly recover a partition by submitting queries of the form ``are $u$ and $v$ in the same group?'' for any pair of elements $u$ and $v$ in the set. Moreover, because the learner only has access to faulty sources of information, they require an error-tolerant algorithm for this task: i.e. they must fully recover the correct partition, even if up to $\ell$ answers are incorrect, for some error-tolerance parameter $\ell$. We study the question: for any given error-tolerance $\ell$, what is the minimum number of queries needed to learn a finite set partition of $n$ elements into $k$ groups? We design algorithms for this task and prove that they achieve optimal query complexity. To analyze our algorithms, we first highlight a connection between this task and correlation clustering. We then use this connection to build a Rényi-Ulam style analytical framework for this problem, which yields matching lower bounds. Our analysis also reveals an inherent asymmetry between the query complexity necessary to be robust against false negative errors as opposed to false positive errors.