How can we subsample graph data so that a graph neural network (GNN) trained on the subsample achieves performance comparable to training on the full dataset? This question is of fundamental interest, as smaller datasets reduce labeling costs, storage requirements, and computational resources needed for training. Selecting an effective subset is challenging: a poorly chosen subsample can severely degrade model performance, and empirically testing multiple subsets for quality obviates the benefits of subsampling. Therefore, it is critical that subsampling comes with guarantees on model performance. In this work, we introduce new subsampling methods for graph datasets that leverage the Tree Mover's Distance to reduce both the number of graphs and the size of individual graphs. To our knowledge, our approach is the first that is supported by rigorous theoretical guarantees: we prove that training a GNN on the subsampled data results in a bounded increase in loss compared to training on the full dataset. Unlike existing methods, our approach is both model-agnostic, requiring minimal assumptions about the GNN architecture, and label-agnostic, eliminating the need to label the full training set. This enables subsampling early in the model development pipeline (before data annotation, model selection, and hyperparameter tuning) reducing costs and resources needed for storage, labeling, and training. We validate our theoretical results with experiments showing that our approach outperforms existing subsampling methods across multiple datasets.

Reinforcement Learning (RL) has become a popular learning framework to solve sequential decision making problems in unknown environments, and has achieved tremendous success in a wide array of domains such as playing the game of Go (Silver et al., 2016), robotic control (Kober et al., 2013), and autonomous driving (Kiran et al., 2022; Lin et al., 2023). A critical feature of most real-world systems is their uncertain nature, and consequently RL has emerged as a powerful tool for learning optimal policies for multi-agent systems to operate in unknown environments (Kim & Giannakis, 2017; Zhang et al., 2021; Lin et al., 2024; Anand & Qu, 2024). While the early literature on RL predominantly focused on the single-agent setting, multi-agent reinforcement learning (MARL) has also recently achieved impressive successes in a broad range of areas, such as coordination of robotic swarms (Preiss et al., 2017), self-driving vehicles (DeWeese & Qu, 2024), real-time bidding (Jin et al., 2018), ride-sharing (Li et al., 2019), and stochastic games (Jin et al., 2020). Despite growing interest in multi-agent RL (MARL), extending RL to multi-agent settings poses significant computational challenges due to the curse of dimensionality (Sayin et al., 2021). Even if the individual agents' state or action spaces are small, the global state space or action space can take values from a set with size that is exponentially large as a function of the number of agents.

We consider the problem of estimating the spectral density of the normalized adjacency matrix of an $n$-node undirected graph. We provide a randomized algorithm that, with $O(n\epsilon^{-2})$ queries to a degree and neighbor oracle and in $O(n\epsilon^{-3})$ time, estimates the spectrum up to $\epsilon$ accuracy in the Wasserstein-1 metric. This improves on previous state-of-the-art methods, including an $O(n\epsilon^{-7})$ time algorithm from [Braverman et al., STOC 2022] and, for sufficiently small $\epsilon$, a $2^{O(\epsilon^{-1})}$ time method from [Cohen-Steiner et al., KDD 2018]. To achieve this result, we introduce a new notion of graph sparsification, which we call nuclear sparsification. We provide an $O(n\epsilon^{-2})$-query and $O(n\epsilon^{-2})$-time algorithm for computing $O(n\epsilon^{-2})$-sparse nuclear sparsifiers. We show that this bound is optimal in both its sparsity and query complexity, and we separate our results from the related notion of additive spectral sparsification. Of independent interest, we show that our sparsification method also yields the first deterministic algorithm for spectral density estimation that scales linearly with $n$ (sublinear in the representation size of the graph).

We provide faster randomized algorithms for computing an $\epsilon$-optimal policy in a discounted Markov decision process with $A_{\text{tot}}$-state-action pairs, bounded rewards, and discount factor $\gamma$. We provide an $\tilde{O}(A_{\text{tot}}[(1 - \gamma)^{-3}\epsilon^{-2} + (1 - \gamma)^{-2}])$-time algorithm in the sampling setting, where the probability transition matrix is unknown but accessible through a generative model which can be queried in $\tilde{O}(1)$-time, and an $\tilde{O}(s + (1-\gamma)^{-2})$-time algorithm in the offline setting where the probability transition matrix is known and $s$-sparse. These results improve upon the prior state-of-the-art which either ran in $\tilde{O}(A_{\text{tot}}[(1 - \gamma)^{-3}\epsilon^{-2} + (1 - \gamma)^{-3}])$ time [Sidford, Wang, Wu, Ye 2018] in the sampling setting, $\tilde{O}(s + A_{\text{tot}} (1-\gamma)^{-3})$ time [Sidford, Wang, Wu, Yang, Ye 2018] in the offline setting, or time at least quadratic in the number of states using interior point methods for linear programming. We achieve our results by building upon prior stochastic variance-reduced value iteration methods [Sidford, Wang, Wu, Yang, Ye 2018]. We provide a variant that carefully truncates the progress of its iterates to improve the variance of new variance-reduced sampling procedures that we introduce to implement the steps. Our method is essentially model-free and can be implemented in $\tilde{O}(A_{\text{tot}})$-space when given generative model access. Consequently, our results take a step in closing the sample-complexity gap between model-free and model-based methods.

In clustering algorithm selection, we are given a massive dataset and must efficiently select which clustering algorithm to use. We study this problem in a semi-supervised setting, with an unknown ground-truth clustering that we can only access through expensive oracle queries. Ideally, the clustering algorithm's output will be structurally close to the ground truth. We approach this problem by introducing a notion of size generalization for clustering algorithm accuracy. We identify conditions under which we can (1) subsample the massive clustering instance, (2) evaluate a set of candidate algorithms on the smaller instance, and (3) guarantee that the algorithm with the best accuracy on the small instance will have the best accuracy on the original big instance. We provide theoretical size generalization guarantees for three classic clustering algorithms: single-linkage, k-means++, and (a smoothed variant of) Gonzalez's k-centers heuristic. We validate our theoretical analysis with empirical results, observing that on real-world clustering instances, we can use a subsample of as little as 5% of the data to identify which algorithm is best on the full dataset.