We study the fundamental classification problem of computing a separating hyperplane for a binary-labeled dataset of size $n$ with normalized $d$-dimensional features. Letting $\Phi \in \mathbb{R}^{n \times d}$ denote the feature matrix and $\gamma$ the margin of the maximum-margin separating hyperplane, we present a randomized algorithm that solves this problem in $\tilde{O}(\gamma^{-2/3}\, \operatorname{nnz}(\Phi) + \gamma^{-2(\omega+1)/3})$-sequential running time (work), $\tilde{O}(\gamma^{-2/3})$-parallel (computational) depth, and accesses $\Phi$ only through $\tilde{O}(\gamma^{-2/3})$-matrix-vector queries (matvecs). We also present a second, faster randomized algorithm with a $\tilde{O}(\gamma^{-2/3}\, \operatorname{nnz}(\Phi) + \gamma^{-2})$-sequential running time that uses $\tilde{O}(\gamma^{-2/3})$-matvecs to $\Phi$, but achieves only $\tilde{O}(\gamma^{-4/3})$-parallel depth. Both algorithms match the near-optimal deterministic matvec complexity recently established by Kornowski and Shamir [2025], Karmarkar et al. [2026] and achieve improved sequential runtime and parallel depth, albeit at the expense of using randomness.

We study the problem of computing an є-approximate Nash equilibrium of a two-player, bilinear game with a bounded payoff matrix A ∈ ℝm × n, when the players’ strategies are constrained to lie in simple sets. We provide algorithms which solve this problem in Õ(є−2/3) matrix-vector multiplies (matvecs) in two well-studied cases: ℓ1-ℓ1 (or zero-sum) games, where the players’ strategies are both in the probability simplex, and ℓ2-ℓ1 games (encompassing hard-margin SVMs), where the players’ strategies are in the unit Euclidean ball and probability simplex respectively. These results improve upon the previous state-of-the-art complexities of Õ(є−8/9) for ℓ1-ℓ1 and Õ(є−7/9) for ℓ2-ℓ1 due to [KOS ’25]. In both settings our results are nearly-optimal as they match lower bounds of [KS ’25] up to polylogarithmic factors.

In this paper we consider the problem of computing an $\epsilon$-approximate Nash Equilibrium of a zerosum game in a payoff matrix $A \in \mathbb{R}^{m \times n}$ with $O(1)$-bounded entries given access to a matrix-vector product oracle for A and its transpose $A^{\top}$. We provide a deterministic algorithm that solves the problem using $\tilde{O}\left(\epsilon^{-8 / 9}\right)$-oracle queries, where $\tilde{O}(\cdot)$ hides factors polylogarithmic in m, n, and $\epsilon^{-1}$. Our result improves upon the state-of-the-art query complexity of $\tilde{O}\left(\epsilon^{-1}\right)$ established by [Nemirovski, 2004] and [Nesterov, 2005]. We obtain this result through a general framework that yields improved deterministic query complexities for solving a broader class of minimax optimization problems which includes computing a linear classifier (hard-margin support vector machine) as well as linear regression.11This paper is an extended abstract. The full paper can be accessed at https://arxiv.org/abs/2509.04426.

Designing efficient algorithms for multi-agent reinforcement learning (MARL) is fundamentally challenging because the size of the joint state and action spaces grows exponentially in the number of agents. These difficulties are exacerbated when balancing sequential global decision-making with local agent interactions. In this work, we propose a new algorithm $\texttt{SUBSAMPLE-MFQ}$ ($\textbf{Subsample}$-$\textbf{M}$ean-$\textbf{F}$ield-$\textbf{Q}$-learning) and a decentralized randomized policy for a system with $n$ agents. For any $k\leq n$, our algorithm learns a policy for the system in time polynomial in $k$. We prove that this learned policy converges to the optimal policy on the order of $\tilde{O}(1/\sqrt{k})$ as the number of subsampled agents $k$ increases. In particular, this bound is independent of the number of agents $n$.

Data-driven algorithm selection is a powerful approach for choosing effective heuristics for computational problems. It operates by evaluating a set of candidate algorithms on a collection of representative training instances and selecting the one with the best empirical performance. However, running each algorithm on every training instance is computationally expensive, making scalability a central challenge. In practice, a common workaround is to evaluate algorithms on smaller proxy instances derived from the original inputs. However, this practice has remained largely ad hoc and lacked theoretical grounding. We provide the first theoretical foundations for this practice by formalizing the notion of size generalization: predicting an algorithm's performance on a large instance by evaluating it on a smaller, representative instance, subsampled from the original instance. We provide size generalization guarantees for three widely used clustering algorithms (single-linkage, k-means++, and Gonzalez's k-centers heuristic) and two canonical max-cut algorithms (Goemans-Williamson and Greedy). We characterize the subsample size sufficient to ensure that performance on the subsample reflects performance on the full instance, and our experiments support these findings.

We consider the problem of estimating the spectral density of a normalized graph adjacency matrix. Concretely, given an undirected graph $G = (V, E, w)$ with $n$ nodes and positive edge weights $w \in \mathbb{R}^{E}_{> 0}$, the goal is to return eigenvalue estimates $\widehat{\lambda}_1 \le \cdots\le \widehat{\lambda}_n$ such that \begin{align*} \frac{1}{n} \sum_{i\in\{1,\ldots, n\}}|\widehat{\lambda}_i-\lambda_i(N_G)|\le \varepsilon, \end{align*} where ${\lambda}_1(N_G)\le \cdots\le{\lambda}_n(N_G)$ are the eigenvalues of $G$’s normalized adjacency matrix, $N_G$. This goal is equivalent to requiring that the Wasserstein-1 distance between the uniform distribution on $\lambda_1, \ldots, \lambda_n$ and the uniform distribution on $\widehat{\lambda}_1, \ldots, \widehat{\lambda}_n$ is less than $\varepsilon$. We provide a randomized algorithm that achieves the guarantee above with $O(n\varepsilon^{-2})$ queries to a degree and neighbor oracle and in $O(n\varepsilon^{-3})$ time. This improves on previous state-of-the-art methods, including an $O(n\varepsilon^{-7})$ time algorithm from [Braverman et al., STOC 2022] and, for sufficiently small $\varepsilon$, a $2^{O(\varepsilon^{-1})}$ time method from [Cohen-Steiner et al., KDD 2018]. To achieve this result, we introduce a new notion of graph sparsification, which we call \emph{nuclear sparsification}. We provide an $O(n\varepsilon^{-2})$-query and $O(n\varepsilon^{-2})$-time algorithm for computing $O(n\varepsilon^{-2})$-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 \emph{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.

We provide new algorithms and conditional hardness for the problem of estimating effective resistances in $n$-node $m$-edge undirected, expander graphs. We provide an $\widetilde{O}(m\epsilon^{-1})$-time algorithm that produces with high probability, an $\widetilde{O}(n\epsilon^{-1})$-bit sketch from which the effective resistance between any pair of nodes can be estimated, to $(1 \pm \epsilon)$-multiplicative accuracy, in $\widetilde{O}(1)$-time. Consequently, we obtain an $\widetilde{O}(m\epsilon^{-1})$-time algorithm for estimating the effective resistance of all edges in such graphs, improving (for sparse graphs) on the previous fastest runtimes of $\widetilde{O}(m\epsilon^{-3/2})$ [Chu et. al. 2018] and $\widetilde{O}(n^2\epsilon^{-1})$ [Jambulapati, Sidford, 2018] for general graphs and $\widetilde{O}(m + n\epsilon^{-2})$ for expanders [Li, Sachdeva 2022]. We complement this result by showing a conditional lower bound that a broad set of algorithms for computing such estimates of the effective resistances between all pairs of nodes require $\widetilde{\Omega}(n^2 \epsilon^{-1/2})$-time, improving upon the previous best such lower bound of $\widetilde{\Omega}(n^2 \epsilon^{-1/13})$ [Musco et. al. 2017]. Further, we leverage the tools underlying these results to obtain improved algorithms and conditional hardness for more general problems of sketching the pseudoinverse of positive semidefinite matrices and estimating functions of their eigenvalues.