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Free, publiclyaccessible full text available May 20, 2025

In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is approximating cuts in balanced directed graphs, where the goal is to build a data structure to provide a $(1 \pm \epsilon)$estimation of the cut values of a graph on $n$ vertices. For this problem, there are tight bounds for undirected graphs, but for directed graphs, such a data structure requires $\Omega(n^2)$ bits even for constant $\epsilon$. To cope with this, recent works consider $\beta$balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most $\beta$ times the total weight in the other direction. We consider the foreach model, where the goal is to approximate a fixed cut with high probability, and the forall model, where the data structure must simultaneously preserve all cuts. We improve the previous $\Omega(n \sqrt{\beta/\epsilon})$ lower bound in the foreach model to $\tilde\Omega(n \sqrt{\beta}/\epsilon)$ and we improve the previous $\Omega(n \beta/\epsilon)$ lower bound in the forall model to $\Omega(n \beta/\epsilon^2)$. This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is approximating the global minimum cut in the local query model where we can only access the graph through degree, edge, and adjacency queries. We prove an $\Omega(\min\{m, \frac{m}{\epsilon^2 k}\})$ lower bound for this problem, which improves the previous $\Omega(\frac{m}{k})$ lower bound, where $m$ is the number of edges of the graph, $k$ is the minimum cut size, and we seek a $(1+\epsilon)$approximation. In addition, we observe that existing upper bounds with minor modifications match our lower bound up to logarithmic factors.more » « less

Free, publiclyaccessible full text available May 30, 2025

Free, publiclyaccessible full text available March 20, 2025

Lowrank matrix recovery is a fundamental problem in machine learning with numerous applications. In practice, the problem can be solved by convex optimization namely nuclear norm minimization, or by nonconvex optimization as it is wellknown that for lowrank matrix problems like matrix sensing and matrix completion, all local optima of the natural nonconvex objectives are also globally optimal under certain ideal assumptions. In this paper, we study new approaches for matrix sensing in a semirandom model where an adversary can add any number of arbitrary sensing matrices. More precisely, the problem is to recover a lowrank matrix $X^\star$ from linear measurements $b_i = \langle A_i, X^\star \rangle$, where an unknown subset of the sensing matrices satisfies the Restricted Isometry Property (RIP) and the rest of the $A_i$'s are chosen adversarially. It is known that in the semirandom model, existing nonconvex objectives can have bad local optima. To fix this, we present a descentstyle algorithm that provably recovers the groundtruth matrix $X^\star$. For the closelyrelated problem of semirandom matrix completion, prior work [CG18] showed that all bad local optima can be eliminated by reweighting the input data. However, the analogous approach for matrix sensing requires reweighting a set of matrices to satisfy RIP, which is a condition that is NPhard to check. Instead, we build on the framework proposed in [KLL$^+$23] for semirandom sparse linear regression, where the algorithm in each iteration reweights the input based on the current solution, and then takes a weighted gradient step that is guaranteed to work well locally. Our analysis crucially exploits the connection between sparsity in vector problems and lowrankness in matrix problems, which may have other applications in obtaining robust algorithms for sparse and lowrank problems.more » « less

Free, publiclyaccessible full text available December 13, 2024

Finding an approximate secondorder stationary point (SOSP) is a wellstudied and fundamental problem in stochastic nonconvex optimization with many applications in machine learning. However, this problem is poorly understood in the presence of outliers, limiting the use of existing nonconvex algorithms in adversarial settings. In this paper, we study the problem of finding SOSPs in the strong contamination model, where a constant fraction of datapoints are arbitrarily corrupted. We introduce a general framework for efficiently finding an approximate SOSP with dimensionindependent accuracy guarantees, using $\widetilde{O}({D^2}/{\epsilon})$ samples where $D$ is the ambient dimension and $\epsilon$ is the fraction of corrupted datapoints. As a concrete application of our framework, we apply it to the problem of low rank matrix sensing, developing efficient and provably robust algorithms that can tolerate corruptions in both the sensing matrices and the measurements. In addition, we establish a Statistical Query lower bound providing evidence that the quadratic dependence on $D$ in the sample complexity is necessary for computationally efficient algorithms.more » « less

Liu, Zhiwen ; Psaltis, Demetri ; Shi, Kebin (Ed.)