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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 for-each model, where the goal is to approximate a fixed cut with high probability, and the for-all model, where the data structure must simultaneously preserve all cuts. We improve the previous $\Omega(n \sqrt{\beta/\epsilon})$ lower bound in the for-each model to $\tilde\Omega(n \sqrt{\beta}/\epsilon)$ and we improve the previous $\Omega(n \beta/\epsilon)$ lower bound in the for-all 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. 

Finding an approximate second-order stationary point (SOSP) is a well-studied 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 dimension-independent 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. 

Low-rank 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 non-convex optimization as it is well-known that for low-rank matrix problems like matrix sensing and matrix completion, all local optima of the natural non-convex objectives are also globally optimal under certain ideal assumptions. In this paper, we study new approaches for matrix sensing in a semi-random model where an adversary can add any number of arbitrary sensing matrices. More precisely, the problem is to recover a low-rank 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 semi-random model, existing non-convex objectives can have bad local optima. To fix this, we present a descent-style algorithm that provably recovers the ground-truth matrix $X^\star$. For the closely-related problem of semi-random 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 NP-hard to check. Instead, we build on the framework proposed in [KLL$^+$23] for semi-random 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 low-rankness in matrix problems, which may have other applications in obtaining robust algorithms for sparse and low-rank problems. 

Sparse coding, which refers to modeling a signal as sparse linear combinations of the elements of a learned dictionary, has proven to be a successful (and interpretable) approach in applications such as signal processing, computer vision, and medical imaging. While this success has spurred much work on provable guarantees for dictionary recovery when the learned dictionary is the same size as the ground-truth dictionary, work on the setting where the learned dictionary is larger (or over-realized) with respect to the ground truth is comparatively nascent. Existing theoretical results in this setting have been constrained to the case of noise-less data. We show in this work that, in the presence of noise, minimizing the standard dictionary learning objective can fail to recover the elements of the ground-truth dictionary in the over-realized regime, regardless of the magnitude of the signal in the data-generating process. Furthermore, drawing from the growing body of work on self-supervised learning, we propose a novel masking objective for which recovering the ground-truth dictionary is in fact optimal as the signal increases for a large class of data-generating processes. We corroborate our theoretical results with experiments across several parameter regimes showing that our proposed objective also enjoys better empirical performance than the standard reconstruction objective. 

We study equilibrium computation with extensive-form correlation in two-player turn-taking stochastic games. Our main results are two-fold: (1) We give an algorithm for computing a Stackelberg extensive-form correlated equilibrium (SEFCE), which runs in time polynomial in the size of the game, as well as the number of bits required to encode each input number. (2) We give an efficient algorithm for approximately computing an optimal extensive-form correlated equilibrium (EFCE) up to machine precision, i.e., the algorithm achieves approximation error 𝜀 in time polynomial in the size of the game, as well as log(1/𝜀). Our algorithm for SEFCE is the first polynomial-time algorithm for equilibrium computation with com- mitment in such a general class of stochastic games. Existing algorithms for SEFCE typically make stronger assumptions such as no chance moves, and are designed for extensive-form games in the less succinct tree form. Our algorithm for approximately optimal EFCE is, to our knowledge, the first algorithm that achieves 3 desiderata simultaneously: approximate optimality, polylogarithmic dependency on the approximation error, and compatibility with stochastic games in the more succinct graph form. Existing algorithms achieve at most 2 of these desiderata, often also relying on additional technical assumptions. 

We consider the problem of planning with participation constraints introduced in [24]. In this problem, a principal chooses actions in a Markov decision process, resulting in separate utilities for the principal and the agent. However, the agent can and will choose to end the process whenever his expected onward utility becomes negative. The principal seeks to compute and commit to a policy that maximizes her expected utility, under the constraint that the agent should always want to continue participating. We provide the first polynomial-time exact algorithm for this problem for finite-horizon settings, where previously only an additive ε-approximation algorithm was known. Our approach can also be extended to the (discounted) infinite-horizon case, for which we give an algorithm that runs in time polynomial in the size of the input and log(1/ε), and returns a policy that is optimal up to an additive error of ε. 

We pose and study the problem of planning in Markov decision processes (MDPs), subject to participation constraints as studied in mechanism design. In this problem, a planner must work with a self-interested agent on a given MDP. Each action in the MDP provides an immediate reward to the planner and a (possibly different) reward to the agent. The agent has no control in choosing the actions, but has the option to end the entire process at any time. The goal of the planner is to find a policy that maximizes her cumulative reward, taking into consideration the agent's ability to terminate. We give a fully polynomial-time approximation scheme for this problem. En route, we present polynomial-time algorithms for computing (exact) optimal policies for important special cases of this problem, including when the time horizon is constant, or when the MDP exhibits a "definitive decisions" property. We illustrate our algorithms with two different game-theoretic applications: the problem of assigning rides in ride-sharing and the problem of designing screening policies. Our results imply efficient algorithms for computing (approximately) optimal policies in both applications. 

We explore the connection between outlier-robust high-dimensional statistics and non-convex optimization in the presence of sparsity constraints, with a focus on the fundamental tasks of robust sparse mean estimation and robust sparse PCA. We develop novel and simple optimization formulations for these problems such that any approximate stationary point of the associated optimization problem yields a near-optimal solution for the underlying robust estimation task. As a corollary, we obtain that any first-order method that efficiently converges to stationarity yields an efficient algorithm for these tasks. The obtained algorithms are simple, practical, and succeed under broader distributional assumptions compared to prior work.