arXiv subjects

Antares Chen

Publications and source records attributed to Antares Chen.

6 recordsLinked to original sources

arXiv2024

Top-$K$ ranking with a monotone adversary

In this paper, we address the top-$K$ ranking problem with a monotone adversary. We consider the scenario where a comparison graph is randomly generated and the adversary is allowed to add arbitrary edges. The statistician's goal is then to accurately identify the top-$K$ preferred items based on pairwise comparisons derived from this semi-random comparison graph. The main contribution of this paper is to develop a weighted maximum likelihood estimator (MLE) that achieves near-optimal sample complexity, up to a $\log^2(n)$ factor, where $n$ denotes the number of items under comparison. This is made possible through a combination of analytical and algorithmic innovations. On the analytical front, we provide a refined~$\ell_\infty$ error analysis of the weighted MLE that is more explicit and tighter than existing analyses. It relates the~$\ell_\infty$ error with the spectral properties of the weighted comparison graph. Motivated by this, our algorithmic innovation involves the development of an SDP-based approach to reweight the semi-random graph and meet specified spectral properties. Additionally, we propose a first-order method based on the Matrix Multiplicative Weight Update (MMWU) framework. This method efficiently solves the resulting SDP in nearly-linear time relative to the size of the semi-random comparison graph.

stat.ML↗

arXiv2023

Local algorithms and the failure of log-depth quantum advantage on sparse random CSPs

We construct and analyze a message-passing algorithm for random constraint satisfaction problems (CSPs) at large clause density, generalizing work of El Alaoui, Montanari, and Sellke for Maximum Cut [arXiv:2111.06813] through a connection between random CSPs and mean-field Ising spin glasses. For CSPs with even predicates, the algorithm asymptotically solves a stochastic optimal control problem dual to an extended Parisi variational principle. This gives an optimal fraction of satisfied constraints among algorithms obstructed by the branching overlap gap property of Huang and Sellke [arXiv:2110.07847], notably including the Quantum Approximate Optimization Algorithm and all quantum circuits on a bounded-degree architecture of up to $ε\cdot \log n$ depth.

quant-ph↗

arXiv2023

Submodular Hypergraph Partitioning: Metric Relaxations and Fast Algorithms via an Improved Cut-Matching Game

Despite there being significant work on developing spectral, and metric embedding based approximation algorithms for hypergraph generalizations of conductance, little is known regarding the approximability of hypergraph partitioning objectives beyond this. This work proposes algorithms for a general model of hypergraph partitioning that unifies both undirected and directed versions of many well-studied partitioning objectives. The first contribution of this paper introduces polymatroidal cut functions, a large class of cut functions amenable to approximation algorithms via metric embeddings and routing multicommodity flows. We demonstrate an $O(\sqrt{\log n})$-approximation, where $n$ is the number of vertices in the hypergraph, for these problems by rounding relaxations to metrics of negative-type. The second contribution of this paper generalizes the cut-matching game framework of Khandekar et. al. to tackle polymatroidal cut functions. This yields the first almost-linear time $O(\log n)$-approximation algorithm for standard versions of undirected and directed hypergraph partitioning. A technical consequence of our construction is that a cut-matching game which greatly relaxes the set of allowed actions for both players can be used to partition hypergraphs with negligible impact on the approximation ratio. We believe this to be of independent interest.

cs.DS↗

arXiv2023

Hypergraph Diffusions and Resolvents for Norm-Based Hypergraph Laplacians

The development of simple and fast hypergraph spectral methods has been hindered by the lack of numerical algorithms for simulating heat diffusions and computing fundamental objects, such as Personalized PageRank vectors, over hypergraphs. In this paper, we overcome this challenge by designing two novel algorithmic primitives. The first is a simple, easy-to-compute discrete-time heat diffusion that enjoys the same favorable properties as the discrete-time heat diffusion over graphs. This diffusion can be directly applied to speed up existing hypergraph partitioning algorithms. Our second contribution is the novel application of mirror descent to compute resolvents of non-differentiable squared norms, which we believe to be of independent interest beyond hypergraph problems. Based on this new primitive, we derive the first nearly-linear-time algorithm that simulates the discrete-time heat diffusion to approximately compute resolvents of the hypergraph Laplacian operator, which include Personalized PageRank vectors and solutions to the hypergraph analogue of Laplacian systems. Our algorithm runs in time that is linear in the size of the hypergraph and inversely proportional to the hypergraph spectral gap $λ_G$, matching the complexity of analogous diffusion-based algorithms for the graph version of the problem.

cs.DS↗

arXiv2021

Cut Sparsification of the Clique Beyond the Ramanujan Bound: A Separation of Cut Versus Spectral Sparsification

We prove that a random $d$-regular graph, with high probability, is a cut sparsifier of the clique with approximation error at most $\left(2\sqrt{\frac 2 π} + o_{n,d}(1)\right)/\sqrt d$, where $2\sqrt{\frac 2 π} = 1.595\ldots$ and $o_{n,d}(1)$ denotes an error term that depends on $n$ and $d$ and goes to zero if we first take the limit $n\rightarrow \infty$ and then the limit $d \rightarrow \infty$. This is established by analyzing linear-size cuts using techniques of Jagannath and Sen derived from ideas in statistical physics, and analyzing small cuts via martingale inequalities. We also prove new lower bounds on spectral sparsification of the clique. If $G$ is a spectral sparsifier of the clique and $G$ has average degree $d$, we prove that the approximation error is at least the "Ramanujan bound'' $(2-o_{n,d}(1))/\sqrt d$, which is met by $d$-regular Ramanujan graphs, provided that either the weighted adjacency matrix of $G$ is a (multiple of) a doubly stochastic matrix, or that $G$ satisfies a certain high "odd pseudo-girth" property. The first case can be seen as an "Alon-Boppana theorem for symmetric doubly stochastic matrices," showing that a symmetric doubly stochastic matrix with $dn$ non-zero entries has a non-trivial eigenvalue of magnitude at least $(2-o_{n,d}(1))/\sqrt d$; the second case generalizes a lower bound of Srivastava and Trevisan, which requires a large girth assumption. Together, these results imply a separation between spectral sparsification and cut sparsification. If $G$ is a random $\log n$-regular graph on $n$ vertices, we show that, with high probability, $G$ admits a (weighted subgraph) cut sparsifier of average degree $d$ and approximation error at most $(1.595\ldots + o_{n,d}(1))/\sqrt d$, while every (weighted subgraph) spectral sparsifier of $G$ having average degree $d$ has approximation error at least $(2-o_{n,d}(1))/\sqrt d$.

cs.DS↗

arXiv2020

Partial resampling to approximate covering integer programs

We consider column-sparse covering integer programs, a generalization of set cover, which have a long line of research of (randomized) approximation algorithms. We develop a new rounding scheme based on the Partial Resampling variant of the Lovász Local Lemma developed by Harris & Srinivasan (2019). This achieves an approximation ratio of $1 + \frac{\ln (Δ_1+1)}{a_{\min}} + O\Big( \log(1 + \sqrt{ \frac{\log (Δ_1+1)}{a_{\min}}} \Big)$, where $a_{\min}$ is the minimum covering constraint and $Δ_1$ is the maximum $\ell_1$-norm of any column of the covering matrix (whose entries are scaled to lie in $[0,1]$). When there are additional constraints on the variable sizes, we show an approximation ratio of $\ln Δ_0 + O(\log \log Δ_0)$ (where $Δ_0$ is the maximum number of non-zero entries in any column of the covering matrix). These results improve asymptotically, in several different ways, over results of Srinivasan (2006) and Kolliopoulos & Young (2005). We show nearly-matching inapproximability and integrality-gap lower bounds. We also show that the rounding process leads to negative correlation among the variables, which allows us to handle multi-criteria programs.

cs.DS↗