A Parallel Solver for Graph Laplacians

Problems from graph drawing, spectral clustering, network flow and graph partitioning can all be expressed in terms of graph Laplacian matrices. There are a variety of practical approaches to solving these problems in serial. However, as problem sizes increase and single core speeds stagnate, parallelism is essential to solve such problems quickly. We present an unsmoothed aggregation multigrid method for solving graph Laplacians in a distributed memory setting. We introduce new parallel aggregation and low degree elimination algorithms targeted specifically at irregular degree graphs. These algorithms are expressed in terms of sparse matrix-vector products using generalized sum and product operations. This formulation is amenable to linear algebra using arbitrary distributions and allows us to operate on a 2D sparse matrix distribution, which is necessary for parallel scalability. Our solver outperforms the natural parallel extension of the current state of the art in an algorithmic comparison. We demonstrate scalability to 576 processes and graphs with up to 1.7 billion edges.Comment: PASC '18, Code: https://github.com/ligmg/ligm

A new projection method for finding the closest point in the intersection of convex sets

In this paper we present a new iterative projection method for finding the closest point in the intersection of convex sets to any arbitrary point in a Hilbert space. This method, termed AAMR for averaged alternating modified reflections, can be viewed as an adequate modification of the Douglas--Rachford method that yields a solution to the best approximation problem. Under a constraint qualification at the point of interest, we show strong convergence of the method. In fact, the so-called strong CHIP fully characterizes the convergence of the AAMR method for every point in the space. We report some promising numerical experiments where we compare the performance of AAMR against other projection methods for finding the closest point in the intersection of pairs of finite dimensional subspaces

Distributionally Robust Optimization: A Review

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization

Exponential inequalities for unbounded functions of geometrically ergodic Markov chains. Applications to quantitative error bounds for regenerative Metropolis algorithms

The aim of this note is to investigate the concentration properties of unbounded functions of geometrically ergodic Markov chains. We derive concentration properties of centered functions with respect to the square of the Lyapunov's function in the drift condition satisfied by the Markov chain. We apply the new exponential inequalities to derive confidence intervals for MCMC algorithms. Quantitative error bounds are providing for the regenerative Metropolis algorithm of [5]

Communication Complexity of Inner Product in Symmetric Normed Spaces

We introduce and study the communication complexity of computing the inner product of two vectors, where the input is restricted w.r.t. a norm $N$ on the space $\mathbb{R}^n$. Here, Alice and Bob hold two vectors $v,u$ such that $\|v\|_N\le 1$ and $\|u\|_{N^*}\le 1$, where $N^*$ is the dual norm. They want to compute their inner product $\langle v,u \rangle$ up to an $\varepsilon$ additive term. The problem is denoted by $\mathrm{IP}_N$. We systematically study $\mathrm{IP}_N$, showing the following results: - For any symmetric norm $N$, given $\|v\|_N\le 1$ and $\|u\|_{N^*}\le 1$ there is a randomized protocol for $\mathrm{IP}_N$ using $\tilde{\mathcal{O}}(\varepsilon^{-6} \log n)$ bits -- we will denote this by $\mathcal{R}_{\varepsilon,1/3}(\mathrm{IP}_{N}) \leq \tilde{\mathcal{O}}(\varepsilon^{-6} \log n)$. - One way communication complexity $\overrightarrow{\mathcal{R}}(\mathrm{IP}_{\ell_p})\leq\mathcal{O}(\varepsilon^{-\max(2,p)}\cdot \log\frac n\varepsilon)$, and a nearly matching lower bound $\overrightarrow{\mathcal{R}}(\mathrm{IP}_{\ell_p}) \geq \Omega(\varepsilon^{-\max(2,p)})$ for $\varepsilon^{-\max(2,p)} \ll n$. - One way communication complexity $\overrightarrow{\mathcal{R}}(N)$ for a symmetric norm $N$ is governed by embeddings $\ell_\infty^k$ into $N$. Specifically, while a small distortion embedding easily implies a lower bound $\Omega(k)$, we show that, conversely, non-existence of such an embedding implies protocol with communication $k^{\mathcal{O}(\log \log k)} \log^2 n$. - For arbitrary origin symmetric convex polytope $P$, we show $\mathcal{R}(\mathrm{IP}_{N}) \le\mathcal{O}(\varepsilon^{-2} \log \mathrm{xc}(P))$, where $N$ is the unique norm for which $P$ is a unit ball, and $\mathrm{xc}(P)$ is the extension complexity of $P$.Comment: Accepted to ITCS 202