Randomized Low-Memory Singular Value Projection

Affine rank minimization algorithms typically rely on calculating the gradient of a data error followed by a singular value decomposition at every iteration. Because these two steps are expensive, heuristic approximations are often used to reduce computational burden. To this end, we propose a recovery scheme that merges the two steps with randomized approximations, and as a result, operates on space proportional to the degrees of freedom in the problem. We theoretically establish the estimation guarantees of the algorithm as a function of approximation tolerance. While the theoretical approximation requirements are overly pessimistic, we demonstrate that in practice the algorithm performs well on the quantum tomography recovery problem.Comment: 13 pages. This version has a revised theorem and new numerical experiment

On the quality of randomized approximations of Tukey's depth

Tukey's depth (or halfspace depth) is a widely used measure of centrality for multivariate data. However, exact computation of Tukey's depth is known to be a hard problem in high dimensions. As a remedy, randomized approximations of Tukey's depth have been proposed. In this paper we explore when such randomized algorithms return a good approximation of Tukey's depth. We study the case when the data are sampled from a log-concave isotropic distribution. We prove that, if one requires that the algorithm runs in polynomial time in the dimension, the randomized algorithm correctly approximates the maximal depth $1/2$ and depths close to zero. On the other hand, for any point of intermediate depth, any good approximation requires exponential complexity

High-performance Kernel Machines with Implicit Distributed Optimization and Randomization

In order to fully utilize "big data", it is often required to use "big models". Such models tend to grow with the complexity and size of the training data, and do not make strong parametric assumptions upfront on the nature of the underlying statistical dependencies. Kernel methods fit this need well, as they constitute a versatile and principled statistical methodology for solving a wide range of non-parametric modelling problems. However, their high computational costs (in storage and time) pose a significant barrier to their widespread adoption in big data applications. We propose an algorithmic framework and high-performance implementation for massive-scale training of kernel-based statistical models, based on combining two key technical ingredients: (i) distributed general purpose convex optimization, and (ii) the use of randomization to improve the scalability of kernel methods. Our approach is based on a block-splitting variant of the Alternating Directions Method of Multipliers, carefully reconfigured to handle very large random feature matrices, while exploiting hybrid parallelism typically found in modern clusters of multicore machines. Our implementation supports a variety of statistical learning tasks by enabling several loss functions, regularization schemes, kernels, and layers of randomized approximations for both dense and sparse datasets, in a highly extensible framework. We evaluate the ability of our framework to learn models on data from applications, and provide a comparison against existing sequential and parallel libraries.Comment: Work presented at MMDS 2014 (June 2014) and JSM 201

Simple Approximations of Semialgebraic Sets and their Applications to Control

Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance the solution set of linear matrix inequalities or the Schur/Hurwitz stability domains. These sets often have very complicated shapes (non-convex, and even non-connected), which renders very difficult their manipulation. It is therefore of considerable importance to find simple-enough approximations of these sets, able to capture their main characteristics while maintaining a low level of complexity. For these reasons, in the past years several convex approximations, based for instance on hyperrect-angles, polytopes, or ellipsoids have been proposed. In this work, we move a step further, and propose possibly non-convex approximations , based on a small volume polynomial superlevel set of a single positive polynomial of given degree. We show how these sets can be easily approximated by minimizing the L1 norm of the polynomial over the semialgebraic set, subject to positivity constraints. Intuitively, this corresponds to the trace minimization heuristic commonly encounter in minimum volume ellipsoid problems. From a computational viewpoint, we design a hierarchy of linear matrix inequality problems to generate these approximations, and we provide theoretically rigorous convergence results, in the sense that the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set. Two main applications of the proposed approach are considered. The first one aims at reconstruction/approximation of sets from a finite number of samples. In the second one, we show how the concept of polynomial superlevel set can be used to generate samples uniformly distributed on a given semialgebraic set. The efficiency of the proposed approach is demonstrated by different numerical examples

An Adversarial Interpretation of Information-Theoretic Bounded Rationality

Recently, there has been a growing interest in modeling planning with information constraints. Accordingly, an agent maximizes a regularized expected utility known as the free energy, where the regularizer is given by the information divergence from a prior to a posterior policy. While this approach can be justified in various ways, including from statistical mechanics and information theory, it is still unclear how it relates to decision-making against adversarial environments. This connection has previously been suggested in work relating the free energy to risk-sensitive control and to extensive form games. Here, we show that a single-agent free energy optimization is equivalent to a game between the agent and an imaginary adversary. The adversary can, by paying an exponential penalty, generate costs that diminish the decision maker's payoffs. It turns out that the optimal strategy of the adversary consists in choosing costs so as to render the decision maker indifferent among its choices, which is a definining property of a Nash equilibrium, thus tightening the connection between free energy optimization and game theory.Comment: 7 pages, 4 figures. Proceedings of AAAI-1

Generalized Network Dismantling

Finding the set of nodes, which removed or (de)activated can stop the spread of (dis)information, contain an epidemic or disrupt the functioning of a corrupt/criminal organization is still one of the key challenges in network science. In this paper, we introduce the generalized network dismantling problem, which aims to find the set of nodes that, when removed from a network, results in a network fragmentation into subcritical network components at minimum cost. For unit costs, our formulation becomes equivalent to the standard network dismantling problem. Our non-unit cost generalization allows for the inclusion of topological cost functions related to node centrality and non-topological features such as the price, protection level or even social value of a node. In order to solve this optimization problem, we propose a method, which is based on the spectral properties of a novel node-weighted Laplacian operator. The proposed method is applicable to large-scale networks with millions of nodes. It outperforms current state-of-the-art methods and opens new directions in understanding the vulnerability and robustness of complex systems.Comment: 6 pages, 5 figure