Generalizing p-Laplacian: spectral hypergraph theory and a partitioning algorithm

For hypergraph clustering, various methods have been proposed to defne hypergraph p-Laplacians in the literature. This work proposes a general framework for an abstract class of hypergraph p-Laplacians from a diferential-geometric view. This class includes previously proposed hypergraph p-Laplacians and also includes previously unstudied novel generalizations. For this abstract class, we extend current spectral theory by providing an extension of nodal domain theory for the eigenvectors of our hypergraph p-Laplacian. We use this nodal domain theory to provide bounds on the eigenvalues via a higher-order Cheeger inequality. Following our extension of spectral theory, we propose a novel hypergraph partitioning algorithm for our generalized p-Laplacian. Our empirical study shows that our algorithm outperforms spectral methods based on existing p-Laplacians

Multi-class Graph Clustering via Approximated Effective p-Resistance

This paper develops an approximation to the (effective) p-resistance and applies it to multi-class clustering. Spectral methods based on the graph Laplacian and its generalization to the graph p-Laplacian have been a backbone of non-euclidean clustering techniques. The advantage of the p-Laplacian is that the parameter p induces a controllable bias on cluster structure. The drawback of p-Laplacian eigenvector based methods is that the third and higher eigenvectors are difficult to compute. Thus, instead, we are motivated to use the p-resistance induced by the p-Laplacian for clustering. For p-resistance, small p biases towards clusters with high internal connectivity while large p biases towards clusters of small “extent,” that is a preference for smaller shortest-path distances between vertices in the cluster. However, the p-resistance is expensive to compute. We overcome this by developing an approximation to the p-resistance. We prove upper and lower bounds on this approximation and observe that it is exact when the graph is a tree. We also provide theoretical justification for the use of p-resistance for clustering. Finally, we provide experiments comparing our approximated p-resistance clustering to other p-Laplacian based methods

Mistake Bounds for Binary Matrix Completion

We study the problem of completing a binary matrix in an online learning setting.On each trial we predict a matrix entry and then receive the true entry. We propose a Matrix Exponentiated Gradient algorithm [1] to solve this problem. We provide a mistake bound for the algorithm, which scales with the margin complexity [2, 3] of the underlying matrix. The bound suggests an interpretation where each row of the matrix is a prediction task over a finite set of objects, the columns. Using this we show that the algorithm makes a number of mistakes which is comparable up to a logarithmic factor to the number of mistakes made by the Kernel Perceptron with an optimal kernel in hindsight. We discuss applications of the algorithm to predicting as well as the best biclustering and to the problem of predicting the labeling of a graph without knowing the graph in advance

Online Similarity Prediction of Networked Data from Known and Unknown Graphs

We consider online similarity prediction problems over networked data. We begin by relating this task to the more standard class prediction problem, showing that, given an arbitrary algorithm for class prediction, we can construct an algorithm for similarity prediction with "nearly" the same mistake bound, and vice versa. After noticing that this general construction is computationally infeasible, we target our study to {\em feasible} similarity prediction algorithms on networked data. We initially assume that the network structure is {\em known} to the learner. Here we observe that Matrix Winnow \cite{w07} has a near-optimal mistake guarantee, at the price of cubic prediction time per round. This motivates our effort for an efficient implementation of a Perceptron algorithm with a weaker mistake guarantee but with only poly-logarithmic prediction time. Our focus then turns to the challenging case of networks whose structure is initially {\em unknown} to the learner. In this novel setting, where the network structure is only incrementally revealed, we obtain a mistake-bounded algorithm with a quadratic prediction time per round

A Gang of Adversarial Bandits

We consider running multiple instances of multi-armed bandit (MAB) problems in parallel. A main motivation for this study are online recommendation systems, in which each of N users is associated with a MAB problem and the goal is to exploit users' similarity in order to learn users' preferences to K items more efficiently. We consider the adversarial MAB setting, whereby an adversary is free to choose which user and which loss to present to the learner during the learning process. Users are in a social network and the learner is aided by a-priori knowledge of the strengths of the social links between all pairs of users. It is assumed that if the social link between two users is strong then they tend to share the same action. The regret is measured relative to an arbitrary function which maps users to actions. The smoothness of the function is captured by a resistance-based dispersion measure Ψ. We present two learning algorithms, GABA-I and GABA-II which exploit the network structure to bias towards functions of low Ψ values. We show that GABA-I has an expected regret bound of O(pln(N K/Ψ)ΨKT) and per-trial time complexity of O(K ln(N)), whilst GABA-II has a weaker O(pln(N/Ψ) ln(N K/Ψ)ΨKT) regret, but a better O(ln(K) ln(N)) per-trial time complexity. We highlight improvements of both algorithms over running independent standard MABs across users

Online Learning of Facility Locations

In this paper, we provide a rigorous theoretical investigation of an online learning version of the Facility Location problem which is motivated by emerging problems in real-world applications. In our formulation, we are given a set of sites and an online sequence of user requests. At each trial, the learner selects a subset of sites and then incurs a cost for each selected site and an additional cost which is the price of the user’s connection to the nearest site in the selected subset. The problem may be solved by an application of the well-known Hedge algorithm. This would, however, require time and space exponential in the number of the given sites, which motivates our design of a novel quasi-linear time algorithm for this problem, with good theoretical guarantees on its performance

Service Placement with Provable Guarantees in Heterogeneous Edge Computing Systems

Mobile edge computing (MEC) is a promising technique for providing low-latency access to services at the network edge. The services are hosted at various types of edge nodes with both computation and communication capabilities. Due to the heterogeneity of edge node characteristics and user locations, the performance of MEC varies depending on where the service is hosted. In this paper, we consider such a heterogeneous MEC system, and focus on the problem of placing multiple services in the system to maximize the total reward. We show that the problem is NP-hard via reduction from the set cover problem, and propose a deterministic approximation algorithm to solve the problem, which has an approximation ratio that is not worse than (1 − e−1)/4. The proposed algorithm is based on two subroutines that are suitable for small and arbitrarily sized services, respectively. The algorithm is designed using a novel way of partitioning each edge node into multiple slots, where each slot contains one service. The approximation guarantee is obtained via a specialization of the method of conditional expectations, which uses a randomized procedure as an intermediate step. In addition to theoretical guarantees, simulation results also show that the proposed algorithm outperforms other state-of-the-art approache

MaxHedge: Maximising a Maximum Online

We introduce a new online learning framework where, at each trial, the learner is required to select a subset of actions from a given known action set. Each action is associated with an energy value, a reward and a cost. The sum of the energies of the actions selected cannot exceed a given energy budget. The goal is to maximise the cumulative profit, where the profit obtained on a single trial is defined as the difference between the maximum reward among the selected actions and the sum of their costs. Action energy values and the budget are known and fixed. All rewards and costs associated with each action change over time and are revealed at each trial only after the learner’s selection of actions. Our framework encompasses several online learning problems where the environment changes over time; and the solution trades-off between minimising the costs and maximising the maximum reward of the selected subset of actions, while being constrained to an action energy budget. The algorithm that we propose is efficient and general that may be specialised to multiple natural online combinatorial problems