Reasoning about Independence in Probabilistic Models of Relational Data

We extend the theory of d-separation to cases in which data instances are not independent and identically distributed. We show that applying the rules of d-separation directly to the structure of probabilistic models of relational data inaccurately infers conditional independence. We introduce relational d-separation, a theory for deriving conditional independence facts from relational models. We provide a new representation, the abstract ground graph, that enables a sound, complete, and computationally efficient method for answering d-separation queries about relational models, and we present empirical results that demonstrate effectiveness.Comment: 61 pages, substantial revisions to formalisms, theory, and related wor

A new property of the Lov\'asz number and duality relations between graph parameters

We show that for any graph $G$, by considering "activation" through the strong product with another graph $H$, the relation $\alpha(G) \leq \vartheta(G)$ between the independence number and the Lov\'{a}sz number of $G$ can be made arbitrarily tight: Precisely, the inequality $$\alpha(G \times H) \leq \vartheta(G \times H) = \vartheta(G)\,\vartheta(H)$$ becomes asymptotically an equality for a suitable sequence of ancillary graphs $H$. This motivates us to look for other products of graph parameters of $G$ and $H$ on the right hand side of the above relation. For instance, a result of Rosenfeld and Hales states that $$\alpha(G \times H) \leq \alpha^*(G)\,\alpha(H),$$ with the fractional packing number $\alpha^*(G)$, and for every $G$ there exists $H$ that makes the above an equality; conversely, for every graph $H$ there is a $G$ that attains equality. These findings constitute some sort of duality of graph parameters, mediated through the independence number, under which $\alpha$ and $\alpha^*$ are dual to each other, and the Lov\'{a}sz number $\vartheta$ is self-dual. We also show duality of Schrijver's and Szegedy's variants $\vartheta^-$ and $\vartheta^+$ of the Lov\'{a}sz number, and explore analogous notions for the chromatic number under strong and disjunctive graph products.Comment: 16 pages, submitted to Discrete Applied Mathematics for a special issue in memory of Levon Khachatrian; v2 has a full proof of the duality between theta+ and theta- and a new author, some new references, and we corrected several small errors and typo

From Bandits to Experts: A Tale of Domination and Independence

We consider the partial observability model for multi-armed bandits, introduced by Mannor and Shamir. Our main result is a characterization of regret in the directed observability model in terms of the dominating and independence numbers of the observability graph. We also show that in the undirected case, the learner can achieve optimal regret without even accessing the observability graph before selecting an action. Both results are shown using variants of the Exp3 algorithm operating on the observability graph in a time-efficient manner