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Learning graphical models from the Glauber dynamics

arXiv.org Machine Learning

In this paper we consider the problem of learning undirected graphical models from data generated according to the Glauber dynamics. The Glauber dynamics is a Markov chain that sequentially updates individual nodes (variables) in a graphical model and it is frequently used to sample from the stationary distribution (to which it converges given sufficient time). Additionally, the Glauber dynamics is a natural dynamical model in a variety of settings. This work deviates from the standard formulation of graphical model learning in the literature, where one assumes access to i.i.d. samples from the distribution. Much of the research on graphical model learning has been directed towards finding algorithms with low computational cost. As the main result of this work, we establish that the problem of reconstructing binary pairwise graphical models is computationally tractable when we observe the Glauber dynamics. Specifically, we show that a binary pairwise graphical model on $p$ nodes with maximum degree $d$ can be learned in time $f(d)p^2\log p$, for a function $f(d)$, using nearly the information-theoretic minimum number of samples.


Approximation algorithms for stochastic clustering

Neural Information Processing Systems

We consider stochastic settings for clustering, and develop provably-good (approximation) algorithms for a number of these notions. These algorithms allow one to obtain better approximation ratios compared to the usual deterministic clustering setting. Additionally, they offer a number of advantages including providing fairer clustering and clustering which has better long-term behavior for each user. In particular, they ensure that *every user* is guaranteed to get good service (on average). We also complement some of these with impossibility results.


Algorithms for Generalized Cluster-wise Linear Regression

arXiv.org Machine Learning

Cluster-wise linear regression (CLR), a clustering problem intertwined with regression, is to find clusters of entities such that the overall sum of squared errors from regressions performed over these clusters is minimized, where each cluster may have different variances. We generalize the CLR problem by allowing each entity to have more than one observation, and refer to it as generalized CLR. We propose an exact mathematical programming based approach relying on column generation, a column generation based heuristic algorithm that clusters predefined groups of entities, a metaheuristic genetic algorithm with adapted Lloyd's algorithm for K-means clustering, a two-stage approach, and a modified algorithm of Sp{\"a}th \cite{Spath1979} for solving generalized CLR. We examine the performance of our algorithms on a stock keeping unit (SKU) clustering problem employed in forecasting halo and cannibalization effects in promotions using real-world retail data from a large supermarket chain. In the SKU clustering problem, the retailer needs to cluster SKUs based on their seasonal effects in response to promotions. The seasonal effects are the results of regressions with predictors being promotion mechanisms and seasonal dummies performed over clusters generated. We compare the performance of all proposed algorithms for the SKU problem with real-world and synthetic data.


Stern

AAAI Conferences

Most work in heuristic search considers problems where a low cost solution is preferred (MIN problems). In this paper, we investigate the complementary setting where a solution of high reward is preferred (MAX problems). Example MAX problems include finding a longest simple path in a graph, maximal coverage, and various constraint optimization problems. We examine several popular search algorithms for MIN problems and discover the curious ways in which they misbehave on MAX problems. We propose modifications that preserve the original intentions behind the algorithms but allow them to solve MAX problems, and compare them theoretically and empirically. Interesting results include the failure of bidirectional search and close relationships between Dijkstra's algorithm, weighted A*, and depth-first search.