arXiv subjects

Tony Jebara

Publications and source records attributed to Tony Jebara.

24 records · Page 2Linked to original sources

arXiv2014

Semistochastic Quadratic Bound Methods

Partition functions arise in a variety of settings, including conditional random fields, logistic regression, and latent gaussian models. In this paper, we consider semistochastic quadratic bound (SQB) methods for maximum likelihood inference based on partition function optimization. Batch methods based on the quadratic bound were recently proposed for this class of problems, and performed favorably in comparison to state-of-the-art techniques. Semistochastic methods fall in between batch algorithms, which use all the data, and stochastic gradient type methods, which use small random selections at each iteration. We build semistochastic quadratic bound-based methods, and prove both global convergence (to a stationary point) under very weak assumptions, and linear convergence rate under stronger assumptions on the objective. To make the proposed methods faster and more stable, we consider inexact subproblem minimization and batch-size selection schemes. The efficacy of SQB methods is demonstrated via comparison with several state-of-the-art techniques on commonly used datasets.

stat.ML↗

arXiv2013

Approximating the Bethe partition function

When belief propagation (BP) converges, it does so to a stationary point of the Bethe free energy $F$, and is often strikingly accurate. However, it may converge only to a local optimum or may not converge at all. An algorithm was recently introduced for attractive binary pairwise MRFs which is guaranteed to return an $ε$-approximation to the global minimum of $F$ in polynomial time provided the maximum degree $Δ=O(\log n)$, where $n$ is the number of variables. Here we significantly improve this algorithm and derive several results including a new approach based on analyzing first derivatives of $F$, which leads to performance that is typically far superior and yields a fully polynomial-time approximation scheme (FPTAS) for attractive models without any degree restriction. Further, the method applies to general (non-attractive) models, though with no polynomial time guarantee in this case, leading to the important result that approximating $\log$ of the Bethe partition function, $\log Z_B=-\min F$, for a general model to additive $ε$-accuracy may be reduced to a discrete MAP inference problem. We explore an application to predicting equipment failure on an urban power network and demonstrate that the Bethe approximation can perform well even when BP fails to converge.

cs.LG↗

arXiv2013

Stochastic Bound Majorization

Recently a majorization method for optimizing partition functions of log-linear models was proposed alongside a novel quadratic variational upper-bound. In the batch setting, it outperformed state-of-the-art first- and second-order optimization methods on various learning tasks. We propose a stochastic version of this bound majorization method as well as a low-rank modification for high-dimensional data-sets. The resulting stochastic second-order method outperforms stochastic gradient descent (across variations and various tunings) both in terms of the number of iterations and computation time till convergence while finding a better quality parameter setting. The proposed method bridges first- and second-order stochastic optimization methods by maintaining a computational complexity that is linear in the data dimension and while exploiting second order information about the pseudo-global curvature of the objective function (as opposed to the local curvature in the Hessian).

cs.LG↗

arXiv2013

$\propto$SVM for learning with label proportions

We study the problem of learning with label proportions in which the training data is provided in groups and only the proportion of each class in each group is known. We propose a new method called proportion-SVM, or $\propto$SVM, which explicitly models the latent unknown instance labels together with the known group label proportions in a large-margin framework. Unlike the existing works, our approach avoids making restrictive assumptions about the data. The $\propto$SVM model leads to a non-convex integer programming problem. In order to solve it efficiently, we propose two algorithms: one based on simple alternating optimization and the other based on a convex relaxation. Extensive experiments on standard datasets show that $\propto$SVM outperforms the state-of-the-art, especially for larger group sizes.

cs.LG↗

arXiv2012

Bethe Bounds and Approximating the Global Optimum

Inference in general Markov random fields (MRFs) is NP-hard, though identifying the maximum a posteriori (MAP) configuration of pairwise MRFs with submodular cost functions is efficiently solvable using graph cuts. Marginal inference, however, even for this restricted class, is in #P. We prove new formulations of derivatives of the Bethe free energy, provide bounds on the derivatives and bracket the locations of stationary points, introducing a new technique called Bethe bound propagation. Several results apply to pairwise models whether associative or not. Applying these to discretized pseudo-marginals in the associative case we present a polynomial time approximation scheme for global optimization provided the maximum degree is $O(\log n)$, and discuss several extensions.

cs.LG↗

arXiv2009

Approximating the Permanent with Belief Propagation

This work describes a method of approximating matrix permanents efficiently using belief propagation. We formulate a probability distribution whose partition function is exactly the permanent, then use Bethe free energy to approximate this partition function. After deriving some speedups to standard belief propagation, the resulting algorithm requires $(n^2)$ time per iteration. Finally, we demonstrate the advantages of using this approximation.

cs.LG↗