Computing Upper and Lower Bounds on Likelihoods in Intractable Networks

We present techniques for computing upper and lower bounds on the likelihoods of partial instantiations of variables in sigmoid and noisy-OR networks. The bounds determine confidence intervals for the desired likelihoods and become useful when the size of the network (or clique size) precludes exact computations. We illustrate the tightness of the obtained bounds by numerical experiments

Mean Field Theory for Sigmoid Belief Networks

We develop a mean field theory for sigmoid belief networks based on ideas from statistical mechanics. Our mean field theory provides a tractable approximation to the true probability distribution in these networks; it also yields a lower bound on the likelihood of evidence. We demonstrate the utility of this framework on a benchmark problem in statistical pattern recognition---the classification of handwritten digits.Comment: See http://www.jair.org/ for any accompanying file

Discrete Temporal Models of Social Networks

We propose a family of statistical models for social network evolution over time, which represents an extension of Exponential Random Graph Models (ERGMs). Many of the methods for ERGMs are readily adapted for these models, including maximum likelihood estimation algorithms. We discuss models of this type and their properties, and give examples, as well as a demonstration of their use for hypothesis testing and classification. We believe our temporal ERG models represent a useful new framework for modeling time-evolving social networks, and rewiring networks from other domains such as gene regulation circuitry, and communication networks

Efficient Optimization of Loops and Limits with Randomized Telescoping Sums

We consider optimization problems in which the objective requires an inner loop with many steps or is the limit of a sequence of increasingly costly approximations. Meta-learning, training recurrent neural networks, and optimization of the solutions to differential equations are all examples of optimization problems with this character. In such problems, it can be expensive to compute the objective function value and its gradient, but truncating the loop or using less accurate approximations can induce biases that damage the overall solution. We propose randomized telescope (RT) gradient estimators, which represent the objective as the sum of a telescoping series and sample linear combinations of terms to provide cheap unbiased gradient estimates. We identify conditions under which RT estimators achieve optimization convergence rates independent of the length of the loop or the required accuracy of the approximation. We also derive a method for tuning RT estimators online to maximize a lower bound on the expected decrease in loss per unit of computation. We evaluate our adaptive RT estimators on a range of applications including meta-optimization of learning rates, variational inference of ODE parameters, and training an LSTM to model long sequences